Outer approximation with conic certificates for mixed-integer convex problems

Abstract

A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed. We present a branch-and-bound LP outer approximation algorithm for an MI-convex problem transformed to MI-conic form. The polyhedral relaxations are refined with ${{\mathcal {K}}}^*$cuts derived from conic certificates for continuous primal-dual conic subproblems. Under the assumption that all subproblems are well-posed, the algorithm detects infeasibility or unboundedness or returns an optimal solution in finite time. Using properties of the conic certificates, we show that the ${{\mathcal {K}}}^*$ cuts imply certain practically-relevant guarantees about the quality of the polyhedral relaxations, and demonstrate how to maintain helpful guarantees when the LP solver uses a positive feasibility tolerance. We discuss how to disaggregate${{\mathcal {K}}}^*$ cuts in order to tighten the polyhedral relaxations and thereby improve the speed of convergence, and propose fast heuristic methods of obtaining useful ${{\mathcal {K}}}^*$ cuts. Our new open source MI-conic solver Pajarito (github.com/JuliaOpt/Pajarito.jl) uses an external mixed-integer linear solver to manage the search tree and an external continuous conic solver for subproblems. Benchmarking on a library of mixed-integer second-order cone (MISOCP) problems, we find that Pajarito greatly outperforms Bonmin (the leading open source alternative) and is competitive with CPLEX’s specialized MISOCP algorithm. We demonstrate the robustness of Pajarito by solving diverse MI-conic problems involving mixtures of positive semidefinite, second-order, and exponential cones, and provide evidence for the practical value of our analyses and enhancements of ${{\mathcal {K}}}^*$ cuts.

Appendices

The standard primitive nonpolyhedral cones

As we discussed in Sect. 5.2, Pajarito recognizes three standard primitive nonpolyhedral cones defined by MathProgBase: exponential, second-order, and positive semidefinite cones. Here, we tailor the general techniques for tightening OAs from Sect. 4 to a primitive cone constraint involving one of these three cones. In particular, we describe the initial fixed OAs (see Sect. 4.2), extreme ray disaggregations (see Sect. 4.1), and separation procedures (see Sect. 4.3) implemented in Pajarito.

These ideas could be adapted to other primitive nonpolyhedral cones if the user desires. For example, consider a convex constraint (in NLP form) $f(\varvec{t}) \le r$, where $f : {\mathbb {R}}^n \rightarrow {\mathbb {R}}$ is a non-homogeneous convex function. If we define a closed convex cone using the closure of the epigraph of the perspective of f:

$$\begin{aligned} {{\mathcal {K}}}_f = {{\,\mathrm{cl}\,}}\{ (r, s, \varvec{t}) \in {\mathbb {R}}^{2+n} : s > 0, r \ge s f ( \varvec{t}/s ) \} , \end{aligned}$$

(28)

then the equivalent conic constraint is $(r, 1, \varvec{t}) \in {{\mathcal {K}}}_f$. From Zhang [43], the dual cone of ${{\mathcal {K}}}_f$ is the closure of the perspective of the epigraph of the convex conjugate of f, $f^*(\varvec{w}) = \sup \{ -\varvec{w}^T \varvec{t}- f(\varvec{t}) : \varvec{t}\in {{\,\mathrm{dom}\,}}f \}$:

$$\begin{aligned} {{\mathcal {K}}}_f^* = {{\,\mathrm{cl}\,}}\{ (u, v, \varvec{w}) \in {\mathbb {R}}^{2+n} : u > 0, v \ge u f^* ( \varvec{w}/u ) \} . \end{aligned}$$

(29)

Note ${{\mathcal {K}}}_f^* \ne {{\mathcal {K}}}_{f^*}$ because of the permuting of the first two indices.

1.1 Exponential cone

The exponential cone ${\mathcal {E}}$ is defined from the convex univariate exponential function $f(t) = \exp (t)$ in Eq. (28):

$$\begin{aligned} {\mathcal {E}}&= {{\,\mathrm{cl}\,}}\{ (r, s, t) \in {\mathbb {R}}^3 : s> 0, r \ge s \exp ( t/s ) \} \nonumber \\&= \{ (r, 0, t) : r \ge 0, t \le 0 \} \cup \{ (r, s, t) : s > 0, r \ge s \exp ( t/s ) \} . \end{aligned}$$

(30)

The convex conjugate is $f^*(w) = w - w \log (-w)$, so by Eq.  (29), the dual cone of the exponential cone is:

$$\begin{aligned} {\mathcal {E}}^*&= {{\,\mathrm{cl}\,}}\{ (u, v, w) \in {\mathbb {R}}^3 : u> 0, w< 0, v \ge w - w \log ( {-}w/u ) \} \nonumber \\&= \{ (u, v, 0) : u, v \ge 0 \} \cup \{ (u, v, w) : u > 0, w < 0, v \ge w - w \log ( {-}w/u ) \} . \end{aligned}$$

(31)

1.1.1 Initial fixed polyhedral relaxation

Suppose we have a primitive cone constraint $(r, s, t) \in {\mathcal {E}}$. We use the two ${\mathcal {E}}^*$ extreme rays (1, 0, 0), (0, 1, 0) to impose the simple bound constraints $r, s \ge 0$. We use more ${\mathcal {E}}^*$ extreme rays of the form $(1, w - w \log (-w), w)$ by picking several different values $w < 0$. Note the corresponding cuts separate any point (0, 0, t) satisfying $t > 0$.

1.1.2 Extreme ray disaggregation

Suppose we have the ${\mathcal {E}}^*$ point (u, v, w). If $w = 0$, the point is already a nonnegative combination of the initial fixed ${\mathcal {E}}^*$ points from “Appendix A.1.1”, so we discard it. If $w < 0$, we use the ${\mathcal {E}}^*$ extreme ray $(u, w - w \log ({-}w/u), w)$, which when added to some nonnegative multiple of (0, 1, 0), gives (u, v, w).³⁸

1.1.3 Separation of an infeasible point

Suppose we want to separate a point $(r, s, t) \notin {\mathcal {E}}$ that satisfies the initial fixed cuts. Then $r, s \ge 0$ and if $r = s = 0$ then $t \le 0$. If $s = 0$, then $t > 0$ and $r > 0$, and we use the ${\mathcal {E}}^*$ extreme ray $(t/r, -2 + 2 \log (2r/t), -2)$. If $s > 0$, then $r < s \exp (t/s)$, and we use the ${\mathcal {E}}^*$ extreme ray $(1, (t/s - 1) \exp (t/s), -\exp (t/s))$.

1.2 Second-order cone

For $n \ge 2$, the second-order cone is the epigraph of the $\ell _2$-norm, which is convex and homogeneous:

$$\begin{aligned} {\mathcal {L}}^{1+n} = \{ (r, \varvec{t}) \in {\mathbb {R}}^{1+n} : r \ge \Vert \varvec{t}\Vert _2 \} . \end{aligned}$$

(32)

We sometimes drop the dimension $1 + n$ when implied by context. This cone is self-dual (${\mathcal {L}}^* = {\mathcal {L}}$). We also define the (self-dual) rotated second-order cone:

$$\begin{aligned} {\mathcal {V}}^{2+n} = \{ (r, s, \varvec{t}) \in {\mathbb {R}}^{2+n} : r, s \ge 0, 2rs \ge \Vert \varvec{t}\Vert _2^2 \} . \end{aligned}$$

(33)

Note that ${\mathcal {V}}$ is an invertible linear transformation of ${\mathcal {L}}$, since $(r, s, \varvec{t}) \in {\mathcal {V}}^{2+n}$ if and only if $(r + s, r - s, \sqrt{2} t_1, \ldots , \sqrt{2} t_n ) \in {\mathcal {L}}^{2+n}$, so for simplicity we restrict attention to ${\mathcal {L}}$.³⁹

1.2.1 Initial fixed polyhedral relaxation

Suppose we have a primitive cone constraint $(r, \varvec{t}) \in {\mathcal {L}}^{1+n}$. First, we note that the $\ell _{\infty }$-norm lower-bounds the $\ell _2$-norm, since for any $\varvec{t}\in {\mathbb {R}}^n$ we have:

$$\begin{aligned} \Vert \varvec{t}\Vert _{\infty } = \max \limits _{{i \in \llbracket n \rrbracket }} \, |t_i |\le \Vert \varvec{t}\Vert _{2} . \end{aligned}$$

(34)

Let $\varvec{e}(i) \in {\mathbb {R}}^n$ be the ith unit vector in n dimensions. We use the 2n${\mathcal {L}}^*$ extreme rays $(1, \pm \varvec{e}(i)), \forall i \in \llbracket n \rrbracket$, which imply the conditions $r \ge |t_i |, \forall i \in \llbracket n \rrbracket$, equivalent to the homogenized box relaxation $r \ge \Vert \varvec{t}\Vert _{\infty }$. Second, we note that the $\ell _1$-norm also provides a lower bound for the $\ell _2$-norm, since for any $\varvec{t}\in {\mathbb {R}}^n$ we have:

$$\begin{aligned} \Vert \varvec{t}\Vert _1 = \sum \limits _{{i \in \llbracket n \rrbracket }} \, |t_i |\le \sqrt{n} \Vert \varvec{t}\Vert _2 . \end{aligned}$$

(35)

We use the $2^n$${\mathcal {L}}^*$ extreme rays $(1, \varvec{\sigma }/\sqrt{n}), \forall \varvec{\sigma } \in \{-1, 1\}^n$, which imply the homogenized diamond relaxation $r \ge \Vert \varvec{t}\Vert _1/\sqrt{n}$. Although the number of initial fixed cuts is exponential in the dimension n, in “Appendix B” we describe how to use an extended formulation introduced by Vielma et al. [40] with n auxiliary variables to imply an initial fixed OA that is no weaker but uses only a polynomial number of cuts. Note that the ${\mathcal {L}}^*$ point $(1, \varvec{0})$, which corresponds to the simple variable bound $r \ge 0$, is a nontrivial conic combination of these initial fixed ${\mathcal {L}}^*$ extreme rays.

1.2.2 Extreme ray disaggregation

Suppose we have the ${\mathcal {L}}^*$ point $(u, \varvec{w})$. If $\varvec{w}= \varvec{0}$, the point is already a nonnegative multiple of the ${\mathcal {L}}^*$ point $(1, \varvec{0})$, so we discard it. Otherwise, we use the ${\mathcal {L}}^*$ extreme ray $(\Vert \varvec{w}\Vert _2, \varvec{w})$, which when added to some nonnegative multiple of $(1, \varvec{0})$, gives the original point $(u, \varvec{w})$.⁴⁰

1.2.3 Separation of an infeasible point

Suppose we want to separate a point $(r, \varvec{t}) \notin {\mathcal {L}}$ that satisfies the initial fixed cuts. Then $r \ge 0$ and so $\varvec{t}\ne \varvec{0}$, and we use the ${\mathcal {L}}^*$ extreme ray $(1, {-}\varvec{t}/\Vert \varvec{t}\Vert _2)$.

1.3 Positive semidefinite cone

For $n \ge 2$, we define the $n \times n$-dimensional positive semidefinite (PSD) matrix cone ${\mathbb {S}}^n_+$ as a subset of the symmetric matrices ${\mathbb {S}}^n = \{ \varvec{T}\in {\mathbb {R}}^{n \times n} : \varvec{T}= \varvec{T}^T \}$, avoiding the need to enforce symmetry constraints. From the minimum eigenvalue function $\lambda _{\min } : {\mathbb {S}}^n \rightarrow {\mathbb {R}}$, we have:

$$\begin{aligned} {\mathbb {S}}^n_+ = \{ \varvec{T}\in {\mathbb {S}}^n : \lambda _{\min } (\varvec{T}) \ge 0 \} . \end{aligned}$$

(36)

For $\varvec{W}, \varvec{T}\in {\mathbb {S}}^n$, we use the trace inner product $\langle \varvec{W}, \varvec{T}\rangle = \sum _{i,j \in \llbracket n \rrbracket } W_{i,j} T_{i,j}$. ${\mathbb {S}}^n_+$ is self-dual and its extreme rays are the rank-1 PSD matrices [4], i.e. any $\varvec{\omega }\varvec{\omega }^T$ for $\varvec{\omega }\in {\mathbb {R}}^n$. An extreme ray $({\mathbb {S}}^n_+)^*$ cut has the form:

$$\begin{aligned} \langle \varvec{\omega }\varvec{\omega }^T, \varvec{T}\rangle = \varvec{\omega }^T \varvec{T}\varvec{\omega }\ge 0 . \end{aligned}$$

(37)

In “Appendix C”, we describe how to strengthen extreme ray ${\mathbb {S}}_+^*$ cuts to rotated second-order cone constraints (for MISOCP OA; see Sect. 5.4).

Recall that our MI-conic form ${\mathfrak {M}}$ uses vector cone definitions, as does MathProgBase. Mosek ApS [30] refers to the matrix cone ${\mathbb {S}}^n_+ \subset {\mathbb {S}}^n$ as the smat PSD cone, and to its equivalent vectorized definition ${\mathcal {S}}^{n (n {+} 1)/2} \subset {\mathbb {R}}^{n (n {+} 1)/2}$ as the svec PSD cone. In svec space, we use the usual vector inner product, and ${\mathcal {S}}$ is also self-dual. The invertible linear transformations for an smat-space point $\varvec{T}\in {\mathbb {S}}^n$ and an svec-space point $\varvec{t}\in {\mathbb {R}}^{n (n {+} 1)/2}$ are:

$$\begin{aligned} {{\,\mathrm{svec}\,}}(\varvec{T})&= ( T_{1,1}, \sqrt{2} T_{2,1}, \ldots , \sqrt{2} T_{n,1}, T_{2,2}, \sqrt{2} T_{3,2}, \ldots , T_{n,n} ) , \end{aligned}$$

(38a)

$$\begin{aligned} {{\,\mathrm{smat}\,}}(\varvec{t})&= \begin{bmatrix} t_1 &{} \quad t_1/\sqrt{2} &{}\quad \cdots &{}\quad t_n/\sqrt{2} \\ t_2/\sqrt{2} &{}\quad t_{n+1} &{}\quad \cdots &{}\quad t_{2n-1}/\sqrt{2} \\ \vdots &{}\quad \vdots &{}\quad \ddots &{} \quad \vdots \\ t_n/\sqrt{2} &{}\quad t_{n-1}/\sqrt{2} &{}\quad \cdots &{}\quad t_{n (n {+} 1)/2} \end{bmatrix} . \end{aligned}$$

(38b)

1.3.1 Initial fixed polyhedral relaxation

Suppose we have a primitive cone constraint $\varvec{T}\in {\mathbb {S}}_+^n$. Let $\varvec{e}(i) \in {\mathbb {R}}^n$ be the ith unit vector in n dimensions. For each $i \in \llbracket n \rrbracket$, we let $\varvec{\omega }= \varvec{e}(i)$ in the extreme ray ${\mathbb {S}}^*_+$ cut (37), which imposes the diagonal nonnegativity condition $T_{i,i} \ge 0$ necessary for PSDness. For each $i,j \in \llbracket n \rrbracket : i > j$, we let $\varvec{\omega }= \varvec{e}(i) \pm \varvec{e}(j)$ in (37), which enforces the condition $T_{i,i} + T_{j,j} \ge 2 |T_{i,j} |$ necessary for PSDness. Ahmadi and Hall [2] discuss an LP inner approximation of the PSD cone called the cone of diagonally dominant (DD) matrices. Our initial fixed ${\mathbb {S}}^*_+$ points $\varvec{\omega }\varvec{\omega }^T$ are exactly the extreme rays of the DD cone, so our initial fixed OA is the dual cone of the DD cone.

1.3.2 Extreme ray disaggregation

Suppose we have the ${\mathbb {S}}_+^*$ point $\varvec{W}$, not necessarily and extreme ray of ${\mathbb {S}}_+^*$. We perform an eigendecomposition $\varvec{W}= \sum _{i \in \llbracket n \rrbracket } \lambda _i \tilde{\varvec{\omega }}_i \tilde{\varvec{\omega }}_i^T$, where for all $i \in \llbracket n \rrbracket$, $\lambda _i$ is the ith eigenvalue and $\tilde{\varvec{\omega }}_i$ is its corresponding eigenvector.⁴¹ Since $\varvec{W}$ is PSD, every eigenvalue is nonnegative, and there are ${{\,\mathrm{rank}\,}}(\varvec{W}) \le n$ positive eigenvalues. For each $i \in \llbracket n \rrbracket : \lambda _i > 0$, we let $\varvec{\omega }= \sqrt{\lambda _i} \tilde{\varvec{\omega }}_i$ in (37).⁴² These extreme ray ${\mathbb {S}}^*_+$ cuts aggregate to imply the original ${\mathbb {S}}_+^*$ cut $\langle \varvec{W}, \varvec{T}\rangle \ge 0$.

1.3.3 Separation of an infeasible point

Suppose we want to separate a point $\varvec{T}\in {\mathbb {S}}^n \backslash {\mathbb {S}}^n_+$. We perform an eigendecomposition $\varvec{T}= \sum _{i \in \llbracket n \rrbracket } \lambda _i \varvec{\tau }_i \varvec{\tau }_i^T$, for which at least one eigenvalue is negative. For each $i \in \llbracket n \rrbracket : \lambda _i < 0$, we let $\varvec{\omega }= \varvec{\tau }_i$ in (37) (note $\langle \varvec{\tau }_i \varvec{\tau }_i^T, \varvec{T}\rangle = \varvec{\tau }_i^T \varvec{T}\varvec{\tau }_i = \lambda _i < 0$).

The second-order cone extended formulation

Recall the definitions of the second-order cone ${\mathcal {L}}$ and the rotated-second-order cone ${\mathcal {V}}$ in “Appendix A.2”. Both ${\mathcal {L}}$ and ${\mathcal {V}}$ are self-dual. As discussed in Sect. 5.4, Pajarito can optionally use an extended formulation (EF) for second-order cone constraints, leading to tighter polyhedral relaxations. Vielma et al. [40] show that the constraint $(r, \varvec{t}) \in {\mathcal {L}}^{1+n}$ is equivalent to the following $1 + n$ constraints on r, $\varvec{t}$, and the auxiliary variables $\varvec{\pi }\in {\mathbb {R}}^n$:

$$\begin{aligned}&\sum \limits _{{i \in \llbracket n \rrbracket }} 2 \pi _i \le r \end{aligned}$$

(39a)

$$\begin{aligned}&(r, \pi _i, t_i) \in {\mathcal {V}}^3 \quad \forall i \,\, \in \llbracket n \rrbracket . \end{aligned}$$

(39b)

By projecting out the $\varvec{\pi }$ variables, the equivalence is obvious. Constraints (39b) imply $r \ge 0$ and $\pi _i \ge 0$ and $2 r \pi _i \ge t_i^2$ for all $i \in \llbracket n \rrbracket$. Aggregating the latter conditions and using the linear inequality (39a), we see $r^2 \ge \sum _{i \in \llbracket n \rrbracket } 2 r \pi _i \ge \sum _{i \in \llbracket n \rrbracket } t_i^2$, which is equivalent to the original constraint (for $r \ge 0$). We only use ${\mathcal {L}}^{1+n}$ and ${\mathcal {V}}^3$ here, so for convenience we drop the dimensions.⁴³

Suppose $(u, \varvec{w})$ is a ${\mathcal {L}}^*$ extreme ray, so from “Appendix A.2.2”, we have $\varvec{w}\ne \varvec{0}$ and $u = \Vert \varvec{w}\Vert _2 > 0$. Then $u r + \varvec{w}^T \varvec{t}\ge 0$ is a ${\mathcal {L}}^*$ cut. Note that the linear constraint (39a) in the EF implies:

$$\begin{aligned} u r + \varvec{w}^T \varvec{t}\ge \frac{u r}{2} + u \sum \limits _{{i \in \llbracket n \rrbracket }} \pi _i + \varvec{w}^T \varvec{t}= \sum \limits _{{i \in \llbracket n \rrbracket }} \left( \frac{w_i^2 r}{2u} + u \pi _i + w_i t_i \right) . \end{aligned}$$

(40)

For each $i \in \llbracket n \rrbracket$, consider the ${\mathcal {V}}^*$ extreme ray $(w_i^2/2u, u, w_i)$, which implies a ${\mathcal {V}}^*$ cut for the ith constraint (39b) in the EF. The RHS of (40) is an aggregation of these n${\mathcal {V}}^*$ cuts, which means the ${\mathcal {V}}^*$ cuts imply the ${\mathcal {L}}^*$ cut condition $u r + \varvec{w}^T \varvec{t}\ge 0$. Therefore, there is no loss of strength in the polyhedral relaxations, and we maintain the certificate ${{\mathcal {K}}}^*$ cut guarantees from Sect. 3.1.⁴⁴

We apply this lifting procedure to the initial fixed ${\mathcal {L}}^*$ points described in Sect. A.2.1. The ${\mathcal {L}}^*$ points for the $\ell _{\infty }$-norm relaxation are $(1, \pm \varvec{e}(i)), \forall i \in \llbracket n \rrbracket$; for each $i \in \llbracket n \rrbracket$, we get three unique ${\mathcal {V}}^*$ extreme rays (0, 1, 0) (for $w_i = 0$) and $(1/2, 1, \pm 1)$ (for $w_i = \pm 1$). The ${\mathcal {L}}^*$ points for the $\ell _1$-norm relaxation are $(1, \varvec{\sigma }/\sqrt{n}), \forall \varvec{\sigma } \in \{-1, 1\}^n$; for each $i \in \llbracket n \rrbracket$, we get two unique ${\mathcal {V}}^*$ extreme rays $(1/2n, 1, {\pm } 1/\sqrt{n})$ (for $w_i = {\pm } 1/\sqrt{n}$). The polyhedral relaxation implied by these 5n${\mathcal {V}}^*$ points in the EF (39b) and (39a) is at least as strong as that implied by the $2n + 2^n$${\mathcal {L}}^*$ points from Sect. A.2.1, so our initial fixed OA can be imposed much more economically with the EF.

SOCP outer approximation for PSD cones

Recall the definitions of the self-dual smat-space PSD cone ${\mathbb {S}}_+$ in “Appendix A.3” and the self-dual rotated-second-order cone ${\mathcal {V}}$ in “Appendix A.2”.⁴⁵ For a primitive cone constraint $\varvec{T}\in {\mathbb {S}}_+^n$, we demonstrate how to strengthen an $({\mathbb {S}}^n_+)^*$ extreme ray cut $\langle \varvec{\omega }\varvec{\omega }^T, \varvec{T}\rangle \ge 0$ to up to n different ${\mathcal {V}}^3$ constraints. As discussed in Sect. 5.4, Pajarito can optionally solve an MISOCP OA model including these ${\mathcal {V}}^3$ constraints, leading to tighter relaxations of a challenging ${\mathbb {S}}_+^n$ constraint.

Fix the index $i \in \llbracket n \rrbracket$. Let $\underline{\omega } = \omega _i$ be the ith element of $\varvec{\omega }$, and $\underline{\varvec{\omega }} = (\omega _j)_{j \in \llbracket n \rrbracket \backslash \{i\}} \in {\mathbb {R}}^{n-1}$ be the (column) subvector of $\varvec{\omega }$ with the ith element removed. Similarly, let $\underline{t} = T_{i,i}$ and $\underline{\varvec{t}} = (T_{i,j})_{j \in \llbracket n \rrbracket \backslash \{i\}} \in {\mathbb {R}}^{n-1}$, and let $\underline{\varvec{T}} = (T_{k,j})_{k,j \in \llbracket n \rrbracket \backslash \{i\}} \in {\mathbb {S}}^{n-1}$ be the submatrix of $\varvec{T}$ with the ith column and row removed. Kim et al. [22] prove a variant of the standard Schur-complement result that $\varvec{T}\in {\mathbb {S}}^n_+$ if and only if $\underline{t} \ge 0$ and:

$$\begin{aligned} \underline{\varvec{T}}&\in {\mathbb {S}}^{n-1}_+ \end{aligned}$$

(41a)

$$\begin{aligned} \underline{t} \underline{\varvec{T}} - \underline{\varvec{t}} \underline{\varvec{t}}^T&\in {\mathbb {S}}^{n-1}_{+}\,. \end{aligned}$$

(41b)

Consider the 3-dimensional rotated second-order cone constraint:

$$\begin{aligned} ( \underline{t}, \underline{\varvec{\omega }}^T \underline{\varvec{T}} \underline{\varvec{\omega }}, \sqrt{2} \underline{\varvec{\omega }}^T \underline{\varvec{t}} ) \in {\mathcal {V}}^3 . \end{aligned}$$

(42)

By the definition of ${\mathcal {V}}$, constraint (42) is equivalent to the conditions $\underline{t} \ge 0$ and:

$$\begin{aligned} \underline{\varvec{\omega }}^T \underline{\varvec{T}} \underline{\varvec{\omega }}&\ge 0 \end{aligned}$$

(43a)

$$\begin{aligned} \underline{\varvec{\omega }}^T (\underline{t} \underline{\varvec{T}}) \underline{\varvec{\omega }}&\ge (\underline{\varvec{\omega }}^T \underline{\varvec{t}})^2 . \end{aligned}$$

(43b)

Condition (41a) implies condition (43a), by the dual cone definition (5). Since $(\underline{\varvec{\omega }}^T \underline{\varvec{t}})^2 = \underline{\varvec{\omega }}^T \underline{\varvec{t}} \underline{\varvec{t}}^T \underline{\varvec{\omega }}$, condition (43b) is equivalent to $\underline{\varvec{\omega }}^T (\underline{t} \underline{\varvec{T}} - \underline{\varvec{t}} \underline{\varvec{t}}^T) \underline{\varvec{\omega }} \ge 0$. Thus, by the dual cone definition again, condition (41b) implies condition (43b). Therefore, constraint (42) is a valid relaxation of the PSD constraint $\varvec{T}\in {\mathbb {S}}^n_+$. Furthermore, from Theorem 3.3 of Kim et al. [22], constraint (42) holds if and only if:

$$\begin{aligned} \langle \varvec{W}, \varvec{T}\rangle \ge 0 \qquad \forall \varvec{W}\in {\mathbb {S}}^n_+ : ( W_{k,j} = \omega _k \omega _j, \forall k,j \in \llbracket n \rrbracket \backslash \{i\} ) . \end{aligned}$$

(44)

Thus constraint (42) potentially implies an infinite family of $({\mathbb {S}}^n_+)^*$ cuts, including the original $({\mathbb {S}}^n_+)^*$ cut $\langle \varvec{\omega }\varvec{\omega }^T, \varvec{T}\rangle \ge 0$.⁴⁶ Note that the choice of $i \in \llbracket n \rrbracket$ is arbitrary, so we can derive n different ${\mathcal {V}}$ constraints of the form (42).⁴⁷

We now apply this strengthening procedure to the initial fixed $({\mathbb {S}}^n_+)^*$ extreme rays described in “Appendix A.3.1”. Letting $\varvec{\omega }= \varvec{e}(i) \pm \varvec{e}(j)$ for each $i,j \in \llbracket n \rrbracket : j > i$ in constraint (42), we get the $n (n {-} 1)/2$ initial fixed ${\mathcal {V}}$ constraints:

$$\begin{aligned} ( T_{i,i}, T_{j,j}, \sqrt{2} T_{i,j} ) \in {\mathcal {V}}^3 \quad \forall i,j \in \llbracket n \rrbracket : j > i . \end{aligned}$$

(45)

These constraints enforce that every $2 \times 2$ principal matrix of $\varvec{T}$ is PSD, a necessary but insufficient condition for $\varvec{T}\in {\mathbb {S}}^n_+$. Ahmadi and Hall [2] discuss an SOCP inner approximation of the PSD cone called the cone of scaled diagonally dominant (SDD) matrices. Our initial fixed SOCP OA is the dual SDD matrix cone, a strict subset of the polyhedral dual DD matrix cone that corresponds to our initial fixed LP OA from “Appendix A.3.1”.