[Paper Review] Solving unconstrained 0-1 polynomial programs through quadratic convex reformulation

Sourour Elloumi, Amélie Lambert+1 more|arXiv (Cornell University)|Jan 22, 2019

Advanced Optimization Algorithms Research46 references21 citations

TL;DR

This paper proposes PQCR, a three-phase exact method for solving unconstrained binary polynomial programs by reformulating them into convex quadratic programs via quadratic convex reformulation. It achieves superior bounds and optimality gaps on challenging instances, including solving 6 previously unsolved low auto-correlation binary sequence problems and improving bounds on 27 others.

ABSTRACT

We propose a solution approach for the problem (P) of minimizing an unconstrained binary polynomial optimization problem. We call this method PQCR (Polynomial Quadratic Convex Reformulation). The resolution is based on a 3-phase method. The first phase consists in reformulating (P) into a quadratic program (QP). For this, we recursively reduce the degree of (P) to two, by use of the standard substitution of the product of two variables by a new one. We then obtain a linearly constrained binary program. In the second phase, we rewrite the quadratic objective function into an equivalent and parametrized quadratic function using the equality x 2 i = x i and new valid quadratic equalities. Then, we focus on finding the best parameters to get a quadratic convex program which continuous relaxation's optimal value is maximized. For this, we build a semidefinite relaxation (SDP) of (QP). Then, we prove that the standard linearization inequalities, used for the quadratization step, are redundant in (SDP) in presence of the new quadratic equalities. Next, we deduce our optimal parameters from the dual optimal solution of (SDP). The third phase consists in solving (QP *), the optimal reformulated problem, with a standard solver. In particular, at each node of the branch-and-bound, the solver computes the optimal value of a continuous quadratic convex program. We present computational results on instances of the image restoration problem and of the low autocorrelation binary sequence problem. We compare PQCR with other convexification methods, and with the general solver Baron 17.4.1 [39]. We observe that most of the considered instances can be solved with our approach combined with the use of Cplex [24].

Motivation & Objective

• Address the NP-hard problem of minimizing unconstrained binary polynomial programs with non-convex, discrete objectives.
• Develop a tailored convexification method that outperforms general solvers and existing convexification techniques on difficult instances.
• Improve the continuous relaxation bound through a novel semidefinite relaxation tailored to the quadratized problem structure.
• Enable exact global optimization of large-scale binary polynomial programs using standard solvers after optimal reformulation.
• Investigate the impact of different quadratization strategies on the quality of the convex reformulation and solution efficiency.

Proposed method

• Phase 1: Quadratize the original polynomial problem by recursively replacing products of variables with new binary variables, transforming the problem into a linearly constrained binary quadratic program (QP).
• Phase 2: Introduce new valid quadratic equalities (e.g., $x_i^2 = x_i$ and additional identities derived from the quadratization) to parameterize the objective function of QP.
• Construct a new semidefinite programming (SDP) relaxation of QP to compute optimal parameters that maximize the continuous relaxation bound.
• Prove that standard linearization inequalities are redundant when these new quadratic equalities are included, improving the SDP formulation.
• Derive optimal parameters from the dual solution of the SDP relaxation to create an equivalent convex quadratic program (QP*).
• Phase 3: Solve the optimally reformulated convex QP* using a standard branch-and-bound solver, leveraging the convexity for efficient node optimization.

Experimental results

Research questions

• RQ1 Can a tailored quadratic convex reformulation strategy significantly improve the continuous relaxation bound for unconstrained binary polynomial programs?
• RQ2 How does the choice of quadratization method affect the tightness of the resulting convex relaxation and solution performance?
• RQ3 Can the proposed PQCR method outperform general-purpose global solvers like Baron on benchmark instances of binary polynomial programs?
• RQ4 What is the impact of preprocessing to identify 'important' quadratic equalities on the efficiency of the SDP solution phase?
• RQ5 To what extent can PQCR solve previously unsolved instances of the low auto-correlation binary sequence problem?

Key findings

• PQCR solved 6 of the 45 unsolved instances in the low auto-correlation binary sequence (LABS) problem set from minlplib, which were previously unreported as solved.
• For the remaining 27 unsolved or partially solved LABS instances, PQCR improved both primal and dual bounds, demonstrating enhanced bound quality.
• The continuous relaxation bound of PQCR was consistently tighter than those obtained with alternative quadratization methods (PC1, PC2, PC3), with the best results achieved using the proposed quadratization strategy.
• On average, solving the SDP relaxation consumed 75% of the total CPU time, indicating a need for preprocessing to identify critical equalities and accelerate this phase.
• The method significantly outperformed the general-purpose solver Baron in terms of bound improvement and solution quality on the tested instances.
• The authors observed that the number of variables in the quadratized problem (N) and the quality of the SDP bound (LB_i) were strongly influenced by the choice of quadratization, with PC3 showing competitive performance in some cases.