Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs

Zhu, Yuzixuan; Pataki, Gábor; Tran-Dinh, Quoc

doi:10.1007/s12532-019-00164-4

Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs

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Published: 01 July 2019

Volume 11, pages 503–586, (2019)
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Yuzixuan Zhu¹,
Gábor Pataki¹ &
Quoc Tran-Dinh¹

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Abstract

We introduce Sieve-SDP, a simple facial reduction algorithm to preprocess semidefinite programs (SDPs). Sieve-SDP inspects the constraints of the problem to detect lack of strict feasibility, deletes redundant rows and columns, and reduces the size of the variable matrix. It often detects infeasibility. It does not rely on any optimization solver: the only subroutine it needs is Cholesky factorization, hence it can be implemented in a few lines of code in machine precision. We present extensive computational results on several problem collections from the literature, with many SDPs coming from polynomial optimization.

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Facial reduction for symmetry reduced semidefinite and doubly nonnegative programs

Article Open access 27 September 2022

Partial facial reduction: simplified, equivalent SDPs via approximations of the PSD cone

Article 28 June 2017

Accelerated first-order methods for a class of semidefinite programs

Article 22 March 2024

Notes

More precisely, when (P) is strictly feasible, strong duality holds between (P) and its dual, i.e., their values agree and the latter is attained.
That is, $F \ne \mathcal{S}_+^{n}, \,$F is convex, and $X, Y \in \mathcal{S}_+^{n}, \, \frac{1}{2}(X+Y) \in F$ implies that X and Y are in F.
This SDP is from the DIW dataset.
The description of the DIMACS errors is given in Appendix C.
See Remark 1 about how the primal and dual are defined in [34].

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Acknowledgements

We thank the Technical Editor and the referees for their helpful comments. The second author, Gábor Pataki, is supported by the National Science Foundation, Award DMS-1817272. The third author, Quoc Tran-Dinh, is supported in part by the National Science Foundation, Award DMS-1619884. We are very grateful to Erling Andersen at Mosek for running several SDPs, and explaining the results; to Joachim Dahl at Mosek for helpful discussions on converting SDPs, and for providing his conversion code; to Didier Henrion and Kim-Chuan Toh for providing us with some of the datasets; to Frank Permenter and Johan Löfberg for helpful comments; to Oktay Günlük for helping us to invent the name “Sieve-SDP”; and to Hans Mittelmann for helping us with some of the large-scale SDPs.

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Department of Statistics and Operations Research, The University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599-3260, USA
Yuzixuan Zhu, Gábor Pataki & Quoc Tran-Dinh

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Appendices

A Very detailed results

We now give very detailed computational results on all problems, separately for the five datasets. We only report on problems that were reduced by at least one of the five preprocessors.

In all tables the first column gives the number of the SDP, the second gives the name, and the third gives the names of the preprocessing methods.

The next two columns describe the size of the problem. The entry “f; l; s” describes the size of the variables of the problem, where

the number “f” is the number of free variables;
the number “l” is the number of linear nonnegative variables;
the number, or numbers “s” describe the size of the semidefinite variable blocks, possibly with multiplicity.

For example, 3; 5; 6 means that a problem has 3 free variables; 5 linear nonnegative variables; and a semidefinite matrix variable block of order 6. The tuple $3; 5; 6, 5_3$ means that a problem has 3 free variables; 5 linear nonnegative variables; and four semidefinite matrix variable blocks, which are of order $6, 5, 5, 5, \,$ respectively. The number m is the number of constraints.

In the next three columns we put information about the preprocessors. In the column “red.” we put 1, if a preprocessor reduced a problem, and 0 if it did not. In this column under Sieve-SDP we put the same entries, except when Sieve-SDP actually proved infeasibility. In that case we entered “inf” there. The number $\text {t}_\mathrm{prep}$ is the time spent on preprocessing; the number $\text {t}_\mathrm{conv}$ is the time spent on converting from Mosek format to Sedumi format and back (for the methods pd1, pd2, dd1, dd2).

In the next four columns we show how Mosek performed. In the column “inf” we have a 1 if Mosek detected infeasibility, and 0 if it did not. The column obj (P, D) shows the objective values (primal and dual, respectively). The column DIMACS contains the largest absolute value of the DIMACS errors.

In the last column we show help codes, which show whether a preprocessor helped or hurt to solve an SDP. Although the help codes can be deduced from the previous columns, they still help to quickly evaluate the preprocessors. A positive help code means that a preprocessor helped, and a negative one means that it hurt.

In detail, let us recall from Sect. 2.7 that $\text {DIMACS}_{\mathrm{before}}$ [$\text {DIMACS}_{\mathrm{after}}$] is the absolute value of the DIMACS error that is largest in absolute value before [after] preprocessing. We let $\text {obj}_{\mathrm{before}}$ and $\text {obj}_{\mathrm{after}}$ be the primal objective values before and after preprocessing, respectively.

Given this notation,

the help code is $1, \,$ if
- Sieve-SDP detects infeasibility, or
- Mosek does not detect infeasibility before preprocessing but it does detect infeasibility after preprocessing;
the help code is $-1, \,$ if
- Mosek detects infeasibility before preprocessing but does not detect infeasibility after preprocessing;
the help code is $2, \,$ if
- it is not $\pm 1$ and preprocessing improved the DIMACS error, i.e.,
  $$\begin{aligned} \text {DIMACS}_{\mathrm{before}} > {10}^{-6} \, ~~\mathrm{and}~~ \, \dfrac{\text {DIMACS}_{\mathrm{after}}}{\text {DIMACS}_{\mathrm{before}}} < \dfrac{1}{10}; \end{aligned}$$
the help code is $-2, \, $ if
- it is not $\pm 1$ and preprocessing made the DIMACS error worse, i.e.,
  $$\begin{aligned} \text {DIMACS}_{\mathrm{after}}> {10}^{-6} \, ~~\mathrm{and}~~ \, \dfrac{\text {DIMACS}_{\mathrm{after}}}{\text {DIMACS}_{\mathrm{before}}} > 10; \end{aligned}$$
the help code is 3, if preprocessing shifted the objective value, i.e.,
- if help codes $\pm 1$ and $-2$ do not apply, and
  $$\begin{aligned} \dfrac{|\text {obj}_{\mathrm{before}} - \text {obj}_{\mathrm{after}}|}{1 + |\text {obj}_{\mathrm{before}}|} > 10^{-6}; \end{aligned}$$
the help code is MM if a code ran out of memory or crashed.

1.1 A.1 Detailed results on the Permenter–Parrilo (PP) dataset

This dataset has 68 problems. From these 59 problems were reduced by at least one of the five methods.

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
1	CompactDim2R1	None	0; 3; 3	5				0	3.79e$+$06, 4.20e$+$06	2.22e$+$01	3.02
		pd1	0; 3; 1	3	1	0.05	0.00	1	0.00e$+$00, 1.00e$+$00	7.07e$-$01	0.64	1
		pd2	0; 3; 1	3	1	0.04	0.00	1	0.00e$+$00, 1.00e$+$00	7.07e$-$01	0.69	1
		dd1			0	0.03	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			inf	0.01					0.00	1
2	CompactDim2R2	None	$0;0;6,{3}_{3}$	14				0	6.41e$-$10, 6.81e$-$10	7.07e$-$01	3.16
		pd1	$0;0;{1}_{3}$	2	1	0.11	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.13	1
		pd2	$0;0;{1}_{3}$	2	1	0.09	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.04	1
		dd1			0	0.03	0.00
		dd2			0	0.04	0.00
		Sieve-SDP			inf	0.01					0.00	1
3	CompactDim2R3	None	$0;0;10,{6}_{3}$	27				0	1.50e$+$00, 1.50e$+$00	1.15e$-$07	2.03
		pd1	$0;0;{1}_{3}$	2	1	0.14	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.07	1
		pd2	$0;0;{1}_{3}$	2	1	0.13	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.09	1
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			inf	0.01					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
5	CompactDim2R5	None	$0;0;21,{15}_{3}$	65				0	1.50e$+$00, 1.50e$+$00	1.83e$-$07	2.05
		pd1	$0;0;{1}_{3}$	2	1	0.25	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.06	1
		pd2	$0;0;{1}_{3}$	2	1	0.27	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.07	1
		dd1			0	0.02	0.00
		dd2			0	0.05	0.00
		Sieve-SDP			inf	0.03					0.00	1
6	CompactDim2R6	None	$0;0;28,{21}_{3}$	90				0	1.50e$+$00, 1.50e$+$00	2.70e$-$07	2.06
		pd1	$0;0;{1}_{3}$	2	1	0.32	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.10	1
		pd2	$0;0;{1}_{3}$	2	1	0.38	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.03	1
		dd1			0	0.05	0.00
		dd2			0	0.05	0.00
		Sieve-SDP			inf	0.04					0.00	1
7	CompactDim2R7	None	$0;0;36,{28}_{3}$	119				0	1.50e$+$00, 1.50e$+$00	3.66e$-$07	2.13
		pd1	$0;0;{1}_{3}$	2	1	0.41	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.08	1
		pd2	$0;0;{1}_{3}$	2	1	0.59	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.06	1
		dd1			0	0.03	0.00
		dd2			0	0.07	0.00
		Sieve-SDP			inf	0.06					0.00	1
8	CompactDim2R8	None	$0;0;45,{36}_{3}$	152				0	1.50e$+$00, 1.50e$+$00	5.61e$-$07	2.07
		pd1	$0;0;{1}_{3}$	2	1	0.56	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.02	1
		pd2	$0;0;{1}_{3}$	2	1	0.86	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.05	1
		dd1			0	0.03	0.00
		dd2			0	0.09	0.00
		Sieve-SDP			inf	0.08					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
9	CompactDim2R9	None	$0;0;55,{45}_{3}$	189				0	1.50e$+$00, 1.50e$+$00	6.27e$-$07	2.11
		pd1	$0;0;{1}_{3}$	2	1	0.71	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.08	1
		pd2	$0;0;{1}_{3}$	2	1	1.28	0.00	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.05	1
		dd1			0	0.03	0.00
		dd2			0	0.14	0.00
		Sieve-SDP			inf	0.11					0.00	1
10	CompactDim2R10	None	$0;0;66,{55}_{3}$	230				0	1.50e$+$00, 1.50e$+$00	5.17e$-$07	2.28
		pd1	$0;0;{1}_{3}$	2	1	0.86	0.01	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.17	1
		pd2	$0;0;{1}_{3}$	2	1	1.90	0.01	1	1.00e$+$00, 2.00e$+$00	7.07e$-$01	1.09	1
		dd1			0	0.04	0.01
		dd2			0	0.18	0.01
		Sieve-SDP			inf	0.15					0.00	1
11	Example1	None	0; 0; 3	2				0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	1.72
		pd1	0; 0; 2	1	1	0.06	0.01	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	1.05
		pd2	0; 0; 2	1	1	0.05	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.96
		dd1	5; 0; 1	2	1	0.07	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.96
		dd2	5; 0; 1	2	1	0.06	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.96
		Sieve-SDP	0; 0; 2	1	1	0.03		0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	1.63
12	Example2	None	0; 0; 3	2				0	3.33e$-$01, 3.33e$-$01	5.05e$-$02	1.73
		pd1	0; 0; 2	1	1	0.05	0.01	0	1.00e$+$00, 1.00e$+$00	0.00e$+$00	0.94	2, 3
		pd2	0; 0; 2	1	1	0.05	0.00	0	1.00e$+$00, 1.00e$+$00	0.00e$+$00	0.97	2, 3
		dd1	3; 0; 2	2	1	0.05	0.00	0	4.73e$-$15, 1.82e$-$14	2.75e$-$14	1.01	2, 3
		dd2	3; 0; 2	2	1	0.02	0.00	0	4.73e$-$15, 1.82e$-$14	2.75e$-$14	1.01	2, 3
		Sieve-SDP	0; 0; 2	1	1	0.01		0	1.00e$+$00, 1.00e$+$00	0.00e$+$00	2.61	2, 3
13	Example3	None	0; 0; 3	4				0	3.33e$-$01, 3.33e$-$01	6.90e$-$02	1.79
		pd1	0; 0; 2	1	1	0.03	0.00	0	1.17e$-$07, 1.69e$-$07	5.14e$-$08	1.44	2, 3
		pd2	0; 0; 2	1	1	0.03	0.00	0	1.17e$-$07, 1.69e$-$07	5.14e$-$08	1.48	2, 3
		dd1	3; 0; 2	4	1	0.02	0.01	0	4.73e$-$15, 1.82e$-$14	2.75e$-$14	1.00	2, 3
		dd2	3; 0; 2	4	1	0.03	0.00	0	4.73e$-$15, 1.82e$-$14	2.75e$-$14	0.99	2, 3
		Sieve-SDP	0; 0; 2	1	1	0.01		0	1.17e$-$07, 1.69e$-$07	5.14e$-$08	1.59	2, 3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
14	Example4	None	0; 0; 3	3				1	0.00e$+$00, 3.74e$-$07	5.00e$-$01	1.43
		pd1	0; 0; 1	1	1	0.03	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.64
		pd2	0; 0; 1	1	1	0.03	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.63
		dd1	5; 0; 1	3	1	0.03	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.96	$-$ 1
		dd2	5; 0; 1	3	1	0.04	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.99	$-$ 1
		Sieve-SDP			inf	0.00					0.00	1
15	Example6	None	0; 0; 8	8				0	1.00e$+$00, 1.00e$+$00	1.95e$-$08	0.66
		pd1	0; 0; 5	4	1	0.04	0.00	0	1.00e$+$00, 1.00e$+$00	0.00e$+$00	0.99
		pd2	0; 0; 5	4	1	0.04	0.00	0	1.00e$+$00, 1.00e$+$00	0.00e$+$00	0.98
		dd1	26; 0; 4	8	1	0.02	0.00	0	1.00e$+$00, 1.00e$+$00	9.75e$-$09	1.02
		dd2	26; 0; 4	8	1	0.02	0.00	0	1.00e$+$00, 1.00e$+$00	9.75e$-$09	1.19
		Sieve-SDP	0; 0; 5	4	1	0.01		0	1.00e$+$00, 1.00e$+$00	0.00e$+$00	0.56
16	Example7	None	0; 0; 5	3				0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.60
		pd1	0; 0; 4	2	1	0.02	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.96
		pd2	0; 0; 4	2	1	0.03	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.97
		dd1	14; 0; 1	3	1	0.03	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.98
		dd2	14; 0; 1	3	1	0.03	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	1.00
		Sieve-SDP	0; 0; 4	2	1	0.00		0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.54
17	Example9size20	None	0; 0; 20	20				1	0.00e$+$00, 3.39e$-$01	5.00e$-$01	2.58
		pd1	0; 0; 1	1	1	0.06	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.63
		pd2	0; 0; 1	1	1	0.04	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.62
		dd1	209; 0; 1	20	1	0.19	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.95	$-$ 1
		dd2	209; 0; 1	20	1	0.24	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.97	$-$ 1
		Sieve-SDP			inf	0.00					0.00	1
18	Example9size100	None	0; 0; 100	100				1	0.00e$+$00, 3.43e$-$01	5.00e$-$01	0.83
		pd1	0; 0; 1	1	1	0.04	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.64
		pd2	0; 0; 1	1	1	0.19	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.63
		dd1	5049; 0; 1	100	1	1.33	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	1.01	$-$ 1
		dd2	5049; 0; 1	100	1	3.50	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	1.00	$-$ 1
		Sieve-SDP			inf	0.00					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
19	RandGen6	None	0; 0; 320	140				0	3.95e$-$06, 3.24e$-$06	2.29e$-$05	24.07
		pd1			0	3.64	1.00
		pd2			0	16.39	1.00
		dd1			0	0.75	1.00
		dd2	19985; 0; 250	140	1	37.13	2.14	0	1.68e$-$07, 1.26e$-$11	8.00e$-$07	5.88	2, 3
		Sieve-SDP	0; 0; 120	70	1	2.10		0	3.73e$-$06, 3.04e$-$06	9.17e$-$06	2.32
20	RandGen7	None	0; 0; 40	27				0	9.42e$-$07, 4.22e$-$07	4.69e$-$06	0.67
		pd1			0	0.03	0.01
		pd2	0; 0; 28	14	1	0.10	0.02	0	9.85e$-$07, 4.53e$-$07	3.27e$-$06	1.04
		dd1			0	0.02	0.01
		dd2	649; 0; 18	27	1	0.11	0.01	0	2.65e$-$11, 4.69e$-$16	7.21e$-$11	1.08	2
		Sieve-SDP	0; 0; 28	14	1	0.02		0	9.85e$-$07, 4.53e$-$07	3.27e$-$06	0.72
21	RandGen8	None	0; 0; 60	40				0	5.41e$-$09, 2.44e$-$09	9.31e$-$08	0.83
		pd1			0	0.04	0.01
		pd2			0	0.22	0.01
		dd1			0	0.02	0.01
		dd2	1269; 0; 33	40	1	0.33	0.02	0	2.15e$-$15, 2.78e$-$19	6.90e$-$14	1.05
		Sieve-SDP	0; 0; 30	20	1	0.03		0	1.52e$-$09, 6.33e$-$10	9.04e$-$09	0.69
22	copos_1	None	0; 0; 35	210				0	0.00e$+$00, 1.11e$-$08	4.40e$-$07	0.66
		pd1			0	0.02	0.00
		pd2	0; 0; 25	160	1	0.06	0.02	0	0.00e$+$00, $-$ 3.86e$-$10	2.12e$-$08	1.01
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP			0	0.02
23	copos_2	None	0; 0; 120	1716				0	0.00e$+$00, 5.76e$-$11	1.69e$-$08	1.83
		pd1			0	0.03	0.00
		pd2	0; 0; 96	1524	1	0.57	0.11	0	0.00e$+$00, $-$ 2.31e$-$13	6.38e$-$11	1.72
		dd1			0	0.02	0.00
		dd2			0	0.13	0.00
		Sieve-SDP			0	0.09

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$
24	copos_3	None	0; 0; 286	8008				0	0.00e$+$00, $-$ 4.93e$-$10	1.59e$-$07	44.68
		pd1			0	0.10	0.01
		pd2	0; 0; 242	7524	1	37.41	0.57	0	0.00e$+$00, $-$ 4.51e$-$11	1.26e$-$08	30.28
		dd1			0	0.06	0.01
		dd2			0	0.85	0.01
		Sieve-SDP			0	0.46
25	copos_4	None	0; 0; 560	27132				0	0.00e$+$00, $-$ 9.00e$-$11	7.21e$-$08	1526.50
		pd1			0	0.46	0.06
		pd2	0; 0; 490	26152	1	26.16	1.98	0	0.00e$+$00, $-$ 1.70e$-$10	6.56e$-$08	1139.18
		dd1			0	0.36	0.06
		dd2			0	5.09	0.06
		Sieve-SDP			0	1.80
26	cprank_1	None	$9;0;{1}_{9},10,9$	46				0	$-$ 3.00e$+$00, $-$ 3.00e$+$00	3.50e$-$08	1.32
		pd1			0	0.08	0.00
		pd2			0	0.03	0.00
		dd1	$30;0;{1}_{7},8,9$	46	1	0.07	0.01	0	$-$ 3.00e$+$00, $-$ 3.00e$+$00	4.62e$-$08	1.16
		dd2	$30;0;{1}_{7},8,9$	46	1	0.06	0.01	0	$-$ 3.00e$+$00, $-$ 3.00e$+$00	3.88e$-$08	1.17
		Sieve-SDP			0	0.01
27	cprank_2	None	$1296;0;{1}_{81},82,81$	3322				0	$-$ 9.00e$+$00, $-$ 9.00e$+$00	6.62e$-$08	15.40
		pd1			0	0.08	0.00
		pd2			0	0.18	0.00
		dd1	$3456;0;{1}_{49},50,81$	3322	1	0.14	0.31	0	$-$ 9.00e$+$00, $-$ 9.00e$+$00	6.64e$-$09	10.50
		dd2	$3456;0;{1}_{49},50,81$	3322	1	0.50	0.32	0	$-$ 9.00e$+$00, $-$ 9.00e$+$00	1.51e$-$09	9.75
		Sieve-SDP			0	0.06
28	hinf12	None	$0;0;{6}_{2},12$	43				0	$-$ 1.45e$-$13, $-$ 1.17e$-$13	1.80e$+$00	1.38
		pd1			0	0.03	0.00
		pd2	0; 0; 6, 2, 6	22	1	0.04	0.00	0	$-$ 2.64e$-$15, $-$ 1.77e$-$15	1.79e$+$00	1.69
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP			0	0.01

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
29	horn2	None	0; 0; 4	7				0	0.00e$+$00, 6.69e$-$13	9.06e$-$13	2.03
		pd1			0	0.02	0.00
		pd2	0; 0; 2	3	1	0.06	0.00	0	0.00e$+$00, 0.00e$+$00	1.57e$-$16	0.99
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			0	0.01
30	horn3	None	0; 0; 10	28				0	0.00e$+$00, 1.46e$-$07	8.62e$-$07	2.00
		pd1			0	0.02	0.00
		pd2	0; 0; 6	16	1	0.05	0.00	0	0.00e$+$00, 3.53e$-$09	2.65e$-$08	0.99
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			0	0.00
31	horn4	None	0; 0; 20	84				0	0.00e$+$00, 1.13e$-$07	1.90e$-$06	2.14
		pd1			0	0.02	0.00
		pd2	0; 0; 14	60	1	0.07	0.01	0	0.00e$+$00, 7.11e$-$09	7.44e$-$08	1.07	2
		dd1			0	0.03	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			0	0.01
32	horn5	None	0; 0; 35	210				0	0.00e$+$00, 1.07e$-$08	2.69e$-$07	2.05
		pd1			0	0.02	0.00
		pd2	0; 0; 25	160	1	0.08	0.01	0	0.00e$+$00, $-$ 2.28e$-$09	2.35e$-$07	0.99
		dd1			0	0.03	0.00
		dd2			0	0.04	0.00
		Sieve-SDP			0	0.03
33	hornD2	None	0; 0; 4	3				0	$-$ 5.25e$-$08, 0.00e$+$00	5.25e$-$08	2.04
		pd1			0	0.02	0.00
		pd2			0	0.03	0.00
		dd1			0	0.03	0.00
		dd2	7; 0; 2	3	1	0.05	0.00	0	$-$ 1.88e$-$16, 0.00e$+$00	1.78e$-$15	1.00
		Sieve-SDP			0	0.00

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
34	hornD3	None	0; 0; 10	27				0	$-$ 5.58e$-$08, 0.00e$+$00	8.62e$-$07	2.01
		pd1			0	0.03	0.00
		pd2			0	0.02	0.00
		dd1			0	0.03	0.00
		dd2	34; 0; 6	27	1	0.05	0.00	0	$-$ 8.68e$-$10, 0.00e$+$00	1.88e$-$08	1.13
		Sieve-SDP			0	0.01
35	hornD4	None	0; 0; 20	126				0	1.77e$-$07, 0.00e$+$00	1.02e$-$06	2.04
		pd1			0	0.02	0.00
		pd2			0	0.04	0.00
		dd1			0	0.02	0.00
		dd2	105; 0; 14	126	1	0.06	0.01	0	7.49e$-$08, 0.00e$+$00	2.38e$-$07	1.12
		Sieve-SDP			0	0.02
36	hornD5	None	0; 0; 35	420				0	2.32e$-$08, 0.00e$+$00	1.83e$-$07	2.06
		pd1			0	0.03	0.00
		pd2			0	0.04	0.00
		dd1			0	0.03	0.00
		dd2	305; 0; 25	420	1	0.09	0.03	0	5.58e$-$10, 0.00e$+$00	2.00e$-$09	1.26
		Sieve-SDP			0	0.04
37	hybridLyap	None	860; 0; $6,108,{11}_{10}$	3093				0	0.00e$+$00, 7.29e$-$07	2.11e$-$04	7.85
		pd1	860; 0; 6, 56, 11, ${1}_{2},11,{1}_{2},{11}_{2}$	1607	1	0.16	0.09	0	0.00e$+$00, 3.48e$-$07	6.61e$-$05	1.49
		pd2	860; 0; 6, 34, $8,{1}_{2},8,{1}_{2},9,7$	1173	1	1.02	0.05	0	0.00e$+$00, 4.24e$-$09	4.86e$-$07	1.23	2
		dd1			0	0.05	0.00
		dd2			0	0.14	0.00
		Sieve-SDP			0	0.05

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
38	leverage_limit	None	0; 18100; ${151}_{100},{30}_{100}$	68195				0	$-$ 8.75e$+$01, $-$ 8.75e$+$01	1.53e$-$05	278.60
		pd1			0	2.10	0.17
		pd2	0; 18100; ${151}_{99},121,{30}_{100}$	67700	1	120.98	7.87	0	$-$ 8.75e$+$01, $-$ 8.75e$+$01	5.63e$-$06	150.78	3
		dd1	958500; 18100; ${61}_{100},{30}_{100}$	68195	1	3.87	7.20	0	$-$ 8.75e$+$01, $-$ 8.75e$+$01	2.45e$-$05	250.27
		dd2	1193505; 18100; ${1}_{99},31$	68195	1	291.58	1.39	$-$ 1	$-$ 3.35e$+$00, 0.00e$+$00	1.03e$+$01	1.97	$-$ 2
		Sieve-SDP	$0;18100;{143}_{97},$ ${141}_{3},{26}_{98},{25}_{2}$	56196	1	253.43		0	$-$ 8.74e$+$01, $-$ 8.74e$+$01	1.73e$-$05	179.26	3
39	long_only	None	0; 9000; ${91}_{100},{30}_{100}$	59095				0	$-$ 4.13e$+$01, $-$ 4.13e$+$01	5.23e$-$06	373.38
		pd1			0	1.18	0.17
		pd2	0; 9000; ${91}_{99},61,{30}_{100}$	58600	1	24.33	6.91	0	$-$ 4.13e$+$01, $-$ 4.13e$+$01	4.64e$-$07	205.50	2, 3
		dd1	229500; 9000; ${61}_{100},{30}_{100}$	59095	1	1.77	6.96	0	$-$ 4.13e$+$01, $-$ 4.13e$+$01	6.47e$-$06	246.03	3
		dd2	229500; 9000; ${61}_{100},{30}_{100}$	59095	1	531.60	6.94	0	$-$ 4.13e$+$01, $-$ 4.13e$+$01	2.80e$-$06	315.18	3
		Sieve-SDP	$0;8573;{83}_{97},$ ${81}_{3},{26}_{98},{25}_{2}$	46670	1	190.92		0	$-$ 4.13e$+$01, $-$ 4.13e$+$01	1.64e$-$06	94.12	3
40	sector_neutral	None	0; 12000; ${121}_{100},{30}_{100}$	62392				0	$-$ 1.21e$+$02, $-$ 1.21e$+$02	8.35e$-$05	152.27
		pd1			0	1.84	0.26
		pd2	0; 12000; ${121}_{99},91,{30}_{100}$	61897	1	183.96	7.17	0	$-$ 1.21e$+$02, $-$ 1.21e$+$02	2.79e$-$04	150.19	3
		dd1	549000; 12000; ${61}_{100},{30}_{100}$	62392	1	2.79	7.05	0	$-$ 1.21e$+$02, $-$ 1.21e$+$02	8.23e$-$05	154.78	3
		dd2	549000; 12000; ${61}_{100},{30}_{100}$	62392	1	217.62	7.13	0	$-$ 1.21e$+$02, $-$ 1.21e$+$02	1.24e$-$04	140.83	3
		Sieve-SDP	$0;12000;{121}_{99},$ $111,{30}_{100}$	62247	1	52.25		0	$-$ 1.21e$+$02, $-$ 1.21e$+$02	1.76e$-$04	151.52	3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
41	unconstrained	None	0; 12000; ${121}_{100},{30}_{100}$	62095				0	$-$ 1.33e$+$02, $-$ 1.33e$+$02	7.89e$-$05	279.82
		pd1			0	1.52	0.15
		pd2	0; 12000; ${121}_{99},91,{30}_{100}$	61600	1	38.70	6.95	0	$-$ 1.33e$+$02, $-$ 1.33e$+$02	1.42e$-$05	282.87	3
		dd1	549000; 12000; ${61}_{100},{30}_{100}$	62095	1	2.63	6.74	0	$-$ 1.33e$+$02, $-$ 1.33e$+$02	3.34e$-$05	258.89	3
		dd2	549000; 12000; ${61}_{100},{30}_{100}$	62095	1	505.61	6.70	0	$-$ 1.33e$+$02, $-$ 1.33e$+$02	3.64e$-$05	260.11	3
		Sieve-SDP	$0;12000;{113}_{97},$ ${111}_{3},{26}_{98},{25}_{2}$	50097	1	213.04		0	$-$ 1.28e$+$02, $-$ 1.28e$+$02	1.64e$-$05	185.98	3
42	unboundDim	None	0; 2; 2	2				0	1.33e$-$09, $-$ 7.05e$-$10	4.38e$-$09	2.89
	1R1	pd1			0	0.05	0.01
		pd2			0	0.03	0.01
		dd1			0	0.03	0.01
		dd2			0	0.02	0.01
		Sieve-SDP	0; 1; 1	1	1	0.01		0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	2.25
43	unboundDim1R2	None	$0;0;3,{2}_{2}$	4				0	$-$ 8.91e$-$15, $-$ 8.01e$-$15	7.07e$-$01	4.52
		pd1	$0;0;{1}_{2}$	1	1	0.11	0.01	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.43	2
		pd2	$0;0;{1}_{2}$	1	1	0.10	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.47	2
		dd1			0	0.03	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	$0;0;{1}_{2}$	1	1	0.00		0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	2.39	2
44	unboundDim1R3	None	$0;0;4,{3}_{2}$	6				0	$-$ 2.04e$-$11, $-$ 2.02e$-$11	7.07e$-$01	4.17
		pd1	$0;0;{1}_{2}$	1	1	0.12	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.50	2
		pd2	$0;0;{1}_{2}$	1	1	0.11	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.47	2
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	$0;0;{1}_{2}$	1	1	0.01		0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	2.31	2

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
45	unboundDim1R4	None	$0;0;5,{4}_{2}$	8				0	$-$ 2.34e$-$10, $-$ 2.32e$-$10	7.07e$-$01	3.79
		pd1	$0;0;{1}_{2}$	1	1	0.14	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.43	2
		pd2	$0;0;{1}_{2}$	1	1	0.14	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.43	2
		dd1			0	0.03	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	$0;0;{1}_{2}$	1	1	0.01		0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	2.41	2
46	unboundDim1R5	None	$0;0;6,{5}_{2}$	10				0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	9.88e$-$08	2.74
		pd1	$0;0;{1}_{2}$	1	1	0.16	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.43	3
		pd2	$0;0;{1}_{2}$	1	1	0.20	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.42	3
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	$0;0;{1}_{2}$	1	1	0.01		0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	2.32	3
47	unboundDim1R6	None	$0;0;7,{6}_{2}$	12				0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	2.15e$-$07	2.78
		pd1	$0;0;{1}_{2}$	1	1	0.20	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.50	3
		pd2	$0;0;{1}_{2}$	1	1	0.22	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.49	3
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	$0;0;{1}_{2}$	1	1	0.01		0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	2.24	3
48	unboundDim1R7	None	$0;0;8,{7}_{2}$	14				0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	5.11e$-$08	2.82
		pd1	$0;0;{1}_{2}$	1	1	0.21	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.45	3
		pd2	$0;0;{1}_{2}$	1	1	0.23	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.44	3
		dd1			0	0.03	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	$0;0;{1}_{2}$	1	1	0.02		0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	2.32	3
49	unboundDim1R8	None	$0;0;9,{8}_{2}$	16				0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	5.43e$-$08	2.29
		pd1	$0;0;{1}_{2}$	1	1	0.52	0.02	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.44	3
		pd2	$0;0;{1}_{2}$	1	1	0.53	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.45	3
		dd1			0	0.07	0.00
		dd2			0	0.04	0.00
		Sieve-SDP	$0;0;{1}_{2}$	1	1	0.13		0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	1.74	3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
50	unboundDim1R9	None	$0;0;10,{9}_{2}$	18				0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	6.50e$-$08	2.09
		pd1	$0;0;{1}_{2}$	1	1	0.31	0.01	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.43	3
		pd2	$0;0;{1}_{2}$	1	1	0.30	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.64	3
		dd1			0	0.04	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	$0;0;{1}_{2}$	1	1	0.05		0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	1.60	3
51	unboundDim1R10	None	$0;0;11,{10}_{2}$	20				0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	1.41e$-$07	2.76
		pd1	$0;0;{1}_{2}$	1	1	0.28	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.44	3
		pd2	$0;0;{1}_{2}$	1	1	0.32	0.00	0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	0.45	3
		dd1			0	0.03	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	$0;0;{1}_{2}$	1	1	0.04		0	0.00e$+$00, 0.00e$+$00	0.00e$+$00	2.33	3
52	vamos_5_34	None	0; 0; 52	721				0	0.00e$+$00, $-$ 4.18e$-$09	5.21e$-$08	2.10
		pd1			0	0.07	0.00
		pd2				MM						MM
		dd1			0	0.05	0.00
		dd2			0	0.07	0.00
		Sieve-SDP			0	0.06
53	wei_wagner_F7_minus_4	None	0; 0; 8	31				0	0.00e$+$00, $-$ 9.60e$-$13	1.11e$-$11	1.87
		pd1			0	0.02	0.00
		pd2	0; 0; 5	14	1	0.08	0.01	0	0.00e$+$00, $-$ 5.80e$-$11	2.12e$-$10	0.99
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			0	0.01
54	wei_wagner_P7	None	0; 0; 8	32				0	0.00e$+$00, $-$ 1.46e$-$08	9.09e$-$08	1.99
		pd1			0	0.02	0.00
		pd2	0; 0; 4	10	1	0.05	0.00	0	0.00e$+$00, $-$ 3.02e$-$10	1.31e$-$09	1.04
		dd1			0	0.03	0.00
		dd2			0	0.04	0.00
		Sieve-SDP			0	0.01

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$
55	wei_wagner_W3Plus	None	0; 0; 8	31				0	0.00e$+$00, $-$ 6.06e$-$09	5.47e$-$08	1.95
		pd1			0	0.02	0.00
		pd2	0; 0; 3	6	1	0.05	0.00	0	0.00e$+$00, $-$ 4.77e$-$09	1.11e$-$08	1.01
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP			0	0.00
56	wei_wagner_W3_PlusE	None	0; 0; 9	38				0	0.00e$+$00, $-$ 9.18e$-$09	5.53e$-$08	1.98
		pd1			0	0.02	0.00
		pd2	0; 0; 5	15	1	0.06	0.00	0	0.00e$+$00, $-$ 7.21e$-$09	3.21e$-$08	1.02
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			0	0.00
57	wei_wagner_nP_minus_1_24	None	0; 0; 12	64				0	0.00e$+$00, $-$ 5.50e$-$09	8.80e$-$08	2.03
		pd1			0	0.02	0.00
		pd2	0; 0; 6	21	1	0.07	0.00	0	0.00e$+$00, $-$ 1.08e$-$11	5.60e$-$11	1.02
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			0	0.01
58	wei_wagner_nP_minus_9_12	None	0; 0; 12	64				0	0.00e$+$00, $-$ 3.92e$-$09	4.87e$-$08	1.98
		pd1			0	0.02	0.00
		pd2	0; 0; 5	15	1	0.05	0.00	0	0.00e$+$00, $-$ 4.11e$-$15	2.34e$-$14	1.02
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP			0	0.01
59	wei_wagner_vamos_12	None	0; 0; 16	103				0	0.00e$+$00, $-$ 1.59e$-$08	1.38e$-$07	2.10
		pd1			0	0.02	0.00
		pd2	0; 0; 13	74	1	0.06	0.01	0	0.00e$+$00, $-$ 2.54e$-$10	1.62e$-$09	1.03
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP			0	0.01

1.2 A.2 Detailed results on the Mittelmann dataset

This dataset has 31 problems. From these 8 problems were reduced by at least one of the five methods. There were 5 problems on which pd2 or dd2 ran out of memory or crashed.

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
1	diamond_patch	None	0; 0; 5477	5478				0	1.63e$+$01, 1.63e$+$01	3.56e$-$04	10854.97
		pd1			0	31.05	0.06
		pd2				MM						MM
		dd1			0	27.94	0.06
		dd2			0	3008.86	0.06
		Sieve-SDP			0	1.12
2	e_moment_stable_17_0.5_2_2	None	0; 342; $171,{18}_{17}$	5984				0	$-$ 1.98e$-$01, $-$ 1.98e$-$01	1.14e$-$05	38.53
		pd1	$0;342;{18}_{18}$	1139	1	0.71	0.16	0	$-$ 1.98e$-$01, $-$ 1.98e$-$01	8.44e$-$06	1.64
		pd2	$0;342;{18}_{18}$	1139	1	0.86	0.13	0	$-$ 1.98e$-$01, $-$ 1.98e$-$01	8.44e$-$06	1.66
		dd1			0	0.08	0.01
		dd2			0	0.34	0.01
		Sieve-SDP	$0;342;{18}_{18}$	1139	1	0.52		0	$-$ 1.98e$-$01, $-$ 1.98e$-$01	8.44e$-$06	1.63
3	ice_2.0	None	0; 0; 8113	8113				0	6.81e$+$03, 6.81e$+$03	4.58e$-$07	17680.82
		pd1			0	65.58	0.01
		pd2				MM						MM
		dd1			0	64.03	0.01
		dd2				MM						MM
		Sieve-SDP			0	0.80
4	G60_mb	None	0; 0; 7000	7001				0	1.93e$+$03, 1.93e$+$03	6.64e$-$05	29138.79
		pd1			0	107.66	10.87
		pd2				MM						MM
		dd1			0	72.76	10.87
		dd2				MM						MM
		Sieve-SDP			0	22.42

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
5	maxG60	None	0; 0; 7000	7000				0	$-$ 1.52e$+$04, $-$ 1.52e$+$04	6.73e$-$07	5217.88
		pd1			0	47.47	0.01
		pd2				MM						MM
		dd1			0	45.65	0.01
		dd2				MM						MM
		Sieve-SDP			0	0.47
6	neu3	None	0; 2; 418	7364				0	7.10e$-$08, 1.12e$-$08	2.01e$-$06	153.03
		pd1	0; 2; 87	1152	1	0.94	0.11	0	4.69e$-$08, 3.50e$-$08	1.94e$-$07	3.01	2
		pd2	0; 2; 87	1152	1	5.41	0.10	0	4.69e$-$08, 3.50e$-$08	1.94e$-$07	2.97	2
		dd1			0	0.16	0.02
		dd2			0	2.29	0.02
		Sieve-SDP	0; 2; 87	1152	1	2.34		0	4.69e$-$08, 3.50e$-$08	1.94e$-$07	2.99	2
7	neu3g	None	0; 0; 462	8007				0	4.58e$-$08, $-$ 2.89e$-$09	8.67e$-$07	151.22
		pd1	0; 0; 87	1151	1	1.32	0.11	0	8.91e$-$08, 5.65e$-$08	2.91e$-$07	3.00
		pd2	0; 0; 87	1151	1	10.68	0.11	0	8.91e$-$08, 5.65e$-$08	2.91e$-$07	3.09
		dd1			0	0.19	0.03
		dd2			0	2.66	0.03
		Sieve-SDP	0; 0; 87	1151	1	2.26		0	8.91e$-$08, 5.65e$-$08	2.91e$-$07	3.03
8	p_auss2_3.0	None	0; 0; 9115	9115				0	8.62e$+$03, 8.62e$+$03	2.36e$-$07	25651.19
		pd1			0	93.91	0.02
		pd2				MM						MM
		dd1			0	97.11	0.02
		dd2				MM						MM
		Sieve-SDP			0	0.76
9	rose13	None	0; 0; 105	2379				0	1.20e$+$01, 1.20e$+$01	1.65e$-$06	7.63
		pd1	0; 0; 92	1911	1	0.11	0.14	0	1.20e$+$01, 1.20e$+$01	4.86e$-$07	5.26
		pd2	0; 0; 80	1523	1	0.51	0.11	0	1.20e$+$01, 1.20e$+$01	1.98e$-$07	2.94
		dd1			0	0.05	0.01
		dd2			0	0.12	0.01
		Sieve-SDP	0; 0; 92	1911	1	0.39		0	1.20e$+$01, 1.20e$+$01	4.86e$-$07	5.28

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
10	rose15	None	0; 2; 135	3860				0	$-$ 3.11e$-$06, $-$ 2.94e$-$06	1.83e$-$05	19.47
		pd1	0; 2; 121	3181	1	0.08	0.24	0	$-$ 3.52e$-$07, $-$ 1.52e$-$07	5.07e$-$05	11.73	3
		pd2	0; 2; 107	2593	1	0.66	0.19	0	$-$ 1.59e$-$09, $-$ 1.57e$-$09	1.10e$-$08	5.74	2, 3
		dd1			0	0.07	0.00
		dd2			0	0.18	0.00
		Sieve-SDP	0; 2; 121	3181	1	0.52		0	$-$ 3.52e$-$07, $-$ 1.52e$-$07	5.07e$-$05	11.71	3
11	taha1a	None	$0;0;252,{56}_{3},{126}_{10}$	3002				0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	9.39e$-$07	37.54
		pd1	$0;0;126,{56}_{3},{126}_{10}$	2001	1	10.57	0.72	0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	1.20e$-$07	21.55
		pd2	$0;0;126,{56}_{3},{126}_{10}$	2001	1	18.98	0.75	0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	1.20e$-$07	21.50
		dd1			0	0.21	0.06
		dd2			0	21.47	0.06
		Sieve-SDP	$0;0;126,{56}_{3},{126}_{10}$	2001	1	1.75		0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	1.20e$-$07	21.70
12	taha1b	None	$0;3;286,{66}_{20}$	8007				0	$-$ 7.73e$-$01, $-$ 7.73e$-$01	1.59e$-$07	148.99
		pd1	$0;3;{66}_{21}$	3002	1	13.97	0.87	0	$-$ 7.73e$-$01, $-$ 7.73e$-$01	1.32e$-$07	34.29
		pd2	$0;3;{66}_{21}$	3002	1	18.37	0.85	0	$-$ 7.73e$-$01, $-$ 7.73e$-$01	1.32e$-$07	33.03
		dd1			0	0.16	0.04
		dd2			0	1.82	0.04
		Sieve-SDP	$0;3;{66}_{21}$	3002	1	1.97		0	$-$ 7.73e$-$01, $-$ 7.73e$-$01	1.32e$-$07	32.97
13	taha1c	None	$0;0;462,{126}_{3},{252}_{10}$	6187				0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	3.12e$-$07	314.61
		pd1	$0;0;252,{126}_{3},{252}_{10}$	4367	1	148.36	2.11	0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	4.37e$-$07	178.22
		pd2	$0;0;252,{126}_{3},{252}_{10}$	4367	1	187.99	2.01	0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	4.37e$-$07	177.80
		dd1			0	0.75	0.25
		dd2			0	156.72	0.25
		Sieve-SDP	$0;0;252,{126}_{3},{252}_{10}$	4367	1	10.85		0	$-$ 1.00e$+$00, $-$ 1.00e$+$00	4.37e$-$07	182.86

1.3 A.3 Detailed results on the Dressler-Illiman-de Wolff (DIW) dataset

This is a collection of 155 SDP relaxations from polynomial optimization generated by Gloptipoly 3 based on the paper [11].

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
1	ex3.3_order4	None	0; 1; 15	44				0	3.54e$-$10, 3.56e$-$10	6.87e$-$01	1.69
		pd1	0; 1; 2	3	1	0.10	0.00	1	0.00e$+$00, 5.00e$-$01	5.27e$-$01	0.63	1
		pd2	0; 1; 2	3	1	0.35	0.00	1	0.00e$+$00, 5.00e$-$01	5.27e$-$01	0.73	1
		dd1			0	0.04	0.00
		dd2			0	0.04	0.00
		Sieve-SDP			inf	0.03					0.00	1
2	ex3.3_order5	None	0; 0; 21, 3	65				0	6.16e$-$02, 6.16e$-$02	1.18e$-$06	1.25
		pd1	0; 0; 2, 1	3	1	0.15	0.00	1	0.00e$+$00, 5.00e$-$01	5.27e$-$01	0.64	1
		pd2	0; 0; 2, 1	3	1	0.15	0.00	1	0.00e$+$00, 5.00e$-$01	5.27e$-$01	0.75	1
		dd1			0	0.04	0.00
		dd2			0	0.04	0.00
		Sieve-SDP			inf	0.03					0.00	1
3	ex3.3_order6	None	0; 0; 28, 6	90				0	6.16e$-$02, 6.16e$-$02	1.60e$-$06	1.00
		pd1	0; 0; 8, 2	22	1	0.10	0.00	0	6.16e$-$02, 6.16e$-$02	4.23e$-$08	1.08	2
		pd2	0; 0; 8, 2	22	1	0.16	0.00	0	6.16e$-$02, 6.16e$-$02	4.23e$-$08	0.69	2
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 8, 2	22	1	0.03		0	6.16e$-$02, 6.16e$-$02	4.23e$-$08	0.69	2
4	ex3.3_order7	None	0; 0; 36, 10	119				0	6.16e$-$02, 6.16e$-$02	3.07e$-$06	0.65
		pd1	0; 0; 10, 4	31	1	0.13	0.00	0	6.16e$-$02, 6.16e$-$02	6.03e$-$08	0.59	2, 3
		pd2	0; 0; 10, 4	31	1	0.24	0.00	0	6.16e$-$02, 6.16e$-$02	6.03e$-$08	0.65	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 10, 4	31	1	0.05		0	6.16e$-$02, 6.16e$-$02	6.03e$-$08	0.59	2, 3
5	ex3.3_order8	None	0; 0; 45, 15	152				0	6.16e$-$02, 6.16e$-$02	2.45e$-$06	1.46
		pd1	0; 0; 12, 6	38	1	0.25	0.01	0	6.16e$-$02, 6.16e$-$02	5.23e$-$08	0.65	2, 3
		pd2	0; 0; 12, 6	38	1	0.37	0.01	0	6.16e$-$02, 6.16e$-$02	5.23e$-$08	0.67	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.04	0.00
		Sieve-SDP	0; 0; 12, 6	38	1	0.07		0	6.16e$-$02, 6.16e$-$02	5.23e$-$08	0.58	2, 3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
6	ex3.3_order9	None	0; 0; 55, 21	189				0	6.16e$-$02, 6.16e$-$02	4.63e$-$06	0.94
		pd1	0; 0; 13, 7	41	1	0.18	0.01	0	6.16e$-$02, 6.16e$-$02	5.54e$-$08	0.60	2, 3
		pd2	0; 0; 13, 7	41	1	0.36	0.01	0	6.16e$-$02, 6.16e$-$02	5.54e$-$08	0.61	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.05	0.00
		Sieve-SDP	0; 0; 13, 7	41	1	0.09		0	6.16e$-$02, 6.16e$-$02	5.54e$-$08	0.69	2, 3
7	ex3.3_order10	None	0; 0; 66, 28	230				0	6.16e$-$02, 6.16e$-$02	4.33e$-$06	1.29
		pd1	0; 0; 14, 8	44	1	0.33	0.03	0	6.16e$-$02, 6.16e$-$02	6.73e$-$08	1.06	2, 3
		pd2	0; 0; 14, 8	44	1	0.56	0.01	0	6.16e$-$02, 6.16e$-$02	6.73e$-$08	1.00	2, 3
		dd1			0	0.06	0.00
		dd2			0	0.08	0.00
		Sieve-SDP	0; 0; 14, 8	44	1	0.15		0	6.16e$-$02, 6.16e$-$02	6.73e$-$08	1.24	2, 3
8	ex3.3_order11	None	0; 0; 78, 36	275				0	6.16e$-$02, 6.16e$-$02	1.12e$-$05	1.38
		pd1	0; 0; 16, 10	53	1	0.31	0.01	0	6.16e$-$02, 6.16e$-$02	9.05e$-$08	1.06	2, 3
		pd2	0; 0; 16, 10	53	1	0.70	0.01	0	6.16e$-$02, 6.16e$-$02	9.05e$-$08	1.06	2, 3
		dd1			0	0.04	0.00
		dd2			0	0.10	0.00
		Sieve-SDP	0; 0; 16, 10	53	1	0.18		0	6.16e$-$02, 6.16e$-$02	9.05e$-$08	1.22	2, 3
9	ex3.3_order12	None	0; 0; 91, 45	324				0	6.16e$-$02, 6.16e$-$02	2.34e$-$05	1.50
		pd1	0; 0; 18, 12	60	1	0.38	0.01	0	6.16e$-$02, 6.16e$-$02	1.12e$-$07	1.38	2, 3
		pd2	0; 0; 18, 12	60	1	1.17	0.01	0	6.16e$-$02, 6.16e$-$02	1.12e$-$07	0.63	2, 3
		dd1			0	0.04	0.00
		dd2			0	0.11	0.00
		Sieve-SDP	0; 0; 18, 12	60	1	0.22		0	6.16e$-$02, 6.16e$-$02	1.12e$-$07	0.68	2, 3
10	ex3.3_order13	None	0; 0; 105, 55	377				0	6.15e$-$02, 6.15e$-$02	3.47e$-$05	1.38
		pd1	0; 0; 19, 13	63	1	0.45	0.01	0	6.16e$-$02, 6.16e$-$02	1.21e$-$07	0.61	2, 3
		pd2	0; 0; 19, 13	63	1	1.50	0.01	0	6.16e$-$02, 6.16e$-$02	1.21e$-$07	0.59	2, 3
		dd1			0	0.03	0.01
		dd2			0	0.15	0.01
		Sieve-SDP	0; 0; 19, 13	63	1	0.28		0	6.16e$-$02, 6.16e$-$02	1.21e$-$07	0.57	2, 3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
11	ex3.3_order14	None	0; 0; 120, 66	434				0	6.16e$-$02, 6.16e$-$02	1.41e$-$05	1.50
		pd1	0; 0; 20, 14	66	1	0.58	0.01	0	6.16e$-$02, 6.16e$-$02	1.31e$-$07	0.57	2, 3
		pd2	0; 0; 20, 14	66	1	2.07	0.01	0	6.16e$-$02, 6.16e$-$02	1.31e$-$07	1.39	2, 3
		dd1			0	0.05	0.00
		dd2			0	0.22	0.00
		Sieve-SDP	0; 0; 20, 14	66	1	0.38		0	6.16e$-$02, 6.16e$-$02	1.31e$-$07	1.27	2, 3
12	ex3.3_order15	None	0; 0; 136, 78	495				0	6.16e$-$02, 6.16e$-$02	1.06e$-$05	2.72
		pd1	0; 0; 22, 16	75	1	0.73	0.01	0	6.16e$-$02, 6.16e$-$02	1.47e$-$07	1.06	2, 3
		pd2	0; 0; 22, 16	75	1	2.89	0.02	0	6.16e$-$02, 6.16e$-$02	1.47e$-$07	1.06	2, 3
		dd1			0	0.05	0.00
		dd2			0	0.24	0.00
		Sieve-SDP	0; 0; 22, 16	75	1	0.47		0	6.16e$-$02, 6.16e$-$02	1.47e$-$07	1.10	2, 3
13	ex3.3_order16	None	0; 0; 153, 91	560				0	6.16e$-$02, 6.16e$-$02	1.18e$-$05	3.64
		pd1	0; 0; 24, 18	82	1	0.92	0.01	0	6.16e$-$02, 6.16e$-$02	1.79e$-$07	0.95	2, 3
		pd2	0; 0; 24, 18	82	1	3.88	0.01	0	6.16e$-$02, 6.16e$-$02	1.79e$-$07	0.92	2, 3
		dd1			0	0.05	0.00
		dd2			0	0.29	0.00
		Sieve-SDP	0; 0; 24, 18	82	1	0.58		0	6.16e$-$02, 6.16e$-$02	1.79e$-$07	0.93	2, 3
14	ex3.3_order17	None	0; 0; 171, 105	629				0	6.15e$-$02, 6.15e$-$02	2.84e$-$05	4.62
		pd1	0; 0; 25, 19	85	1	1.16	0.02	0	6.16e$-$02, 6.16e$-$02	1.89e$-$07	0.94	2, 3
		pd2	0; 0; 25, 19	85	1	5.37	0.02	0	6.16e$-$02, 6.16e$-$02	1.89e$-$07	0.94	2, 3
		dd1			0	0.06	0.01
		dd2			0	0.41	0.01
		Sieve-SDP	0; 0; 25, 19	85	1	0.73		0	6.16e$-$02, 6.16e$-$02	1.89e$-$07	0.92	2, 3
15	ex3.3_order18	None	0; 0; 190, 120	702				0	6.15e$-$02, 6.15e$-$02	6.51e$-$05	6.29
		pd1	0; 0; 26, 20	88	1	1.55	0.02	0	6.16e$-$02, 6.16e$-$02	1.98e$-$07	0.93	2, 3
		pd2	0; 0; 26, 20	88	1	7.78	0.02	0	6.16e$-$02, 6.16e$-$02	1.98e$-$07	0.58	2, 3
		dd1			0	0.05	0.01
		dd2			0	0.50	0.01
		Sieve-SDP	0; 0; 26, 20	88	1	0.94		0	6.16e$-$02, 6.16e$-$02	1.98e$-$07	0.61	2, 3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
16	ex3.3_order19	None	0; 0; 210, 136	779				0	6.15e$-$02, 6.15e$-$02	2.92e$-$04	10.01
		pd1	0; 0; 28, 22	97	1	2.18	0.02	0	6.16e$-$02, 6.16e$-$02	2.10e$-$07	1.22	2, 3
		pd2	0; 0; 28, 22	97	1	10.34	0.03	0	6.16e$-$02, 6.16e$-$02	2.10e$-$07	0.69	2, 3
		dd1			0	0.06	0.01
		dd2			0	0.60	0.01
		Sieve-SDP	0; 0; 28, 22	97	1	1.31		0	6.16e$-$02, 6.16e$-$02	2.10e$-$07	0.67	2, 3
17	ex3.3_order20	None	0; 0; 231, 153	860				0	6.15e$-$02, 6.15e$-$02	2.95e$-$04	21.36
		pd1	0; 0; 30, 24	104	1	2.68	0.03	0	6.16e$-$02, 6.16e$-$02	2.16e$-$07	1.10	2, 3
		pd2	0; 0; 30, 24	104	1	13.41	0.03	0	6.16e$-$02, 6.16e$-$02	2.16e$-$07	1.09	2, 3
		dd1			0	0.09	0.02
		dd2			0	0.86	0.02
		Sieve-SDP	0; 0; 30, 24	104	1	1.35		0	6.16e$-$02, 6.16e$-$02	2.16e$-$07	1.31	2, 3
18	ex4.1_order3	None	0; 1; 10	27				0	1.24e$-$09, 1.30e$-$09	8.29e$-$01	1.20
		pd1	0; 1; 1	2	1	0.07	0.00	1	1.18e$-$12, 4.80e$+$00	8.28e$-$01	0.79	1
		pd2	0; 1; 1	2	1	0.06	0.00	1	1.18e$-$12, 4.80e$+$00	8.28e$-$01	0.62	1
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP			inf	0.01					0.00	1
19	ex4.1_order4	None	0; 0; 15, 3	44				0	2.55e$-$09, 2.60e$-$09	8.29e$-$01	0.84
		pd1	0; 0; 1	1	1	0.10	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.35	1
		pd2	0; 0; 1	1	1	0.08	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.66	1
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			inf	0.02					0.00	1
20	ex4.1_order5	None	0; 0; 21, 6	65				0	3.08e$-$09, 3.12e$-$09	8.29e$-$01	0.94
		pd1	0; 0; 1	1	1	0.10	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.39	1
		pd2	0; 0; 1	1	1	0.11	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.34	1
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			inf	0.02					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
21	ex4.1_order6	None	0; 0; 28, 10	90				0	1.00e$+$00, 1.00e$+$00	6.31e$-$07	0.85
		pd1	0; 0; 1	1	1	0.14	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.39	1
		pd2	0; 0; 1	1	1	0.15	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.36	1
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			inf	0.02					0.00	1
22	ex4.1_order7	None	0; 0; 36, 15	119				0	1.00e$+$00, 1.00e$+$00	1.00e$-$06	0.61
		pd1	0; 0; 1	1	1	0.17	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.34	1
		pd2	0; 0; 1	1	1	0.21	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.61	1
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			inf	0.04					0.00	1
23	ex4.1_order8	None	0; 0; 45, 21	152				0	1.00e$+$00, 1.00e$+$00	1.42e$-$06	0.68
		pd1	0; 0; 1	1	1	0.19	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.39	1
		pd2	0; 0; 1	1	1	0.27	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.35	1
		dd1			0	0.02	0.00
		dd2			0	0.04	0.00
		Sieve-SDP			inf	0.05					0.00	1
24	ex4.1_order9	None	0; 0; 55, 28	189				0	1.00e$+$00, 1.00e$+$00	1.10e$-$06	0.87
		pd1	0; 0; 1	1	1	0.26	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.39	1
		pd2	0; 0; 1	1	1	0.45	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.39	1
		dd1			0	0.02	0.00
		dd2			0	0.05	0.00
		Sieve-SDP			inf	0.07					0.00	1
25	ex4.1_order10	None	0; 0; 66, 36	230				0	1.00e$+$00, 1.00e$+$00	9.94e$-$07	0.69
		pd1	0; 0; 1	1	1	0.32	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.39	1
		pd2	0; 0; 1	1	1	0.50	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.33	1
		dd1			0	0.02	0.00
		dd2			0	0.06	0.00
		Sieve-SDP			inf	0.09					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
26	ex4.1_order11	None	0; 0; 78, 45	275				0	1.00e$+$00, 1.00e$+$00	2.60e$-$06	0.93
		pd1	0; 0; 1	1	1	0.33	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.38	1
		pd2	0; 0; 1	1	1	0.71	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.34	1
		dd1			0	0.02	0.00
		dd2			0	0.09	0.00
		Sieve-SDP			inf	0.13					0.00	1
27	ex4.1_order12	None	0; 0; 91, 55	324				0	1.00e$+$00, 1.00e$+$00	2.29e$-$06	1.06
		pd1	0; 0; 1	1	1	0.50	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.34	1
		pd2	0; 0; 1	1	1	1.02	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.41	1
		dd1			0	0.02	0.00
		dd2			0	0.11	0.00
		Sieve-SDP			inf	0.16					0.00	1
28	ex4.1_order13	None	0; 0; 105, 66	377				0	1.00e$+$00, 1.00e$+$00	6.83e$-$06	1.00
		pd1	0; 0; 1	1	1	0.52	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.77	1
		pd2	0; 0; 1	1	1	1.55	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.37	1
		dd1			0	0.03	0.00
		dd2			0	0.14	0.00
		Sieve-SDP			inf	0.19					0.00	1
29	ex4.1_order14	None	0; 0; 120, 78	434				0	1.00e$+$00, 1.00e$+$00	2.19e$-$06	1.59
		pd1	0; 0; 1	1	1	0.81	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.43	1
		pd2	0; 0; 1	1	1	2.20	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.38	1
		dd1			0	0.03	0.00
		dd2			0	0.20	0.00
		Sieve-SDP			inf	0.25					0.00	1
30	ex4.1_order15	None	0; 0; 136, 91	495				0	1.00e$+$00, 1.00e$+$00	4.32e$-$06	1.66
		pd1	0; 0; 1	1	1	0.87	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.36	1
		pd2	0; 0; 1	1	1	2.79	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.35	1
		dd1			0	0.04	0.00
		dd2			0	0.24	0.00
		Sieve-SDP			inf	0.29					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
31	ex4.1_order16	None	0; 0; 153, 105	560				0	1.00e$+$00, 1.00e$+$00	7.99e$-$07	2.41
		pd1	0; 0; 1	1	1	1.10	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.36	1
		pd2	0; 0; 1	1	1	4.06	0.00	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.53	1
		dd1			0	0.04	0.00
		dd2			0	0.31	0.00
		Sieve-SDP			inf	0.37					0.00	1
32	ex4.1_order17	None	0; 0; 171, 120	629				0	1.00e$+$00, 1.00e$+$00	1.45e$-$06	3.23
		pd1	0; 0; 1	1	1	1.27	0.01	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.34	1
		pd2	0; 0; 1	1	1	5.29	0.01	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.37	1
		dd1			0	0.05	0.01
		dd2			0	0.44	0.01
		Sieve-SDP			inf	0.47					0.00	1
33	ex4.1_order18	None	0; 0; 190, 136	702				0	1.00e$+$00, 1.00e$+$00	2.43e$-$06	4.52
		pd1	0; 0; 1	1	1	1.72	0.01	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.37	1
		pd2	0; 0; 1	1	1	6.99	0.01	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.46	1
		dd1			0	0.08	0.01
		dd2			0	0.52	0.01
		Sieve-SDP			inf	0.67					0.00	1
34	ex4.1_order19	None	0; 0; 210, 153	779				0	1.00e$+$00, 1.00e$+$00	2.76e$-$06	6.04
		pd1	0; 0; 1	1	1	2.01	0.01	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.37	1
		pd2	0; 0; 1	1	1	9.39	0.01	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.37	1
		dd1			0	0.06	0.01
		dd2			0	0.69	0.01
		Sieve-SDP			inf	0.87					0.00	1
35	ex4.1_order20	None	0; 0; 231, 171	860				0	1.00e$+$00, 1.00e$+$00	5.65e$-$06	8.86
		pd1	0; 0; 1	1	1	2.68	0.02	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.35	1
		pd2	0; 0; 1	1	1	12.71	0.02	1	0.00e$+$00, 3.00e$+$00	7.50e$-$01	0.34	1
		dd1			0	0.07	0.02
		dd2			0	0.87	0.02
		Sieve-SDP			inf	0.88					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
36	ex4.2_order4	None	0; 1; 15	44				0	1.00e$-$09, 1.01e$-$09	7.07e$-$01	0.81
		pd1	0; 1; 1	2	1	0.08	0.00	0	1.00e$+$00, 1.00e$+$00	5.00e$-$01	0.57	3
		pd2	0; 1; 1	2	1	0.09	0.00	0	1.00e$+$00, 1.00e$+$00	5.00e$-$01	0.61	3
		dd1			0	0.04	0.00
		dd2			0	0.05	0.00
		Sieve-SDP			inf	0.03					0.00	1
37	ex4.2_order5	None	0; 0; 21, 3	65				0	5.53e$-$01, 5.53e$-$01	9.19e$-$08	1.85
		pd1	0; 0; 1	1	1	0.15	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.65	1
		pd2	0; 0; 1	1	1	0.14	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.44	1
		dd1			0	0.02	0.00
		dd2			0	0.04	0.00
		Sieve-SDP			inf	0.03					0.00	1
38	ex4.2_order6	None	0; 0; 28, 6	90				0	5.53e$-$01, 5.53e$-$01	3.24e$-$07	0.65
		pd1	0; 0; 8, 2	22	1	0.09	0.00	0	5.53e$-$01, 5.53e$-$01	7.64e$-$09	0.63
		pd2	0; 0; 8, 2	22	1	0.14	0.00	0	5.53e$-$01, 5.53e$-$01	7.64e$-$09	0.62
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 8, 2	22	1	0.03		0	5.53e$-$01, 5.53e$-$01	7.64e$-$09	0.74
39	ex4.2_order7	None	0; 0; 36, 10	119				0	5.53e$-$01, 5.53e$-$01	2.87e$-$07	0.65
		pd1	0; 0; 9, 3	25	1	0.13	0.00	0	5.53e$-$01, 5.53e$-$01	4.89e$-$09	0.57
		pd2	0; 0; 9, 3	25	1	0.16	0.00	0	5.53e$-$01, 5.53e$-$01	4.89e$-$09	0.55
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 9, 3	25	1	0.04		0	5.53e$-$01, 5.53e$-$01	4.89e$-$09	0.68
40	ex4.2_order8	None	0; 0; 45, 15	152				0	5.53e$-$01, 5.53e$-$01	1.08e$-$06	0.64
		pd1	0; 0; 10, 4	28	1	0.17	0.00	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.58	2
		pd2	0; 0; 10, 4	28	1	0.23	0.00	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.57	2
		dd1			0	0.02	0.00
		dd2			0	0.04	0.00
		Sieve-SDP	0; 0; 10, 4	28	1	0.06		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.70	2

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
41	ex4.2_order9	None	0; 0; 55, 21	189				0	5.53e$-$01, 5.53e$-$01	1.10e$-$06	0.67
		pd1	0; 0; 10, 4	28	1	0.22	0.00	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.58	2
		pd2	0; 0; 10, 4	28	1	0.33	0.00	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.57	2
		dd1			0	0.02	0.00
		dd2			0	0.04	0.00
		Sieve-SDP	0; 0; 10, 4	28	1	0.08		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.71	2
42	ex4.2_order10	None	0; 0; 66, 28	230				0	5.53e$-$01, 5.53e$-$01	1.42e$-$06	0.74
		pd1	0; 0; 10, 4	28	1	0.39	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.71	2, 3
		pd2	0; 0; 10, 4	28	1	0.56	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.60	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.06	0.00
		Sieve-SDP	0; 0; 10, 4	28	1	0.13		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.76	2, 3
43	ex4.2_order11	None	0; 0; 78, 36	275				0	5.53e$-$01, 5.53e$-$01	1.61e$-$06	1.66
		pd1	0; 0; 10, 4	28	1	0.49	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.80	2, 3
		pd2	0; 0; 10, 4	28	1	0.82	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.73	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.10	0.00
		Sieve-SDP	0; 0; 10, 4	28	1	0.16		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.70	2, 3
44	ex4.2_order12	None	0; 0; 91, 45	324				0	5.53e$-$01, 5.53e$-$01	2.77e$-$06	1.23
		pd1	0; 0; 10, 4	28	1	0.51	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.68	2, 3
		pd2	0; 0; 10, 4	28	1	1.06	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.57	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.12	0.00
		Sieve-SDP	0; 0; 10, 4	28	1	0.19		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.70	2, 3
45	ex4.2_order13	None	0; 0; 105, 55	377				0	5.53e$-$01, 5.53e$-$01	4.58e$-$06	1.53
		pd1	0; 0; 10, 4	28	1	0.51	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.68	2, 3
		pd2	0; 0; 10, 4	28	1	1.54	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.62	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.15	0.00
		Sieve-SDP	0; 0; 10, 4	28	1	0.25		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.73	2, 3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
46	ex4.2_order14	None	0; 0; 120, 66	434				0	5.53e$-$01, 5.53e$-$01	1.61e$-$06	2.24
		pd1	0; 0; 10, 4	28	1	0.82	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.73	2, 3
		pd2	0; 0; 10, 4	28	1	2.36	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.76	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.26	0.00
		Sieve-SDP	0; 0; 10, 4	28	1	0.32		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.61	2, 3
47	ex4.2_order15	None	0; 0; 136, 78	495				0	5.53e$-$01, 5.53e$-$01	3.52e$-$06	2.90
		pd1	0; 0; 10, 4	28	1	1.08	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	1.50	2, 3
		pd2	0; 0; 10, 4	28	1	3.81	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.86	2, 3
		dd1			0	0.04	0.01
		dd2			0	0.34	0.01
		Sieve-SDP	0; 0; 10, 4	28	1	0.59		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.57	2, 3
48	ex4.2_order16	None	0; 0; 153, 91	560				0	5.53e$-$01, 5.53e$-$01	2.54e$-$06	4.55
		pd1	0; 0; 10, 4	28	1	1.50	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.67	2, 3
		pd2	0; 0; 10, 4	28	1	4.37	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.60	2, 3
		dd1			0	0.04	0.01
		dd2			0	0.37	0.01
		Sieve-SDP	0; 0; 10, 4	28	1	0.50		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.62	2, 3
49	ex4.2_order17	None	0; 0; 171, 105	629				0	5.52e$-$01, 5.52e$-$01	1.06e$-$04	4.96
		pd1	0; 0; 10, 4	28	1	1.52	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.57	2, 3
		pd2	0; 0; 10, 4	28	1	5.71	0.01	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.63	2, 3
		dd1			0	0.05	0.01
		dd2			0	0.50	0.01
		Sieve-SDP	0; 0; 10, 4	28	1	0.59		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.62	2, 3
50	ex4.2_order18	None	0; 0; 190, 120	702				0	5.52e$-$01, 5.52e$-$01	1.70e$-$04	9.65
		pd1	0; 0; 10, 4	28	1	2.01	0.02	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.62	2, 3
		pd2	0; 0; 10, 4	28	1	8.06	0.02	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.65	2, 3
		dd1			0	0.06	0.01
		dd2			0	0.65	0.01
		Sieve-SDP	0; 0; 10, 4	28	1	0.78		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.62	2, 3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
51	ex4.2_order19	None	0; 0; 210, 136	779				0	5.52e$-$01, 5.52e$-$01	9.26e$-$04	11.00
		pd1	0; 0; 10, 4	28	1	2.40	0.02	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.59	2, 3
		pd2	0; 0; 10, 4	28	1	9.99	0.02	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.60	2, 3
		dd1			0	0.06	0.02
		dd2			0	0.80	0.02
		Sieve-SDP	0; 0; 10, 4	28	1	0.93		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.75	2, 3
52	ex4.2_order20	None	0; 0; 231, 153	860				0	5.49e$-$01, 5.49e$-$01	4.36e$-$03	17.65
		pd1	0; 0; 10, 4	28	1	2.93	0.03	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.58	2, 3
		pd2	0; 0; 10, 4	28	1	12.77	0.03	0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.57	2, 3
		dd1			0	0.07	0.02
		dd2			0	0.94	0.02
		Sieve-SDP	0; 0; 10, 4	28	1	1.11		0	5.53e$-$01, 5.53e$-$01	3.90e$-$08	0.68	2, 3
53	ex4.3_order2	None	0; 1; 10	34				0	1.73e$-$07, 1.83e$-$07	9.09e$-$01	4.39
		pd1	0; 1; 4	10	1	0.14	0.03	1	0.00e$+$00, 8.00e$+$00	9.09e$-$01	1.24	1
		pd2	0; 1; 4	10	1	0.34	0.00	1	0.00e$+$00, 8.00e$+$00	9.09e$-$01	1.30	1
		dd1			0	0.08	0.00
		dd2			0	0.04	0.00
		Sieve-SDP			inf	0.52					0.00	1
54	ex4.3_order3	None	0; 0; 20, 4	83				0	6.22e$-$09, 6.48e$-$09	9.09e$-$01	3.78
		pd1	0; 0; 1	1	1	0.19	0.01	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.29	1
		pd2	0; 0; 1	1	1	0.13	0.00	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.28	1
		dd1			0	0.05	0.00
		dd2			0	0.04	0.00
		Sieve-SDP			inf	0.05					0.00	1
55	ex4.3_order4	None	0; 0; 35, 10	164				0	2.50e$-$09, 2.53e$-$09	9.09e$-$01	3.88
		pd1	0; 0; 1	1	1	0.11	0.00	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.31	1
		pd2	0; 0; 1	1	1	0.13	0.00	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.29	1
		dd1			0	0.03	0.00
		dd2			0	0.05	0.00
		Sieve-SDP			inf	0.04					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
56	ex4.3_order5	None	0; 0; 56, 20	285				0	5.18e$-$09, 5.22e$-$09	9.09e$-$01	3.30
		pd1	0; 0; 1	1	1	0.15	0.00	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.24	1
		pd2	0; 0; 1	1	1	0.20	0.00	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.22	1
		dd1			0	0.04	0.00
		dd2			0	0.08	0.00
		Sieve-SDP			inf	0.06					0.00	1
57	ex4.3_order6	None	0; 0; 84, 35	454				0	1.60e$+$01, 1.60e$+$01	3.34e$-$06	3.20
		pd1	0; 0; 1	1	1	0.21	0.00	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.16	1
		pd2	0; 0; 1	1	1	0.41	0.00	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.20	1
		dd1			0	0.05	0.00
		dd2			0	0.13	0.00
		Sieve-SDP			inf	0.10					0.00	1
58	ex4.3_order7	None	0; 0; 120, 56	679				0	1.60e$+$01, 1.60e$+$01	5.26e$-$06	4.45
		pd1	0; 0; 1	1	1	0.30	0.01	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.23	1
		pd2	0; 0; 1	1	1	0.83	0.01	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.19	1
		dd1			0	0.06	0.01
		dd2			0	0.23	0.01
		Sieve-SDP			inf	0.17					0.00	1
59	ex4.3_order8	None	0; 0; 165, 84	968				0	1.60e$+$01, 1.60e$+$01	5.17e$-$06	9.40
		pd1	0; 0; 1	1	1	0.48	0.01	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.65	1
		pd2	0; 0; 1	1	1	2.05	0.01	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.71	1
		dd1			0	0.05	0.01
		dd2			0	0.45	0.01
		Sieve-SDP			inf	0.27					0.00	1
60	ex4.3_order9	None	0; 0; 220, 120	1329				0	1.60e$+$01, 1.60e$+$01	5.88e$-$06	17.62
		pd1	0; 0; 1	1	1	0.85	0.02	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.56	1
		pd2	0; 0; 1	1	1	4.46	0.02	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.59	1
		dd1			0	0.07	0.02
		dd2			0	0.88	0.02
		Sieve-SDP			inf	0.53					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
61	ex4.3_order10	None	0; 0; 286, 165	1770				0	1.60e$+$01, 1.60e$+$01	3.92e$-$05	42.57
		pd1	0; 0; 1	1	1	1.66	0.04	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.35	1
		pd2	0; 0; 1	1	1	9.01	0.04	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.34	1
		dd1			0	0.09	0.04
		dd2			0	1.79	0.04
		Sieve-SDP			inf	0.87					0.00	1
62	ex4.3_order11	None	0; 0; 364, 220	2299				0	7.85e$-$06, 7.85e$-$06	9.09e$-$01	116.27
		pd1	0; 0; 1	1	1	3.12	0.07	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.34	1
		pd2	0; 0; 1	1	1	17.61	0.07	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.34	1
		dd1			0	0.16	0.07
		dd2			0	18.38	0.07
		Sieve-SDP			inf	1.51					0.00	1
63	ex4.3_order12	None	0; 0; 455, 286	2924				0	2.62e$-$06, 2.62e$-$06	9.09e$-$01	330.94
		pd1	0; 0; 1	1	1	5.71	0.11	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.52	1
		pd2	0; 0; 1	1	1	34.17	0.11	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.36	1
		dd1			0	0.25	0.11
		dd2			0	39.81	0.11
		Sieve-SDP			inf	2.90					0.00	1
64	ex4.3_order13	None	0; 0; 560, 364	3653				0	4.85e$-$07, 4.85e$-$07	9.09e$-$01	814.60
		pd1	0; 0; 1	1	1	10.54	0.18	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.34	1
		pd2	0; 0; 1	1	1	61.06	0.18	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.35	1
		dd1			0	0.40	0.18
		dd2			0	74.87	0.18
		Sieve-SDP			inf	5.29					0.00	1
65	ex4.3_order14	None	0; 0; 680, 455	4494				0	1.01e$+$01, 1.01e$+$01	9.40e$-$02	1178.45
		pd1	0; 0; 1	1	1	17.38	0.27	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.34	1
		pd2	0; 0; 1	1	1	109.31	0.27	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.47	1
		dd1			0	0.63	0.27
		dd2			0	146.45	0.27
		Sieve-SDP			inf	9.73					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
66	ex4.3_order15	None	0; 0; 816, 560	5455				0	8.94e$+$00, 8.94e$+$00	1.76e$-$01	2010.36
		pd1	0; 0; 1	1	1	33.32	0.41	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.50	1
		pd2	0; 0; 1	1	1	192.22	0.41	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.70	1
		dd1			0	0.99	0.41
		dd2			0	303.20	0.41
		Sieve-SDP			inf	16.20					0.00	1
67	ex4.3_order16	None	0; 0; 969, 680	6544				0	7.95e$+$00, 7.95e$+$00	2.23e$-$01	3158.88
		pd1	0; 0; 1	1	1	46.39	0.55	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.43	1
		pd2	0; 0; 1	1	1	295.70	0.55	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.33	1
		dd1			0	1.49	0.55
		dd2			0	485.30	0.55
		Sieve-SDP			inf	29.21					0.00	1
68	ex4.3_order17	None	0; 0; 1140, 816	7769				0	7.45e$+$00, 7.45e$+$00	2.00e$-$01	5618.65
		pd1	0; 0; 1	1	1	71.92	0.81	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.34	1
		pd2	0; 0; 1	1	1	472.15	0.81	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.33	1
		dd1			0	2.13	0.81
		dd2			0	949.49	0.81
		Sieve-SDP			inf	49.63					0.00	1
69	ex4.3_order18	None	0; 0; 1330, 969	9138				0	7.16e$+$00, 7.16e$+$00	2.14e$-$01	11769.31
		pd1	0; 0; 1	1	1	112.46	1.13	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.40	1
		pd2	0; 0; 1	1	1	753.21	1.13	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.47	1
		dd1			0	3.05	1.13
		dd2			0	1624.34	1.13
		Sieve-SDP			inf	81.60					0.00	1
70	ex4.3_order19	None	0; 0; 1540, 1140	10659				0	6.82e$+$00, 6.82e$+$00	2.63e$-$01	22830.51
		pd1	0; 0; 1	1	1	171.19	1.62	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.26	1
		pd2	0; 0; 1	1	1	1177.71	1.60	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.23	1
		dd1			0	4.53	1.60
		dd2			0	2852.27	1.60
		Sieve-SDP			inf	134.13					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
71	ex4.3_order20	None	0; 0; 1771, 1330	12340				0	6.52e$+$00, 6.52e$+$00	3.66e$-$01	38786.88
		pd1	0; 0; 1	1	1	375.53	2.81	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	1.09	1
		pd2	0; 0; 1	1	1	2479.13	2.79	1	0.00e$+$00, 8.00e$+$00	8.89e$-$01	0.64	1
		dd1			0	6.68	2.79
		dd2			0	6408.87	2.79
		Sieve-SDP			inf	260.68					0.00	1
72	ex4.4_order3	None	0; 0; 20, 10	83				1	9.88e$-$02, 1.18e$-$01	8.66e$-$01	1.72
		pd1	0; 0; 1	1	1	0.09	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.60
		pd2	0; 0; 1	1	1	0.10	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.53
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP			inf	0.01					0.00	1
73	ex4.4_order4	None	0; 0; 35, 20	164				1	1.73e$-$05, 1.84e$-$05	8.66e$-$01	2.17
		pd1	0; 0; 1	1	1	0.12	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.54
		pd2	0; 0; 1	1	1	0.14	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.60
		dd1			0	0.02	0.00
		dd2			0	0.07	0.00
		Sieve-SDP			inf	0.03					0.00	1
74	ex4.4_order5	None	0; 0; 56, 35	285				0	5.66e$-$08, 5.85e$-$08	8.66e$-$01	2.11
		pd1	0; 0; 1	1	1	0.14	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.60	1
		pd2	0; 0; 1	1	1	0.25	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.59	1
		dd1			0	0.03	0.00
		dd2			0	0.14	0.00
		Sieve-SDP			inf	0.04					0.00	1
75	ex4.4_order6	None	0; 0; 84, 56	454				0	1.49e$-$08, 1.50e$-$08	8.66e$-$01	3.11
		pd1	0; 0; 1	1	1	0.20	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.53	1
		pd2	0; 0; 1	1	1	0.47	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.55	1
		dd1			0	0.03	0.00
		dd2			0	0.43	0.00
		Sieve-SDP			inf	0.08					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
76	ex4.4_order7	None	0; 0; 120, 84	679				0	6.64e$-$09, 6.68e$-$09	8.66e$-$01	4.44
		pd1	0; 0; 1	1	1	0.30	0.01	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.58	1
		pd2	0; 0; 1	1	1	0.96	0.01	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.57	1
		dd1			0	0.04	0.01
		dd2			0	1.12	0.01
		Sieve-SDP			inf	0.17					0.00	1
77	ex4.4_order8	None	0; 0; 165, 120	968				0	1.80e$-$09, 1.80e$-$09	8.66e$-$01	12.00
		pd1	0; 0; 1	1	1	0.61	0.01	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.61	1
		pd2	0; 0; 1	1	1	2.09	0.01	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.55	1
		dd1			0	0.05	0.01
		dd2			0	3.86	0.01
		Sieve-SDP			inf	0.32					0.00	1
78	ex4.4_order9	None	0; 0; 220, 165	1329				0	4.13e$-$10, 4.14e$-$10	8.66e$-$01	31.08
		pd1	0; 0; 1	1	1	1.18	0.03	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.80	1
		pd2	0; 0; 1	1	1	5.85	0.03	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.76	1
		dd1			0	0.12	0.03
		dd2			0	8.18	0.03
		Sieve-SDP			inf	0.53					0.00	1
79	ex4.4_order10	None	0; 0; 286, 220	1770				0	2.65e$-$10, 2.65e$-$10	8.66e$-$01	71.26
		pd1	0; 0; 1	1	1	2.24	0.07	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.69	1
		pd2	0; 0; 1	1	1	10.10	0.06	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.56	1
		dd1			0	0.12	0.06
		dd2			0	13.87	0.06
		Sieve-SDP			inf	0.89					0.00	1
80	ex4.4_order11	None	0; 0; 364, 286	2299				0	1.60e$-$05, 1.61e$-$05	4.44e$-$05	64.15
		pd1	0; 0; 1	1	1	4.21	0.10	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.62	1
		pd2	0; 0; 1	1	1	22.13	0.08	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.63	1
		dd1			0	0.20	0.08
		dd2			0	30.60	0.08
		Sieve-SDP			inf	1.77					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
81	ex4.4_order12	None	0; 0; 455, 364	2924				0	1.93e$-$07, 1.78e$-$07	2.52e$-$06	115.81
		pd1	0; 0; 1	1	1	5.82	0.13	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	1.05	1
		pd2	0; 0; 1	1	1	32.88	0.13	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	1.07	1
		dd1			0	0.31	0.12
		dd2			0	53.13	0.12
		Sieve-SDP			inf	2.99					0.00	1
82	ex4.4_order13	None	0; 0; 560, 455	3653				0	1.45e$-$08, 2.31e$-$09	1.10e$-$06	238.42
		pd1	0; 0; 1	1	1	9.81	0.19	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	1.18	1
		pd2	0; 0; 1	1	1	61.06	0.19	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	1.02	1
		dd1			0	0.49	0.19
		dd2			0	100.91	0.19
		Sieve-SDP			inf	5.35					0.00	1
83	ex4.4_order14	None	0; 0; 680, 560	4494				0	$-$ 1.06e$-$08, $-$ 3.67e$-$08	1.87e$-$06	455.22
		pd1	0; 0; 1	1	1	16.86	0.30	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.46	1
		pd2	0; 0; 1	1	1	107.61	0.30	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.45	1
		dd1			0	0.68	0.30
		dd2			0	169.13	0.30
		Sieve-SDP			inf	9.68					0.00	1
84	ex4.4_order15	None	0; 0; 816, 680	5455				0	$-$ 1.65e$-$08, $-$ 4.13e$-$08	1.79e$-$06	923.64
		pd1	0; 0; 1	1	1	28.95	0.45	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.50	1
		pd2	0; 0; 1	1	1	182.62	0.45	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.44	1
		dd1			0	1.13	0.45
		dd2			0	284.93	0.45
		Sieve-SDP			inf	17.79					0.00	1
85	ex4.4_order16	None	0; 0; 969, 816	6544				0	$-$ 2.21e$-$08, $-$ 4.65e$-$08	1.75e$-$06	1906.07
		pd1	0; 0; 1	1	1	45.76	0.63	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.53	1
		pd2	0; 0; 1	1	1	301.76	0.63	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.46	1
		dd1			0	1.70	0.63
		dd2			0	476.46	0.63
		Sieve-SDP			inf	30.87					0.00	1

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
86	ex4.4_order17	None	0; 0; 1140, 969	7769				0	$-$ 7.69e$-$09, $-$ 1.50e$-$08	5.48e$-$07	3654.29
		pd1	0; 0; 1	1	1	69.43	0.88	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	1.18	1
		pd2	0; 0; 1	1	1	474.62	0.89	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	1.22	1
		dd1			0	2.52	0.88
		dd2			0	823.92	0.88
		Sieve-SDP			inf	51.79					0.00	1
87	ex4.4_order18	None	0; 0; 1330, 1140	9138				0	$-$ 9.27e$-$09, $-$ 1.60e$-$08	4.84e$-$07	7063.56
		pd1	0; 0; 1	1	1	103.53	1.24	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	1.55	1
		pd2	0; 0; 1	1	1	723.31	1.23	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	1.69	1
		dd1			0	3.71	1.23
		dd2			0	1328.75	1.23
		Sieve-SDP			inf	86.64					0.00	1
88	ex4.4_order19	None	0; 0; 1540, 1330	10659				0	$-$ 2.12e$-$08, $-$ 3.68e$-$08	1.22e$-$06	12500.09
		pd1	0; 0; 1	1	1	150.44	1.70	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	1.47	1
		pd2	0; 0; 1	1	1	1128.87	1.70	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	1.64	1
		dd1			0	5.13	1.69
		dd2			0	2112.63	1.69
		Sieve-SDP			inf	140.48					0.00	1
89	ex4.4_order20	None	0; 0; 1771, 1540	12340				0	$-$ 3.07e$-$08, $-$ 5.42e$-$08	2.04e$-$06	25422.27
		pd1	0; 0; 1	1	1	249.05	2.70	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.61	1
		pd2	0; 0; 1	1	1	1889.89	2.69	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.63	1
		dd1			0	7.15	2.69
		dd2			0	3796.27	2.69
		Sieve-SDP			inf	256.35					0.00	1
90	ex5.4_order5	None	0; 0; 21	65				0	2.28e$+$00, 2.28e$+$00	5.89e$-$07	0.90
		pd1	0; 0; 10	31	1	0.07	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.86
		pd2	0; 0; 10	31	1	0.09	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.78
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP	0; 0; 10	31	1	0.02		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.77

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
91	ex5.4_order6	None	0; 0; 28	90				0	2.28e$+$00, 2.28e$+$00	1.89e$-$06	0.95
		pd1	0; 0; 10	31	1	0.09	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.78	2, 3
		pd2	0; 0; 10	31	1	0.11	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.77	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 10	31	1	0.03		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.84	2, 3
92	ex5.4_order7	None	0; 0; 36	119				0	2.28e$+$00, 2.28e$+$00	2.55e$-$06	0.98
		pd1	0; 0; 10	31	1	0.10	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.78	2, 3
		pd2	0; 0; 10	31	1	0.13	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.81	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 10	31	1	0.04		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.82	2, 3
93	ex5.4_order8	None	0; 0; 45	152				0	2.28e$+$00, 2.28e$+$00	2.86e$-$06	0.96
		pd1	0; 0; 10	31	1	0.11	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.77	2, 3
		pd2	0; 0; 10	31	1	0.17	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.78	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 10	31	1	0.04		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.85	2, 3
94	ex5.4_order9	None	0; 0; 55	189				0	2.28e$+$00, 2.28e$+$00	4.47e$-$06	0.95
		pd1	0; 0; 10	31	1	0.13	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.78	2, 3
		pd2	0; 0; 10	31	1	0.23	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.85	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.04	0.00
		Sieve-SDP	0; 0; 10	31	1	0.05		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.77	2, 3
95	ex5.4_order10	None	0; 0; 66	230				0	2.28e$+$00, 2.28e$+$00	2.08e$-$06	1.98
		pd1	0; 0; 10	31	1	0.21	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	1.59	2, 3
		pd2	0; 0; 10	31	1	0.33	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	1.62	2, 3
		dd1			0	0.05	0.00
		dd2			0	0.07	0.00
		Sieve-SDP	0; 0; 10	31	1	0.10		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	1.62	2, 3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
96	ex5.4_order11	None	0; 0; 78	275				0	2.28e$+$00, 2.28e$+$00	9.31e$-$06	3.15
		pd1	0; 0; 10	31	1	0.23	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	3.16	2, 3
		pd2	0; 0; 10	31	1	0.43	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	2.25	2, 3
		dd1			0	0.04	0.00
		dd2			0	0.09	0.00
		Sieve-SDP	0; 0; 10	31	1	0.12		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	1.91	2, 3
97	ex5.4_order12	None	0; 0; 91	324				0	2.28e$+$00, 2.28e$+$00	1.09e$-$05	2.52
		pd1	0; 0; 10	31	1	0.29	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	1.72	2, 3
		pd2	0; 0; 10	31	1	0.54	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.82	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.16	0.00
		Sieve-SDP	0; 0; 10	31	1	0.20		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	1.20	2, 3
98	ex5.4_order13	None	0; 0; 105	377				0	2.28e$+$00, 2.28e$+$00	4.58e$-$06	2.22
		pd1	0; 0; 10	31	1	0.42	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	1.13	2, 3
		pd2	0; 0; 10	31	1	0.76	0.00	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	1.22	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.10	0.00
		Sieve-SDP	0; 0; 10	31	1	0.13		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	1.23	2, 3
99	ex5.4_order14	None	0; 0; 120	434				0	2.28e$+$00, 2.28e$+$00	1.14e$-$05	1.78
		pd1	0; 0; 10	31	1	0.31	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.87	2, 3
		pd2	0; 0; 10	31	1	0.98	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.84	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.12	0.00
		Sieve-SDP	0; 0; 10	31	1	0.15		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.77	2, 3
100	ex5.4_order15	None	0; 0; 136	495				0	2.28e$+$00, 2.28e$+$00	6.12e$-$06	2.11
		pd1	0; 0; 10	31	1	0.43	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.79	2, 3
		pd2	0; 0; 10	31	1	1.34	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.75	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.16	0.00
		Sieve-SDP	0; 0; 10	31	1	0.18		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.74	2, 3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
101	ex5.4_order16	None	0; 0; 153	560				0	2.28e$+$00, 2.28e$+$00	6.71e$-$06	3.38
		pd1	0; 0; 10	31	1	0.53	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.83	2, 3
		pd2	0; 0; 10	31	1	1.84	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.85	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.20	0.00
		Sieve-SDP	0; 0; 10	31	1	0.23		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.80	2, 3
102	ex5.4_order17	None	0; 0; 171	629				0	2.28e$+$00, 2.28e$+$00	4.57e$-$06	4.32
		pd1	0; 0; 10	31	1	0.62	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.77	2, 3
		pd2	0; 0; 10	31	1	2.50	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.95	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.26	0.00
		Sieve-SDP	0; 0; 10	31	1	0.27		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	1.28	2, 3
103	ex5.4_order18	None	0; 0; 190	702				0	2.27e$+$00, 2.27e$+$00	3.25e$-$03	7.00
		pd1	0; 0; 10	31	1	0.93	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	1.14	2, 3
		pd2	0; 0; 10	31	1	4.02	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.81	2, 3
		dd1			0	0.08	0.01
		dd2			0	0.33	0.01
		Sieve-SDP	0; 0; 10	31	1	0.33		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.79	2, 3
104	ex5.4_order19	None	0; 0; 210	779				0	2.23e$+$00, 2.23e$+$00	2.09e$-$02	8.09
		pd1	0; 0; 10	31	1	0.98	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.77	2, 3
		pd2	0; 0; 10	31	1	4.38	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.77	2, 3
		dd1			0	0.04	0.00
		dd2			0	0.61	0.00
		Sieve-SDP	0; 0; 10	31	1	0.67		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	1.66	2, 3
105	ex5.4_order20	None	0; 0; 231	860				0	2.14e$+$00, 2.15e$+$00	5.08e$-$02	10.64
		pd1	0; 0; 10	31	1	1.19	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.75	2, 3
		pd2	0; 0; 10	31	1	6.78	0.01	0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.99	2, 3
		dd1			0	0.08	0.01
		dd2			0	0.65	0.01
		Sieve-SDP	0; 0; 10	31	1	0.48		0	2.28e$+$00, 2.28e$+$00	6.97e$-$08	0.75	2, 3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
106	ex5.5_order4	None	0; 0; 15	44				0	1.62e$-$01, 1.62e$-$01	2.66e$-$06	0.88
		pd1	0; 0; 7	20	1	0.05	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.81	2, 3
		pd2	0; 0; 7	20	1	0.07	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.76	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP	0; 0; 7	20	1	0.02		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.75	2, 3
107	ex5.5_order5	None	0; 0; 21	65				0	1.62e$-$01, 1.62e$-$01	1.73e$-$06	0.83
		pd1	0; 0; 7	20	1	0.07	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.75	2, 3
		pd2	0; 0; 7	20	1	0.08	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.75	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP	0; 0; 7	20	1	0.02		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.92	2, 3
108	ex5.5_order6	None	0; 0; 28	90				0	1.62e$-$01, 1.62e$-$01	2.59e$-$06	0.88
		pd1	0; 0; 7	20	1	0.08	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.75	2, 3
		pd2	0; 0; 7	20	1	0.10	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.76	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 7	20	1	0.02		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.75	2, 3
109	ex5.5_order7	None	0; 0; 36	119				0	1.62e$-$01, 1.62e$-$01	4.17e$-$06	0.90
		pd1	0; 0; 7	20	1	0.09	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.75	2, 3
		pd2	0; 0; 7	20	1	0.13	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.75	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 7	20	1	0.03		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.76	2, 3
110	ex5.5_order8	None	0; 0; 45	152				0	1.62e$-$01, 1.62e$-$01	1.97e$-$06	0.96
		pd1	0; 0; 7	20	1	0.12	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.76	2, 3
		pd2	0; 0; 7	20	1	0.18	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.77	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 7	20	1	0.04		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.75	2, 3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
111	ex5.5_order9	None	0; 0; 55	189				0	1.62e$-$01, 1.62e$-$01	1.33e$-$06	0.92
		pd1	0; 0; 7	20	1	0.14	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.77	2, 3
		pd2	0; 0; 7	20	1	0.22	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.76	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.04	0.00
		Sieve-SDP	0; 0; 7	20	1	0.05		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.74	2, 3
112	ex5.5_order10	None	0; 0; 66	230				0	1.62e$-$01, 1.62e$-$01	1.21e$-$05	0.94
		pd1	0; 0; 7	20	1	0.17	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.86	2, 3
		pd2	0; 0; 7	20	1	0.29	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.75	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.05	0.00
		Sieve-SDP	0; 0; 7	20	1	0.06		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.74	2, 3
113	ex5.5_order11	None	0; 0; 78	275				0	1.62e$-$01, 1.62e$-$01	5.86e$-$06	1.06
		pd1	0; 0; 7	20	1	0.19	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.92	2, 3
		pd2	0; 0; 7	20	1	0.39	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.81	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.06	0.00
		Sieve-SDP	0; 0; 7	20	1	0.08		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	1.13	2, 3
114	ex5.5_order12	None	0; 0; 91	324				0	1.62e$-$01, 1.62e$-$01	5.58e$-$06	1.49
		pd1	0; 0; 7	20	1	0.35	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	1.03	2, 3
		pd2	0; 0; 7	20	1	0.55	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.82	2, 3
		dd1			0	0.02	0.00
		dd2			0	0.08	0.00
		Sieve-SDP	0; 0; 7	20	1	0.09		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.85	2, 3
115	ex5.5_order13	None	0; 0; 105	377				0	1.62e$-$01, 1.62e$-$01	1.01e$-$05	1.24
		pd1	0; 0; 7	20	1	0.29	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.88	2, 3
		pd2	0; 0; 7	20	1	0.76	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	1.26	2, 3
		dd1			0	0.04	0.00
		dd2			0	0.17	0.00
		Sieve-SDP	0; 0; 7	20	1	0.15		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.77	2, 3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
116	ex5.5_order14	None	0; 0; 120	434				0	1.62e$-$01, 1.62e$-$01	2.12e$-$05	1.54
		pd1	0; 0; 7	20	1	0.34	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.83	2, 3
		pd2	0; 0; 7	20	1	1.15	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.75	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.12	0.00
		Sieve-SDP	0; 0; 7	20	1	0.15		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.96	2, 3
117	ex5.5_order15	None	0; 0; 136	495				0	1.62e$-$01, 1.62e$-$01	4.22e$-$05	1.85
		pd1	0; 0; 7	20	1	0.44	0.01	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.77	2, 3
		pd2	0; 0; 7	20	1	1.39	0.00	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.79	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.17	0.00
		Sieve-SDP	0; 0; 7	20	1	0.19		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.88	2, 3
118	ex5.5_order16	None	0; 0; 153	560				0	1.62e$-$01, 1.62e$-$01	4.38e$-$05	2.43
		pd1	0; 0; 7	20	1	0.52	0.01	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.76	2, 3
		pd2	0; 0; 7	20	1	1.93	0.01	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.76	2, 3
		dd1			0	0.03	0.01
		dd2			0	0.22	0.01
		Sieve-SDP	0; 0; 7	20	1	0.22		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.77	2, 3
119	ex5.5_order17	None	0; 0; 171	629				0	1.62e$-$01, 1.62e$-$01	6.83e$-$06	3.24
		pd1	0; 0; 7	20	1	0.63	0.01	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.74	2, 3
		pd2	0; 0; 7	20	1	2.52	0.01	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.75	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.26	0.00
		Sieve-SDP	0; 0; 7	20	1	0.28		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.79	2, 3
120	ex5.5_order18	None	0; 0; 190	702				0	1.62e$-$01, 1.62e$-$01	4.05e$-$06	4.51
		pd1	0; 0; 7	20	1	0.92	0.01	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.89	2, 3
		pd2	0; 0; 7	20	1	3.45	0.01	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.89	2, 3
		dd1			0	0.04	0.00
		dd2			0	0.34	0.00
		Sieve-SDP	0; 0; 7	20	1	0.33		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.80	2, 3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
121	ex5.5_order19	None	0; 0; 210	779				0	1.62e$-$01, 1.62e$-$01	4.64e$-$05	7.12
		pd1	0; 0; 7	20	1	1.41	0.01	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	1.23	2, 3
		pd2	0; 0; 7	20	1	5.53	0.01	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.82	2, 3
		dd1			0	0.04	0.01
		dd2			0	0.43	0.01
		Sieve-SDP	0; 0; 7	20	1	0.39		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.80	2, 3
122	ex5.5_order20	None	0; 0; 231	860				0	1.62e$-$01, 1.62e$-$01	6.34e$-$05	8.90
		pd1	0; 0; 7	20	1	1.22	0.01	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.76	2, 3
		pd2	0; 0; 7	20	1	5.88	0.01	0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.76	2, 3
		dd1			0	0.05	0.01
		dd2			0	0.52	0.01
		Sieve-SDP	0; 0; 7	20	1	0.46		0	3.07e$-$01, 3.07e$-$01	1.08e$-$07	0.76	2, 3
123	ex5.6_order4	None	0; 0; 15	44				0	3.04e$-$01, 3.04e$-$01	1.15e$-$07	1.01
		pd1	0; 0; 8	21	1	0.07	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.94
		pd2	0; 0; 8	21	1	0.08	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.84
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP	0; 0; 8	21	1	0.02		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	1.02
124	ex5.6_order5	None	0; 0; 21	65				0	3.04e$-$01, 3.04e$-$01	1.74e$-$07	1.10
		pd1	0; 0; 8	21	1	0.07	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.84
		pd2	0; 0; 8	21	1	0.07	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.90
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP	0; 0; 8	21	1	0.02		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.92
125	ex5.6_order6	None	0; 0; 28	90				0	3.04e$-$01, 3.04e$-$01	3.45e$-$07	0.89
		pd1	0; 0; 8	21	1	0.07	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.78
		pd2	0; 0; 8	21	1	0.09	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.86
		dd1			0	0.03	0.00
		dd2			0	0.07	0.00
		Sieve-SDP	0; 0; 8	21	1	0.05		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	2.02

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$
126	ex5.6_order7	None	0; 0; 36	119				0	3.04e$-$01, 3.04e$-$01	3.21e$-$07	1.17
		pd1	0; 0; 8	21	1	0.12	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.96
		pd2	0; 0; 8	21	1	0.17	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.85
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 8	21	1	0.03		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	1.39
127	ex5.6_order8	None	0; 0; 45	152				0	3.04e$-$01, 3.04e$-$01	4.51e$-$07	0.91
		pd1	0; 0; 8	21	1	0.12	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.79
		pd2	0; 0; 8	21	1	0.16	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.94
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 8	21	1	0.04		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.74
128	ex5.6_order9	None	0; 0; 55	189				0	3.04e$-$01, 3.04e$-$01	7.42e$-$07	0.93
		pd1	0; 0; 8	21	1	0.14	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.79
		pd2	0; 0; 8	21	1	0.24	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.82
		dd1			0	0.03	0.00
		dd2			0	0.05	0.00
		Sieve-SDP	0; 0; 8	21	1	0.06		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.82
129	ex5.6_order10	None	0; 0; 66	230				0	3.04e$-$01, 3.04e$-$01	7.57e$-$07	0.87
		pd1	0; 0; 8	21	1	0.16	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.77
		pd2	0; 0; 8	21	1	0.28	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.74
		dd1			0	0.02	0.00
		dd2			0	0.04	0.00
		Sieve-SDP	0; 0; 8	21	1	0.06		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.75
130	ex5.6_order11	None	0; 0; 78	275				0	3.04e$-$01, 3.04e$-$01	6.11e$-$07	0.90
		pd1	0; 0; 8	21	1	0.18	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.74
		pd2	0; 0; 8	21	1	0.40	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.76
		dd1			0	0.02	0.00
		dd2			0	0.06	0.00
		Sieve-SDP	0; 0; 8	21	1	0.08		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.76

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
131	ex5.6_order12	None	0; 0; 91	324				0	3.04e$-$01, 3.04e$-$01	9.01e$-$07	1.02
		pd1	0; 0; 8	21	1	0.23	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.75
		pd2	0; 0; 8	21	1	0.54	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.75
		dd1			0	0.02	0.00
		dd2			0	0.08	0.00
		Sieve-SDP	0; 0; 8	21	1	0.10		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.74
132	ex5.6_order13	None	0; 0; 105	377				0	3.04e$-$01, 3.04e$-$01	7.94e$-$07	1.20
		pd1	0; 0; 8	21	1	0.28	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.77
		pd2	0; 0; 8	21	1	0.71	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.84
		dd1			0	0.03	0.00
		dd2			0	0.11	0.00
		Sieve-SDP	0; 0; 8	21	1	0.13		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.79
133	ex5.6_order14	None	0; 0; 120	434				0	3.04e$-$01, 3.04e$-$01	1.19e$-$06	1.31
		pd1	0; 0; 8	21	1	0.34	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.82	2
		pd2	0; 0; 8	21	1	1.01	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.74	2
		dd1			0	0.02	0.00
		dd2			0	0.13	0.00
		Sieve-SDP	0; 0; 8	21	1	0.15		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.79	2
134	ex5.6_order15	None	0; 0; 136	495				0	3.04e$-$01, 3.04e$-$01	8.76e$-$07	1.62
		pd1	0; 0; 8	21	1	0.43	0.00	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	1.27
		pd2	0; 0; 8	21	1	1.45	0.01	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	1.21
		dd1			0	0.04	0.00
		dd2			0	0.16	0.00
		Sieve-SDP	0; 0; 8	21	1	0.19		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.76
135	ex5.6_order16	None	0; 0; 153	560				0	3.04e$-$01, 3.04e$-$01	1.79e$-$06	1.81
		pd1	0; 0; 8	21	1	0.52	0.01	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.75	2
		pd2	0; 0; 8	21	1	1.91	0.01	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.74	2
		dd1			0	0.03	0.00
		dd2			0	0.21	0.00
		Sieve-SDP	0; 0; 8	21	1	0.22		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.76	2

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
136	ex5.6_order17	None	0; 0; 171	629				0	3.04e$-$01, 3.04e$-$01	1.78e$-$06	2.29
		pd1	0; 0; 8	21	1	0.60	0.01	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.83	2
		pd2	0; 0; 8	21	1	2.47	0.01	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.84	2
		dd1			0	0.03	0.00
		dd2			0	0.26	0.00
		Sieve-SDP	0; 0; 8	21	1	0.28		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.74	2
137	ex5.6_order18	None	0; 0; 190	702				0	3.04e$-$01, 3.04e$-$01	2.53e$-$06	2.68
		pd1	0; 0; 8	21	1	0.75	0.01	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.76	2
		pd2	0; 0; 8	21	1	3.24	0.01	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.83	2
		dd1			0	0.04	0.00
		dd2			0	0.31	0.00
		Sieve-SDP	0; 0; 8	21	1	0.33		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.74	2
138	ex5.6_order19	None	0; 0; 210	779				0	3.04e$-$01, 3.04e$-$01	2.54e$-$06	3.34
		pd1	0; 0; 8	21	1	1.06	0.01	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	1.52	2
		pd2	0; 0; 8	21	1	4.87	0.01	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.89	2
		dd1			0	0.05	0.00
		dd2			0	0.45	0.00
		Sieve-SDP	0; 0; 8	21	1	0.50		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.92	2
139	ex5.6_order20	None	0; 0; 231	860				0	3.04e$-$01, 3.04e$-$01	3.35e$-$06	5.55
		pd1	0; 0; 8	21	1	1.48	0.01	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	1.02	2
		pd2	0; 0; 8	21	1	6.57	0.01	0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	0.91	2
		dd1			0	0.07	0.01
		dd2			0	0.58	0.01
		Sieve-SDP	0; 0; 8	21	1	0.53		0	3.04e$-$01, 3.04e$-$01	1.01e$-$08	1.02	2
140	ex5.7_order5	None	0; 1; 21	65				0	8.52e$-$09, $-$ 5.99e$-$09	7.10e$-$08	0.93
		pd1	0; 1; 8	22	1	0.08	0.00	0	1.81e$-$08, $-$ 2.87e$-$09	5.26e$-$08	1.20
		pd2	0; 1; 8	22	1	0.14	0.00	0	1.81e$-$08, $-$ 2.87e$-$09	5.26e$-$08	1.12
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP	0; 1; 8	22	1	0.03		0	1.81e$-$08, $-$ 2.87e$-$09	5.26e$-$08	1.17

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$
141	ex5.7_order6	None	0; 0; 28, 3	90				0	2.28e$-$08, $-$ 2.32e$-$09	7.80e$-$08	0.97
		pd1	0; 0; 10, 2	28	1	0.09	0.00	0	7.23e$-$09, $-$ 4.48e$-$09	2.43e$-$08	1.07
		pd2	0; 0; 10, 2	28	1	0.18	0.00	0	7.23e$-$09, $-$ 4.48e$-$09	2.43e$-$08	0.93
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 10, 2	28	1	0.03		0	7.23e$-$09, $-$ 4.48e$-$09	2.43e$-$08	1.02
142	ex5.7_order7	None	0; 0; 36, 6	119				0	2.01e$-$08, $-$ 9.32e$-$09	1.19e$-$07	0.89
		pd1	0; 0; 11, 3	33	1	0.14	0.00	0	7.70e$-$09, $-$ 4.95e$-$09	2.93e$-$08	0.89
		pd2	0; 0; 11, 3	33	1	0.18	0.00	0	7.70e$-$09, $-$ 4.95e$-$09	2.93e$-$08	0.91
		dd1			0	0.02	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 11, 3	33	1	0.04		0	7.70e$-$09, $-$ 4.95e$-$09	2.93e$-$08	0.95
143	ex5.7_order8	None	0; 0; 45, 10	152				0	1.43e$-$08, $-$ 1.11e$-$08	1.17e$-$07	0.93
		pd1	0; 0; 12, 4	36	1	0.16	0.00	0	2.24e$-$08, $-$ 2.72e$-$09	7.97e$-$08	0.91
		pd2	0; 0; 12, 4	36	1	0.26	0.00	0	2.24e$-$08, $-$ 2.72e$-$09	7.97e$-$08	1.02
		dd1			0	0.02	0.00
		dd2			0	0.04	0.00
		Sieve-SDP	0; 0; 12, 4	36	1	0.07		0	2.24e$-$08, $-$ 2.72e$-$09	7.97e$-$08	1.00
144	ex5.7_order9	None	0; 0; 55, 15	189				0	6.00e$-$09, $-$ 4.83e$-$09	5.61e$-$08	1.24
		pd1	0; 0; 13, 5	41	1	0.24	0.01	0	2.15e$-$08, $-$ 3.69e$-$09	8.53e$-$08	1.19
		pd2	0; 0; 13, 5	41	1	0.39	0.01	0	2.15e$-$08, $-$ 3.69e$-$09	8.53e$-$08	0.89
		dd1			0	0.02	0.00
		dd2			0	0.05	0.00
		Sieve-SDP	0; 0; 13, 5	41	1	0.10		0	2.15e$-$08, $-$ 3.69e$-$09	8.53e$-$08	0.88
145	ex5.7_order10	None	0; 0; 66, 21	230				0	1.30e$-$08, $-$ 1.43e$-$08	1.69e$-$07	0.91
		pd1	0; 0; 13, 5	41	1	0.31	0.01	0	2.15e$-$08, $-$ 3.69e$-$09	8.53e$-$08	0.95
		pd2	0; 0; 13, 5	41	1	0.53	0.01	0	2.15e$-$08, $-$ 3.69e$-$09	8.53e$-$08	1.58
		dd1			0	0.03	0.00
		dd2			0	0.07	0.00
		Sieve-SDP	0; 0; 13, 5	41	1	0.15		0	2.15e$-$08, $-$ 3.69e$-$09	8.53e$-$08	0.94

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$
146	ex5.7_order11	None	0; 0; 78, 28	275				0	1.71e$-$08, $-$ 1.50e$-$08	2.11e$-$07	1.11
		pd1	0; 0; 14, 6	45	1	0.36	0.01	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.91
		pd2	0; 0; 14, 6	45	1	0.75	0.01	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.93
		dd1			0	0.02	0.00
		dd2			0	0.07	0.00
		Sieve-SDP	0; 0; 14, 6	45	1	0.17		0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.95
147	ex5.7_order12	None	0; 0; 91, 36	324				0	5.85e$-$09, $-$ 6.21e$-$09	7.87e$-$08	1.04
		pd1	0; 0; 14, 6	45	1	0.44	0.01	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.91
		pd2	0; 0; 14, 6	45	1	1.04	0.01	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.03
		dd1			0	0.03	0.00
		dd2			0	0.12	0.00
		Sieve-SDP	0; 0; 14, 6	45	1	0.24		0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.50
148	ex5.7_order13	None	0; 0; 105, 45	377				0	2.13e$-$08, $-$ 2.19e$-$08	3.23e$-$07	1.15
		pd1	0; 0; 14, 6	45	1	0.49	0.01	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.90
		pd2	0; 0; 14, 6	45	1	1.56	0.01	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.95
		dd1			0	0.03	0.00
		dd2			0	0.14	0.00
		Sieve-SDP	0; 0; 14, 6	45	1	0.31		0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.05
149	ex5.7_order14	None	0; 0; 120, 55	434				0	2.24e$-$08, $-$ 2.28e$-$08	3.72e$-$07	1.69
		pd1	0; 0; 14, 6	45	1	0.77	0.01	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.07
		pd2	0; 0; 14, 6	45	1	2.39	0.01	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.94
		dd1			0	0.03	0.01
		dd2			0	0.22	0.01
		Sieve-SDP	0; 0; 14, 6	45	1	0.37		0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.88
150	ex5.7_order15	None	0; 0; 136, 66	495				0	2.09e$-$08, $-$ 2.22e$-$08	3.87e$-$07	1.56
		pd1	0; 0; 14, 6	45	1	0.82	0.01	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.84
		pd2	0; 0; 14, 6	45	1	2.80	0.01	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.86
		dd1			0	0.03	0.01
		dd2			0	0.26	0.01
		Sieve-SDP	0; 0; 14, 6	45	1	0.45		0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.38

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$
151	ex5.7_order16	None	0; 0; 153, 78	560				0	1.87e$-$08, $-$ 1.91e$-$08	3.70e$-$07	2.13
		pd1	0; 0; 14, 6	45	1	1.13	0.01	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.97
		pd2	0; 0; 14, 6	45	1	3.91	0.01	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.41
		dd1			0	0.04	0.01
		dd2			0	0.34	0.01
		Sieve-SDP	0; 0; 14, 6	45	1	0.60		0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.43
152	ex5.7_order17	None	0; 0; 171, 91	629				0	1.84e$-$08, $-$ 1.83e$-$08	3.89e$-$07	2.56
		pd1	0; 0; 14, 6	45	1	1.47	0.02	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.92
		pd2	0; 0; 14, 6	45	1	5.58	0.02	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.13
		dd1			0	0.05	0.01
		dd2			0	0.46	0.01
		Sieve-SDP	0; 0; 14, 6	45	1	0.95		0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.47
153	ex5.7_order18	None	0; 0; 190, 105	702				0	1.76e$-$08, $-$ 1.75e$-$08	4.12e$-$07	3.01
		pd1	0; 0; 14, 6	45	1	1.76	0.02	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.84
		pd2	0; 0; 14, 6	45	1	7.46	0.02	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.15
		dd1			0	0.07	0.01
		dd2			0	0.58	0.01
		Sieve-SDP	0; 0; 14, 6	45	1	0.94		0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.36
154	ex5.7_order19	None	0; 0; 210, 120	779				0	1.68e$-$08, $-$ 1.67e$-$08	4.25e$-$07	4.46
		pd1	0; 0; 14, 6	45	1	2.49	0.02	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.09
		pd2	0; 0; 14, 6	45	1	10.47	0.02	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.89
		dd1			0	0.05	0.02
		dd2			0	0.84	0.02
		Sieve-SDP	0; 0; 14, 6	45	1	1.11		0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.01
155	ex5.7_order20	None	0; 0; 231, 136	860				0	1.54e$-$08, $-$ 1.69e$-$08	4.40e$-$07	5.84
		pd1	0; 0; 14, 6	45	1	3.28	0.03	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.02
		pd2	0; 0; 14, 6	45	1	13.67	0.03	0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	1.17
		dd1			0	0.08	0.02
		dd2			0	1.04	0.02
		Sieve-SDP	0; 0; 14, 6	45	1	1.71		0	1.73e$-$08, $-$ 3.62e$-$09	8.09e$-$08	0.86

1.4 A.4 Detailed results on the Henrion-Toh dataset

This dataset has 98 problems. From these 18 problems were reduced by at least one of the five methods.

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
1	sedumi-brown	None	925; 0; 56	461				0	$-$ 7.34e$-$09, 0.00e$+$00	3.75e$-$07	0.93
		pd1	925; 0; 21	251	1	0.15	0.05	0	$-$ 9.33e$-$11, 0.00e$+$00	6.25e$-$09	0.80
		pd2	925; 0; 21	251	1	0.23	0.02	0	$-$ 9.33e$-$11, 0.00e$+$00	6.25e$-$09	0.78
		dd1			0	0.03	0.00
		dd2			0	0.06	0.00
		Sieve-SDP	925; 0; 21	251	1	0.04		0	$-$ 9.33e$-$11, 0.00e$+$00	6.25e$-$09	0.79
2	sedumi-conform3	None	630; 0; 56	285				0	2.05e$-$08, 0.00e$+$00	4.54e$-$07	0.66
		pd1	630; 0; 53	273	1	0.04	0.02	0	2.51e$-$08, 0.00e$+$00	4.90e$-$07	0.71
		pd2	630; 0; 53	273	1	0.11	0.03	0	2.51e$-$08, 0.00e$+$00	4.90e$-$07	0.73
		dd1			0	0.03	0.00
		dd2			0	0.06	0.00
		Sieve-SDP	630; 0; 53	273	1	0.02		0	2.51e$-$08, 0.00e$+$00	4.90e$-$07	0.68
3	sedumi-conform4	None	1890; 0; 84	454				0	$-$ 2.51e$-$08, 0.00e$+$00	5.57e$-$06	0.84
		pd1	1890; 0; 81	442	1	0.07	0.04	0	$-$ 6.47e$-$09, 0.00e$+$00	1.74e$-$06	0.81
		pd2	1890; 0; 81	442	1	0.26	0.04	0	$-$ 6.47e$-$09, 0.00e$+$00	1.74e$-$06	0.88
		dd1			0	0.02	0.00
		dd2			0	0.08	0.00
		Sieve-SDP	1890; 0; 81	442	1	0.02		0	$-$ 6.47e$-$09, 0.00e$+$00	1.74e$-$06	0.77
4	sedumi-fp23	None	$0;0;28,{7}_{13}$	209				0	2.13e$+$02, 2.13e$+$02	3.97e$-$06	1.50
		pd1	$0;0;{7}_{14}$	83	1	0.09	0.02	0	2.13e$+$02, 2.13e$+$02	9.96e$-$07	1.60	3
		pd2	$0;0;{7}_{14}$	83	1	0.13	0.02	0	2.13e$+$02, 2.13e$+$02	9.96e$-$07	1.44	3
		dd1			0	0.05	0.00
		dd2			0	0.06	0.00
		Sieve-SDP	$0;0;{7}_{14}$	83	1	0.04		0	2.13e$+$02, 2.13e$+$02	9.96e$-$07	1.33	3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
5	sedumi-fp24	None	$0;0;105,{14}_{35}$	2379				0	1.95e$+$02, 1.95e$+$02	9.68e$-$08	6.36
		pd1	$0;0;{14}_{36}$	559	1	0.32	0.21	0	1.95e$+$02, 1.95e$+$02	1.74e$-$10	1.77
		pd2	$0;0;{14}_{36}$	559	1	0.82	0.20	0	1.95e$+$02, 1.95e$+$02	1.74e$-$10	1.90
		dd1			0	0.06	0.00
		dd2			0	0.25	0.00
		Sieve-SDP	$0;0;{14}_{36}$	559	1	0.23		0	1.95e$+$02, 1.95e$+$02	1.74e$-$10	1.78
6	sedumi-fp25	None	$0;0;28,{7}_{15}$	209				0	1.10e$+$01, 1.10e$+$01	6.63e$-$06	1.39
		pd1	$0;0;{7}_{16}$	83	1	0.12	0.03	0	1.10e$+$01, 1.10e$+$01	1.39e$-$07	1.46	2, 3
		pd2	$0;0;{7}_{16}$	83	1	0.17	0.05	0	1.10e$+$01, 1.10e$+$01	1.39e$-$07	1.28	2, 3
		dd1			0	0.05	0.00
		dd2			0	0.07	0.00
		Sieve-SDP	$0;0;{7}_{16}$	83	1	0.03		0	1.10e$+$01, 1.10e$+$01	1.39e$-$07	1.30	2, 3
7	sedumi-fp26	None	$0;0;66,{11}_{31}$	1000				0	2.68e$+$02, 2.68e$+$02	3.74e$-$08	2.11
		pd1	$0;0;{11}_{32}$	285	1	0.17	0.16	0	2.68e$+$02, 2.68e$+$02	1.18e$-$07	1.46
		pd2	$0;0;{11}_{32}$	285	1	0.53	0.15	0	2.68e$+$02, 2.68e$+$02	1.18e$-$07	1.48
		dd1			0	0.08	0.00
		dd2			0	0.46	0.00
		Sieve-SDP	$0;0;{11}_{32}$	285	1	0.11		0	2.68e$+$02, 2.68e$+$02	1.18e$-$07	1.53
8	sedumi-fp27	None	$0;0;66,{11}_{25}$	1000				0	3.90e$+$01, 3.90e$+$01	1.96e$-$10	2.50
		pd1	$0;0;{11}_{26}$	285	1	0.16	0.10	0	3.90e$+$01, 3.90e$+$01	3.98e$-$09	1.50
		pd2	$0;0;{11}_{26}$	285	1	0.39	0.11	0	3.90e$+$01, 3.90e$+$01	3.98e$-$09	1.45
		dd1			0	0.05	0.01
		dd2			0	0.30	0.01
		Sieve-SDP	$0;0;{11}_{26}$	285	1	0.11		0	3.90e$+$01, 3.90e$+$01	3.98e$-$09	1.46
9	sedumi-fp32	None	$0;0;165,{45}_{22}$	3002				0	$-$ 7.05e$+$00, $-$ 7.05e$+$00	2.79e$-$07	47.43
		pd1	$0;0;{45}_{4},{9}_{3},{45}_{16}$	1286	1	2.17	0.56	0	$-$ 7.05e$+$00, $-$ 7.05e$+$00	2.50e$-$06	8.58	3
		pd2	$0;0;{45}_{4},{9}_{3},{45}_{16}$	1286	1	4.55	0.57	0	$-$ 7.05e$+$00, $-$ 7.05e$+$00	2.50e$-$06	9.26	3
		dd1			0	0.11	0.02
		dd2			0	9.78	0.02
		Sieve-SDP	$0;0;{45}_{4},{9}_{3},{45}_{16}$	1286	1	1.21		0	$-$ 7.05e$+$00, $-$ 7.05e$+$00	2.50e$-$06	12.14	3

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
10	sedumi-fp33	None	$0;0;21,{6}_{16}$	125				0	$-$ 1.01e$+$04, $-$ 1.01e$+$04	3.36e$-$07	0.75
		pd1	$0;0;13,{6}_{16}$	105	1	0.10	0.04	0	$-$ 1.01e$+$04, $-$ 1.01e$+$04	3.01e$-$07	0.94
		pd2	$0;0;13,{6}_{16}$	105	1	0.20	0.03	0	$-$ 1.01e$+$04, $-$ 1.01e$+$04	3.01e$-$07	1.32
		dd1			0	0.04	0.00
		dd2			0	0.12	0.00
		Sieve-SDP	$0;0;14,{6}_{16}$	111	1	0.03		0	$-$ 1.18e$+$04, $-$ 1.18e$+$04	9.28e$-$02	1.05	$-$ 2
11	sedumi-fp34	None	$0;0;28,{7}_{16}$	209				0	1.72e$+$02, 1.72e$+$02	8.10e$-$07	0.94
		pd1	$0;0;7,{1}_{2},{7}_{14}$	83	1	0.14	0.04	0	1.72e$+$02, 1.72e$+$02	3.11e$-$07	0.88
		pd2	$0;0;7,{1}_{2},{7}_{14}$	83	1	0.11	0.03	0	1.72e$+$02, 1.72e$+$02	3.11e$-$07	0.73
		dd1			0	0.03	0.00
		dd2			0	0.07	0.00
		Sieve-SDP	$0;0;7,{1}_{2},{7}_{14}$	83	1	0.02		0	1.72e$+$02, 1.72e$+$02	3.11e$-$07	0.78
12	sedumi-fp35	None	$0;0;35,{20}_{8}$	164				0	4.00e$+$00, 4.00e$+$00	5.76e$-$06	0.86
		pd1	$0;0;{20}_{8},10$	119	1	0.16	0.04	0	4.00e$+$00, 4.00e$+$00	5.66e$-$07	0.80	2, 3
		pd2	$0;0;{20}_{8},10$	119	1	0.36	0.05	0	4.00e$+$00, 4.00e$+$00	5.66e$-$07	0.85	2, 3
		dd1			0	0.03	0.00
		dd2			0	0.28	0.00
		Sieve-SDP	$0;0;{20}_{8},10$	119	1	0.05		0	4.00e$+$00, 4.00e$+$00	5.66e$-$07	0.89	2, 3
13	sedumi-fp44	None	$0;0;4,{3}_{2}$	6				0	4.44e$+$02, 4.44e$+$02	4.67e$-$08	1.02
		pd1	$0;0;{3}_{3}$	5	1	0.04	0.00	0	4.44e$+$02, 4.44e$+$02	1.55e$-$08	0.87
		pd2	$0;0;{3}_{3}$	5	1	0.04	0.00	0	4.44e$+$02, 4.44e$+$02	1.55e$-$08	0.82
		dd1			0	0.03	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	$0;0;{3}_{3}$	5	1	0.01		0	4.44e$+$02, 4.44e$+$02	1.55e$-$08	0.77
14	sedumi-fp46	None	$0;0;10,{6}_{2}$	27				0	6.70e$-$08, $-$ 2.54e$-$07	4.78e$-$07	0.89
		pd1	0; 0; 5, 3, 2	11	1	0.13	0.00	0	1.54e$-$07, $-$ 2.20e$-$08	2.22e$-$07	0.76
		pd2	0; 0; 5, 3, 2	11	1	0.14	0.00	0	1.54e$-$07, $-$ 2.20e$-$08	2.22e$-$07	0.69
		dd1			0	0.03	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	0; 0; 5, 3, 2	11	1	0.01		0	1.54e$-$07, $-$ 2.20e$-$08	2.22e$-$07	0.76

No.	Name	Method	f; l; s	m	red.	$\text {t}_{\mathrm{prep}}$	$\text {t}_{\mathrm{conv}}$	inf	obj (P, D)	DIMACS	$\text {t}_{\mathrm{sol}}$	Help
15	sedumi-fp49	None	$1;0;6,{3}_{4}$	14				0	1.67e$+$01, 1.67e$+$01	5.98e$-$08	1.04
		pd1	$1;0;4,{3}_{4}$	10	1	0.05	0.00	0	1.67e$+$01, 1.67e$+$01	1.40e$-$08	0.62
		pd2	$1;0;4,{3}_{4}$	10	1	0.06	0.00	0	1.67e$+$01, 1.67e$+$01	1.40e$-$08	0.58
		dd1			0	0.02	0.00
		dd2			0	0.02	0.00
		Sieve-SDP	$1;0;4,{3}_{4}$	10	1	0.01		0	1.67e$+$01, 1.67e$+$01	1.40e$-$08	0.60
16	sedumi-fp210	None	$66;0;66,{11}_{10}$	1000				0	3.75e$-$01, 3.75e$-$01	2.15e$-$07	1.59
		pd1	$66;0;{11}_{11}$	285	1	0.12	0.05	0	3.75e$-$01, 3.75e$-$01	2.55e$-$08	1.35
		pd2	$66;0;{11}_{11}$	285	1	0.17	0.03	0	3.75e$-$01, 3.75e$-$01	2.55e$-$08	1.28
		dd1			0	0.05	0.00
		dd2			0	0.09	0.00
		Sieve-SDP	$66;0;{11}_{11}$	285	1	0.05		0	3.75e$-$01, 3.75e$-$01	2.55e$-$08	1.31
17	sedumi-fp410	None	$1;0;6,{3}_{4}$	14				0	1.67e$+$01, 1.67e$+$01	5.98e$-$08	0.87
		pd1	$1;0;4,{3}_{4}$	10	1	0.08	0.01	0	1.67e$+$01, 1.67e$+$01	1.40e$-$08	0.81
		pd2	$1;0;4,{3}_{4}$	10	1	0.08	0.00	0	1.67e$+$01, 1.67e$+$01	1.40e$-$08	0.90
		dd1			0	0.03	0.00
		dd2			0	0.03	0.00
		Sieve-SDP	$1;0;4,{3}_{4}$	10	1	0.02		0	1.67e$+$01, 1.67e$+$01	1.40e$-$08	0.81
18	sedumi-l4	None	0; 0; 45	152				0	3.70e$-$02, 3.70e$-$02	7.10e$-$08	0.83
		pd1	0; 0; 1	1	1	0.15	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.44	1
		pd2	0; 0; 1	1	1	0.19	0.00	1	0.00e$+$00, 1.00e$+$00	5.00e$-$01	0.42	1
		dd1			0	0.02	0.00
		dd2			0	0.05	0.00
		Sieve-SDP			inf	0.04					0.00	1

B Core Matlab code

In this section we provide our core Matlab code of Sieve-SDP (not including input, output, and dual solution recovery) with comments. In our code we physically delete rows and columns of the $A_i$ and of C only at the very end. During the execution of the algorithm we only mark such rows, columns and constraints as deleted.

We use two arrays to keep track of what has been marked deleted:

1.
The m-vector undeleted, whose ith entry is 1 if constraint ihas not been deleted, and 0 if it has been deleted.
2.
The sparse array $\texttt {I} \in \{0, 1 \}^{n \times (m+1)}$ with entries defined as follows.
1. (a)
  For all i and for $1 \le j \le m$
  $$\begin{aligned} I(i,j) \, = \, \left\{ \begin{array}{rll} 0, &{}\quad \hbox { if in} \, A_j \, \mathrm{the }\, i\hbox { th row and column are all zero or have been}\\ &{}\quad \hbox {\quad deleted}; \\ 1, &{}\quad \hbox { otherwise.} \end{array} \right. \, \end{aligned}$$
2. (b)
  For all i
  $$\begin{aligned} I(i,m+1) \, = \, \left\{ \begin{array}{rll} 0, &{}\quad \hbox { if in} \, \mathrm{all} \, A_j \, \mathrm{the }\, i\hbox { th row and column have been deleted}; \\ 1, &{}\quad \hbox { otherwise}. \end{array} \right. \, \end{aligned}$$

C The DIMACS errors

For the sake of completeness in this section we describe the DIMACS errors, which are commonly used to measure the accuracy of approximate solutions X of (P) and y of (D).

Define the operator $\mathcal{A}: \mathbb {R}^{m} \rightarrow \mathcal{S}^{n}$ and its adjoint as

$$\begin{aligned} \mathcal {A}(X)= & {} (A_1 \bullet X, \dots , A_m \bullet X), \\ \mathcal {A}^*(y)= & {} \sum _{i=1}^m y_i A_i. \end{aligned}$$

Suppose we are given an approximate solution X of (P) and an approximate solution y of (D). For brevity, define $Z = C - \mathcal {A}^*(y)$.

Then the DIMACS error measures are defined as follows:

In the above equations we use the following notation. If $M = (m_{ij}) \in \mathcal{S}^{n},$ then we write $\parallel \!\! M \!\! \parallel _F$ for the Frobenius norm of M and $\parallel \!\! M \!\! \parallel _\infty $ for the infinity norm of M, i.e.,

$$\begin{aligned} \parallel \!\! M \!\! \parallel _F= & {} \sqrt{\sum _{i,j} m_{ij}^2} \\ \parallel \!\! M \!\! \parallel _\infty= & {} \max _{i,j} | m_{ij} |. \end{aligned}$$

We also write $\lambda _{\min }(M)$ for the smallest eigenvalue of M.

D Dual solution recovery

In this section we address the following question: suppose we preprocessed the problem (P) by Sieve-SDP, then computed an optimal solution of the preprocessed SDP, $(P_{\mathrm{pre}})$, and of its dual, $(D_{\mathrm{pre}})$. Can we compute an optimal solution of the original primal (P) and of its dual (D)? The answer to the first question (primal solution recovery) is easy, while the issue of dual solution recovery is much more subtle.

First let us look at primal solution recovery. Since Sieve-SDP deletes rows and columns from the variable matrix X that are always zero anyway, if $X^{\mathrm{pre}}$ is an optimal solution of $(P_{\mathrm{pre}}),$ then by simply padding $X^{\mathrm{pre}}$ with zeroes we obtain an optimal solution of (P).

Next we discuss dual solution recovery. For simplicity we first assume that Sieve-SDP performed just one iteration. Further, we also assume that in the Basic Step (in Fig. 1) it eliminated the constraint $A_1 \bullet X = 0, \, $ where

$$\begin{aligned} A_1 = \begin{pmatrix} D &{}\quad 0\\ 0 &{}\quad 0 \end{pmatrix}, \end{aligned}$$

with $D \succ 0$ and we let r be the order of D.

Next, let us write out $(D_{\mathrm{pre}}):$

where the notation means that the lower right $(n-r) \times (n-r)$ principal block of $C - \sum _{i = 2}^m y_i A_i $ is positive semidefinite, and the rest is arbitrary. Thus clearly

$$\begin{aligned} {\text {val}}(D) \le {\text {val}}(D_{\mathrm{pre}}), \end{aligned}$$

(10)

since $(D_{\mathrm{pre}})$ has a feasible region which is at least as large as that of (D) (and usually it is larger). Assume that $y^{\mathrm{pre}} = (y^{\mathrm{pre}}_2, \dots , y^{\mathrm{pre}}_m)$ is an optimal solution of $(D_{\mathrm{pre}})$. Our recovery procedure, which we call Basic-Recovery, fixes $y^{\mathrm{pre}}$ and seeks $y_1$ such that $(y_1, y^{\mathrm{pre}})$ is feasible in (D), i.e.,

$$\begin{aligned} y_1 A_1 + \sum _{i = 2}^m y_i^{\mathrm{pre}} A_i \preceq C. \end{aligned}$$

(11)

We do this by a very basic linesearch: we first try the values $y_1 = 0, -1, \, $ and $-2$. If these all fail, then we try $y_1 = -100$. If we fail with $y_1 = -100, $ we stop; otherwise we test $y_1 = -3, -4, \dots $ and find the largest $y_1$ such that (11) holds.

To test whether (11) holds, again Cholesky factorization comes into play: using it, we test whether

$$\begin{aligned} C - \left( y_1 A_1 + \sum _{i = 2}^m y_i^{\mathrm{pre}} A_i\right) + 10^{-6}I \succ 0 \, {\mathrm{holds.}} \end{aligned}$$

Basic-Recovery is inspired by the dual solution recovery procedure in [34], which builds on the ideas in [32], and it assumes that the dual problem (D) is reduced.^{Footnote 5}

The procedure Basic-Recovery may fail. To see why, first assume it succeeds, i.e., it computes a feasible solution of (D). Since $y_1$ has zero objective coefficient in (D), this solution has objective value ${\text {val}}(D_{\mathrm{pre}}),$ hence by inequality (10) it is optimal in (D), thus ${\text {val}}(D) = {\text {val}}( D_{\mathrm{pre}} )$. Conversely, if ${\text {val}}(D) < {\text {val}}(D_{\mathrm{pre}}), \, $ then Basic-Recovery must fail.

Example 4

(Example 3 continued) When we apply Sieve-SDP to the SDP (4), it deletes the first row and first column in all matrices and it also deletes the first constraint.

Let us write out $(D_{\mathrm{pre}})$ again for this problem (i.e., repeat the SDP (7)):

$$\begin{aligned} \left. \begin{array}{rl} \displaystyle \sup _{y_2} &{} y_2 \\ \mathrm {s.t.} &{} y_2 \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix} \preceq \begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad 0 \end{pmatrix}, \end{array}\right. \end{aligned}$$

(12)

whose optimal solution is $y^{\mathrm{pre}}_2 = 1$.

Thus, Basic-Recovery seeks $y_1$ such that

$$\begin{aligned} \begin{array}{rl} y_1 \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \end{pmatrix} + \begin{pmatrix} 0 &{}\quad 0 &{}\quad 1 \\ 0 &{}\quad 1 &{}\quad 0 \\ 1 &{}\quad 0 &{}\quad 0 \end{pmatrix} \preceq \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \end{pmatrix}, \end{array} \end{aligned}$$

and clearly there is no such $y_1$.

One can construct more sophisticated examples in which ${\text {val}}(D_{\mathrm{pre}}) = {\text {val}}(D), \, $ but Basic-Recovery still fails.

We next look at dual solution recovery when Sieve-SDP deleted several constraints: then we run Basic-Recovery to find the corresponding $y_i$ sequentially. For simplicity assume that Sieve-SDP deleted constraints $1, 2, \dots , k$ and we found an optimal primal and dual solution of the resulting SDP (by Mosek). We then attempt to find an optimal dual solution of the SDP obtained by deleting only constraints $1, \dots , k-1;$ then of the SDP obtained by deleting only constraints $1, \dots , k-2;$ and so on.

Table 12 Dual solution recovery by four methods

Full size table

Table 13 Dual solution recovery assuming the tightest standard for “success”

Full size table

To conclude this section we make the point that dual solution recovery is much more difficult in SDP than in linear programming. We thus implemented an “ideal” recovery procedure, which we call Ideal-Recovery. It works as follows. Suppose $y^{\mathrm{pre}} = (y^{\mathrm{pre}}_{k+1}, \dots , y^{\mathrm{pre}}_{m})$ is an optimal dual solution of the SDP obtained by deleting constraints $1, \dots , k$. Ideal-Recovery fixes $y^{\mathrm{pre}}, \, $ then calls Mosek to find a feasible solution $(y_1, \dots , y_k)$ of

$$\begin{aligned} \sum _{i=1}^k y_i A_i + \sum _{i=k+1}^m y^{\mathrm{pre}}_i A_i \preceq C. \end{aligned}$$

(13)

Table 12 shows on how many instances pd1, pd2, Sieve-SDP+Basic-Recovery and Sieve-SDP+Ideal-Recovery succeeded. (Note that they succeeded on overlapping, but different problem sets, as a preprocessor may reduce an SDP, while another preprocessor may not reduce the same SDP. We do not report results with dd1 and dd2, since they only reduced a very small percentage of the instances.)

What do we mean by “success”? For pd1 and pd2 it means that their dual solution recovery code reported success. For Sieve-SDP+Basic-Recovery it means that it succeeded in every iteration: it computed the $y_i$ for every deleted constraint. For Sieve-SDP+Ideal-Recovery it means that Mosek did not report that (13) is infeasible.

Next we made the criterion of “success” more rigorous: we redefined “success” as returning a pair of primal-dual optimal solutions whose largest DIMACS error in absolute value is at most $10^{-6}$. Table 13 shows the results: now Sieve-SDP+Basic-Recovery is the winner, as it beats the supposedly perfect Sieve-SDP+Ideal-Recovery procedure.

Nevertheless, none of the methods do very well, and Table 13 shows that dual solution recovery in facial reduction remains a challenge, and an interesting area for further research.

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Zhu, Y., Pataki, G. & Tran-Dinh, Q. Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs. Math. Prog. Comp. 11, 503–586 (2019). https://doi.org/10.1007/s12532-019-00164-4

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Received: 20 September 2017
Accepted: 07 February 2019
Published: 01 July 2019
Issue Date: 01 September 2019
DOI: https://doi.org/10.1007/s12532-019-00164-4

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Sieve-SDP: a simple facial reduction algorithm to preprocess semidefinite programs

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