The Chebyshev center as an alternative to the analytic center in the feasibility pump

As a heuristic for obtaining feasible points of mixed integer linear problems, the feasibility pump (FP) generates two sequences of points: one of feasible solutions for the relaxed linear problem; and another of integer points obtained by rounding the linear solutions. In a previous work, the present authors proposed a variant of FP, named analytic center FP, which obtains integer solutions by rounding points in the segment between the linear solution and the analytic center of the polyhedron of the relaxed problem. This work introduces a new FP variant that replaces the analytic center with the Chebyshev center. Two of the benefits of using the Chebyshev center are: (i) it requires the solution of a linear optimization problem (unlike the analytic center, which involves a convex nonlinear optimization problem for its exact solution); and (ii) it is invariant to redundant constraints (unlike the analytic center, which may not be well centered within the polyhedron for problems with highly rank-deficient matrices). The computational results obtained with a set of more than 200 MIPLIB2003 and MIPLIB2010 instances show that the Chebyshev center FP is competitive and can serve as an alternative to other FP variants.


Given the mixed-integer linear problem (MILP)
where A ∈ ℝ m×n , b ∈ ℝ m , c ∈ ℝ n , I ⊆ N = {1, … , n} and P = {x ∈ ℝ n ∶ Ax = b, x ≥ 0} (that is, P is the feasible region of the linear relaxation of (1)), finding a feasible point of ( 1) is a challenging (NP-hard) problem.Many heuristics have been developed for obtaining feasible (hopefully good) solutions of (1).In this paper, we focus on the feasibility pump (FP) [5,12], which has proven to be a successful heuristic, not only for linear problems but also for nonconvex nonlinear problems [4,7,9].
Briefly, FP alternates between two sequences of points: one of feasible solutions for the linear relaxation of (1), and another of integer points, that hopefully converge to a feasible integer solution.The integer point is obtained by applying some rounding procedure to the feasible solution of the linear relaxation.
Several strides have been made to further develop the original FP.In [1], the authors take the objective function of the MILP into account at each iteration of the algorithm in order to find better quality solutions.This approach was named objective FP.In [13], a new improved rounding scheme based on constraint propagation was introduced.Interior-point methods were applied to primal heuristics in [3] (an approach named analytic center FP or AC-FP) and [14] (resulting in the analytic center feasibility method or ACFM).Although both AC-FP and ACFM used the analytic center of P , they are significantly different.In particular, AC-FP (which is briefly outlined in Sect.3.1) relies on FP and it computes only one analytic center, while ACFM is based on a cutting plane method and it computes an analytic center at each iteration of the algorithm.
AC-FP explores non-integer points in the segment where the feasible point of the linear relaxation of (1) joins the analytic center.The motivation behind using the analytic center lies in the fact that rounding an interior point increases the chances of finding a feasible integer solution.AC-FP was proven in [3] to be a successful heuristic, namely by improving the standard FP in several tested MILP instances.In [6] the authors extended the AC-FP idea by enhancing the rounding procedure.
However, in both practice and theory, using the analytic center has two downsides.First, computing the exact analytic center of P means solving the convex nonlinear optimization problem min − ∑ n i=1 ln x i ∶ Ax = b .An approximate solu- tion to this problem was suggested in [3] by applying a path-following (or barrier) interior-point algorithm to the problem min 0 ∶ Ax = b, x ≥ 0 .This excludes using the simplex method for computing the analytic center.The second drawback is that, theoretically, a large number of redundant constraints (in problems with highly rank deficient matrices A) may change the location of the analytic center [10].In this paper, we consider an alternative to the analytic center, named the Chebyshev center.Using the Chebyshev center within FP overcomes the two above drawbacks: the center of P is not affected by redundant constraints in A (this is clearly shown below in Sect.3.2); and the center can be computed using either the simplex or The Chebyshev center as an alternative to the analytic center… barrier algorithms.As this work shows, the Chebyshev center FP (CC-FP), for some instances, provides better solutions than objective FP and AC-FP variants.The paper is organized as follows.Section 2 outlines the FP heuristic.Section 3 describes the use of a generic center point (analytic or Chebyshev center) in the FP heuristic, while Sect.3.2 focuses on the computation of the Chebyshev center.Section 4 presents extensive computational results on a subset of MIPLIB2003 [2] and MIPLIB2010 [11] instances, in order to compare the objective FP, AC-FP and CC-FP, as well as to show the effectiveness of the CC-FP approach.Finally, we present some closing remarks in Sect. 5.

The feasibility pump heuristic
The original feasibility pump heuristic [12] works iteratively with two points: one ( x * ) is feasible for the continuous relaxation of (1), although it is possibly integer infeasible, and the other ( x ) is integral but might not be in P .The point x * is set to the optimal solution of the linear programming relaxation of (1) while x is obtained by rounding x * to the closest integer point, as follows: where [⋅] represents scalar rounding to the nearest integer.Note that the continuous variables x j , j ∉ I , do not play any role.At each iteration of the FP method, x * is updated by minimizing the following linear optimization problem where Δ(x, x) is the distance between x and x using the L 1 norm: Figure 1 illustrates one iteration of the FP method, where t represents the iteration number, and the final point xt+1 is integer feasible.FP ends when the distance Δ(x, x) is 0 (meaning we have obtained an integer feasible solution) or when a predefined termination criterion is reached.One of the main drawbacks of the FP heuristic is the possibility of visiting integer points already visited in previous iterations, thereby causing a cycle.To avoid this, a restart procedure is proposed in [12].
The FP implementation has three stages.In the first stage, the method considers only the set of binary variables by relaxing the integrality conditions on the general integer variables.In the second stage, FP takes all integer variables into account and uses the best point x obtained at Stage 1 as a starting point.Both stages terminate as soon as a feasible solution is found or when some termination criterion is reached (e.g., the best Δ(x, x) is not updated during a certain number of iterations, or the maximum number of iterations is reached-just to name two).The last stage (Stage 3) starts when FP cannot find a feasible solution to (1) within the established time limit.In this stage, a commercial solver is applied to (1) (CPLEX 12.7 is used in this work), for which the best point obtained from Stage 2 is used as a starting point.Stage 3 stops as soon as a feasible solution is found.An outline of the FP algorithm is shown in Fig. 2, and further details can be found in [5,12].Despite the successful results obtained by the original FP heuristic for finding feasible solutions of MILPs in a short computational time, using the objective function of (1) only at the beginning of the procedure often leads to a rather poor solution.To avoid this, a modified FP heuristic called objective FP was proposed in [1], which considers a convex combination of Δ(x, x) and the objective function of (1).The idea is to focus the search for feasible solutions near the region of high-quality points.The modified objective function Δ  (x, x) is defined as Fig. 1 Graphical representation of the FP method Fig. 2 The feasibility pump heuristic (original version) [5,12] 1 3 The Chebyshev center as an alternative to the analytic center… where ‖ ⋅ ‖ is the Euclidean norm of a vector, and Δ is the objective function vec- tor of Δ(x, x) (i.e., the number of either binary variables at Stage 1, or both integer and binary variables at Stage 2).The weight is reduced at every iteration.When = 0 , the original FP heuristic is obtained.Note that the objective FP algorithm is nearly identical to the original FP algorithm in Fig. 2; it simply replaces Δ(x, x) with Δ  t (x, x) in line 5 and adds the proper initialization and updating of .Further details can be found in [1].

Using a center point in the feasibility pump
Let x ∈ ℝ n be an interior point of polyhedron P , that is, Ax = b and x > 0 (strictly positive components).All the points in the segment xx * are feasible, since they are a convex combination of two feasible points, x * and x , and therefore candidates to be rounded.In addition, all the points except x * in the segment xx * are interior, thus increasing the chances for the rounded point to be feasible for P .Of all interior points x , those that are "well centered" inside P (let them be the center points) are the best choices.The above generic approach is named the center point feasibility pump (CP-FP) in this work.
At each iteration CP-FP considers points x( is feasible, then a fea- sible integer solution is found and the procedure is stopped.Otherwise, CP-FP continues to consider the new integral point to be the one that is closest to P from all the points x() (the ∞ distance between P and x() is used to measure closeness).If more than one integer point x() is feasible for P , CP-FP selects the one closest to x * (which may probably have a better objective value).Figure 3 illustrates the behaviour of CP-FP: while the standard objective FP would provide the infeasible yellow point, CP-FP could deliver the feasible green point.An outline of the algorithm is shown in Fig. 4. Note that if x( = 0) is selected at each iteration, CP-FP behaves exactly as the objective FP.Further details are given in [3].
There are several ways to get the center point.One option is the analytic center of the polyhedron, which was used in [3] with promising results.The main drawback of the analytic center is that redundant constraints can push it near the boundary of Fig. 3 Illustration of CP-FP the polyhedron [10], as is shown in Sect.3.2.To overcome this issue, we suggest in this paper using the Chebyshev center.Both center points are briefly outlined below.

The analytic center
The analytic center of P is defined as the point x that minimizes the primal potential function − ∑ n i=1 ln x i , i.e., Constraints x > 0 can be avoided, since the domain of ln are the positive numbers, and then ( 6) is an equality constrained strictly convex optimization problem.It is easily seen that x is also the solution of max ∏ n i=1 x i ∶ Ax = b ; that is, the analytic center attempts to maximize the distance to the hyperplanes x i = 0, i = 1, … , n , and it is thus expected to be well centered in the interior of P .Note that the ana- lytic center is not a topological property of a polytope, and it depends on how P is defined through Ax = b [15].
The analytic center solves the KKT conditions of ( 6), which can be recast as Fig. 4 The center point feasibility pump heuristic (CP-FP) [3] 1 3 The Chebyshev center as an alternative to the analytic center… y ∈ ℝ m and s ∈ ℝ n being, respectively, the Lagrange multipliers of Ax = b and an auxiliary vector (associated to x > 0 ).Alternatively, we can make use of an avail- able highly efficient implementation, in which we compute the analytic center by applying a primal-dual path-following interior-point algorithm to the barrier problem of the linear relaxation of (1) after setting c = 0 , that is, where is a positive parameter (the parameter of the barrier) that tends to zero.The arc of solutions of the barrier problem for every  > 0 is named the central path.The central path converges to the analytic center of the optimal set of a linear optimization problem.When c = 0 (as in (8)) the central path converges to the analytic center of the feasible set P [15].The use of the analytic center in the feasibility pump, introduced in [3], was named the analytic center FP (AC-FP).

The Chebyshev center
Given a convex polyhedron described by linear inequalities the Chebyshev center x is the center of the largest inscribed Euclidean ball in Q .A Euclidean ball of center x ∈ ℝ n and radius r is the set of all points of distance less than or equal to r from x , i.e., B(x, r) = {x + u ∶ ‖u‖ 2 ≤ r} .The optimization prob- lem that finds the Chebyshev center is [8] where Since a T i u ≤ ‖a i ‖ 2 ‖u‖ 2 ≤ ‖a i ‖ 2 r , we can then write (10) as the following linear optimization problem: For the polyhedron P of the linear relaxation of (1), the Chebyshev center is defined only in terms of the inequalities x ≥ 0 and is restricted to Ax = b , which results in the following problem: The Chebyshev center does not change in the presence of redundant constraints, and it is then always well located in a central position inside the polyhedron, thus making it an effective choice for CP-FP.On the other hand, the analytic center can be pushed out near the boundary of P by redundant constraints.Figures 5 and 6 illus- trate this situation with the convex polyhedron Q described by the following linear inequalities: The largest inscribed Euclidean ball centered at x cc is also shown Fig. 6 The Chebyshev ( x ′ cc ) and analytic ( The Chebyshev center as an alternative to the analytic center… Fig. 5 shows the analytic ( x ac ) and the Chebyshev ( x cc ) centers of Q , as well as the largest inscribed Euclidean ball centered at x cc .Let us consider an alternative repre- sentation of the polyhedron , which is obtained by adding the two redundant constraints Figure 6 shows the analytic ( x ′ ac ) and the Chebyshev (

Implementation and instances
Both AC-FP and CC-FP are implemented in C++ using the base code of the objective FP, which is freely available from https:// site.unibo.it/ opera tions-resea rch/ en/ resea rch/ libra ry-of-codes-and-insta nces-1.The optimization solver CPLEX (version 12.7) is used to solve the linear optimization subproblems.All the runs are carried out on a Fujitsu Primergy RX2540 M1 4X server with two 2.6 GHz Intel Xeon E5-2690v3 CPUs (48 cores) and 192 Gigabytes of RAM, under a GNU/Linux operating system (openSuse 13.2), without exploitation of the multithreading capabilities.A one-hour time limit is imposed on all the runs.AC-FP, CC-FP and objective FP are tested on a subset of MIPLIB2003 [2] and MIPLIB2010 [11] instances, whose dimensions are shown in Tables 1, 2 and 3.The columns "rows", "cols", "nnz", "int", "bin" and "con" provide respectively the numbers of constraints, variables, nonzeros, general integer variables, binary variables and continuous variables of the instances.The column "objective" shows the optimal objective function.

Results
We first analyze the results for the subset of MIPLIB2003 instances.Table 4 presents the results obtained.For AC-FP and CC-FP we report the total CPU time spent on stages 0 to 3 ("tFP"); the time for computing the analytic/Chebyshev center ("tAC/tCC"); the stage in which the feasible point is found ("stage"); and the gap between the feasible and the optimal solution ("gap%").For the objective FP, we report columns "gap%", "stage" and "tFP" with the same meaning as before.The primary goal of this preliminary study is to assess the benefits, if any, of using the Chebyshev center as an alternative to the analytic center.We start by comparing AC-FP with CC-FP.Looking at Table 4, we see 15 instances where AC-FP fails . The Chebyshev center as an alternative to the analytic center… The Chebyshev center as an alternative to the analytic center… The Chebyshev center as an alternative to the analytic center… (i.e., it requires stage 3).In five of those 15 instances (33.3%),CC-FP finds a feasible solution ("aflow40b", "harp2", "nsrand-ipx", "protfold" and "tr12-30").In contrast, CC-FP fails in 12 instances.In two of those instances (16.7%),AC-FP finds a feasible solution ("ds" and "nw04").Finally, in 14 of the 35 instances (40%) where both methods find a feasible solution, CC-FP obtains a solution with a lower gap than AC-FP.In another eight instances CC-FP obtains the same gap as AC-FP.Given that the total computational time is really low in both methods (less than one minute on average), CC-FP proves to be a good alternative to AC-FP.The Chebyshev center as an alternative to the analytic center… Next, we focus on comparing the objective FP heuristic with CP-FP (choosing the best option between AC-FP and CC-FP).Table 5 presents the results obtained, with the column "tCP" showing the time needed for computing either AC or CC.In 39 instances, both CP-FP and objective FP find a feasible solution.In six instances ("cap6000", "danoint", "mkc", "msc98-ip", "roll3000" and "swath"), representing a 15.4% of the cases, CP-FP improves the quality of the feasible solution achieved.Furthermore, when the objective FP fails in nine instances ("arki001", "atlanta-ip", "glass4", "mzzv11", "p2756", "protfold", "roll3000", "swath" and "timtab2"), CP-FP finds a feasible solution in three of them ("protfold", "roll3000" and "swath").It is noteworthy that CC-FP efficiently solves the instance "protfold" when both AC-FP and objective FP fail.We also observe that in the six instances where all methods fail ("arki001", "atlanta-ip", "glass4", "mzzv11", "p2756" and "timtab2"), CP-FP obtains a better feasible solution in two of them ("atlanta-ip" and " timtab2"); and in another two ("mzzv11" and "p2756") it provides the same solution as objective FP.
Second, we provide a similar comparison between AC-FP, CC-FP and objective FP for a subset of MIPLIB2010 instances.The original subset contains 215 instances, but 38 of them are removed because either (i) none of the methods find a feasible solution within the one hour time limit; or (ii) they exhaust the available memory.Tables 6 and 7 show the results obtained with the remaining 177 instances.Comparing AC-FP against CC-FP, we note that in four of the 70 instances (5.7%)where AC-FP fails, CC-FP finds a feasible solution.On the other hand, in seven of the 73 instances (9.6%)where CC-AC fails, AC-FP is able to find a feasible solution.It is worth noting that CC-FP gives a higher quality solution in 36 of the 100 instances (36%) in which both methods successfully find a feasible point.In another 37 instances, AC-FP and CC-FP find points with the same objective function.Therefore, in a total of 73 out of 100 instances The Chebyshev center as an alternative to the analytic center… (73%) CC-FP obtains an equal or better result than AC-FP.These results show that the Chebyshev center can be a good alternative to the analytic center for FP variants.Finally, Tables 8 and 9 show results comparing objective FP with the best option between CC-FP or AC-FP (named CP-FP in these tables).From Tables 8 and 9 we can state that: (i) in 19 of the instances where objective FP fails, CP-FP finds a feasible solution; and (ii) CP-FP obtains a better solution than objective FP in 26% of the cases where all methods successfully end (and in 5.4% of the cases, the solutions have the same objective function).Table 10 summarizes the overall results for all the MIPLIB 2003 and 2010 instances, comparing AC-FP vs CC-FP in subtable (a), and the best between AC-FP and CC-FP (referred to as CP-FP) vs objective FP in subtable (b).The first two rows of each subtable provide, for each method, the percentage of successfully solved instances (i.e., a feasible solution is obtained by the heuristic before stage 3) and failures (i.e., stage 3 is reached).Looking at subtable (a) we notice that both AC-FP and CC-FP solve the same number of instances, although the particular set of instances solved by each method is different.In subtable (b) we see that CP-FP (either AC or CC) solves 4% more of instances than objective FP.The last two rows of subtable (a) show that CC-FP provides a solution of better gap than AC-FP in 3.5% more instances; in 19.65% of the instances both AC-FP and CC-FP report a solution of same gap.This information is also given in the last two rows of subtable (b) comparing CP-FP vs objective FP: it is seen that objective FP provides better gaps than CP-FP in many more cases.However, CP-FP is able to compute a solution in 26% of the instances that are not solved by objective FP.The Chebyshev center as an alternative to the analytic center…    The Chebyshev center as an alternative to the analytic center…  The Chebyshev center as an alternative to the analytic center…   The Chebyshev center as an alternative to the analytic center…   The Chebyshev center as an alternative to the analytic center…  The Chebyshev center as an alternative to the analytic center… 1 3 The Chebyshev center as an alternative to the analytic center…

Conclusions
We propose using the Chebyshev center as an alternative center point to the analytic center in the successful FP heuristic.Our extensive computational results show that the CC-FP variant is competitive in some instances.Furthermore we have also shown that, in theory, the Chebyshev center might provide important benefits when the MILP problem has many redundant constraints.Although CP-FP does not always outperform objective FP, using a center point within FP has been shown to provide a competitive advantage in other FP variants that complement CP-FP, such as in [6].In those cases, using CC instead of AC can provide better and faster feasible points.Developing a decision tool to choose a priori the best center point to use within FP could form a part of further work to be done in this field.

Fig. 5
Fig. 5 The Chebyshev ( x cc ) and analytic ( x ac ) centers of polyhedron Q represented by Qx ≤ b (without redundant constraints).The largest inscribed Euclidean ball centered at x cc is also shown

7
Computational results using AC-FP, CC-FP and objective FP for a subset of MILP instances from MIPLIB 2010 (Part II) while x ′ ac has been pushed out towards the boundary opposite to the redundant constraints.The FP variant based on Chebyshev centers introduced in this work is named Chebyshev center FP (CC-FP).

Table 1
Characteristics of the subset of MILP instances from MIPLIB 2003

Table 2
Characteristics of the subset of MILP instances from MIPLIB 2010 (Part I)

Table 3
Characteristics of the subset of MILP instances from MIPLIB 2010 (Part II)

Table 4
Computational results using AC-FP, CC-FP and objective FP for a subset of MILP instances

Table 6
Computational results using AC-FP, CC-FP and objective FP for a subset of MILP instances from MIPLIB 2010 (Part I)

Table 8
Computational results using the best option between AC-FP and CC-FP (CP-FP) against objective FP for a subset of MILP instances from MIPLIB 2010 (Part I)

Table 9
Computational results using the best option between AC-FP and CC-FP (CP-FP) against objective FP for a subset of MILP instances from MIPLIB 2010 (Part II)

Table 10
Summary tables for all MIPLIB 2003 and 2010 instances Subtable (a) compares AC-FP vs CC-FP.Subtable (b) compares the best between AC-FP and CC-FP vs objective FP