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Optimization methods & software

Feasible direction algorithm for solving the SDP relaxations of quadratic {−1, 1} programming problems

Liu Hongwei*, Xinhui Wang, Liu Sanyang

Published Apr 1, 2004• 13 citations
Publisher

Abstract

In this article, we propose a feasible direction algorithm for solving the semidefinite programming (SDP) relaxations of quadratic {−1, 1} programming problems. This algorithm's distinguishing features are that it uses a low rank factorization and searches with a constant step-size. Its convergence is also proven. Finally, we report some numerical examples to compare our method with the low rank factorization method of Burer and Monteiro on the SDP relaxation of the max-cut problem. *E-mail: xdliuhongwei@hotmail.com

DOI
10.1080/10556780410001647203

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In this article, we propose a feasible direction algorithm for solving the semidefinite programming (SDP) relaxations of quadratic {−1, 1} programming problems. This algorithm's distinguishing features are that it uses a low rank factorization and searches with a constant step-size. Its convergence is also proven. Finally, we report some numerical examples to compare our method with the low rank factorization method of Burer and Monteiro on the SDP relaxation of the max-cut problem. *E-mail: xdliuhongwei@hotmail.com
We consider low-rank semidefinite programming (LRSDP) relaxations of ±1 quadratic problems that can be formulated as the nonconvex nonlinear programming problem of minimizing a quadratic function subject to separable quadratic equality constraints. We prove the equivalence of the LRSDP problem with the unconstrained minimization of a new merit function and we define an effcient and globally convergent algorithm for finding critical points of the LRSDP problem. Finally, we test our code on an extended set of instances of the Max-Cut problem and we report comparisons with other existing codes.
Based on the semidefinite programming relaxation of the binary quadratic programming, a rank-two feasible direction algorithm is presented. The proposed algorithm restricts the rank of matrix variable to be two in the semidefinite programming relaxation and yields a quadratic objective function with simple quadratic constraints. A feasible direction algorithm is used to solve the nonlinear programming. The convergent analysis and time complexity of the method is given. Coupled with randomized algorithm, a suboptimal solution is obtained for the binary quadratic programming. At last, we report some numerical examples to compare our algorithm with randomized algorithm based on the interior point method and the feasible direction algorithm on max-cut problem. Simulation results have shown that our method is faster than the other two methods.
ABSTRACT Quadratically constrained quadratic programming (QCQP) problems appear in a wide range of engineering fields, including computer science, communication engineering, and finance. A key difficulty in solving these problems lies in efficiently finding global solutions, especially for large-scale and nonconvex instances. To address this, Wen & Yin (2013) proposed a method that reformulates semidefinite programming (SDP) relaxations of QCQPs into a nonlinear and nonconvex low-rank problem. This reformulated problem can be efficiently solved using a curvilinear search method combined with Barzilai-Borwein (BB) steps, known as the CSBB algorithm. In this study, we compare two approaches for solving QCQPs: the conventional convex SDP relaxation and the nonconvex low-rank reformulation introduced by Wen and Yin. We propose a set of general QCQP models that are compatible with the low-rank framework and conduct a series of numerical experiments to evaluate their performance. This study evaluates performance using several classes of NP-hard problems, including Binary Integer Quadratic (BIQ) problems, Max-Cut problems, Boolean Least Squares (BLS) problems, and 0-1 Quadratic Knapsack Problems (QKP). The results demonstrate that the low-rank approach offers competitive performance and shows strong potential for solving large-scale QCQPs more efficiently than traditional convex methods.
We study the ternary quadratic problem (TQP), a quadratic optimization problem with linear constraints where the variables take values in {0,±1}\{0, \pm 1\}{0,±1}. While semidefinite programming (SDP) techniques are well established for {0,1}\{0,1\}{0,1}- and {±1}\{\pm 1\}{±1}-valued quadratic problems, no dedicated integer semidefinite programming framework exists for the ternary case. In this paper, we introduce a ternary SDP formulation for the TQP that forms the basis of an exact solution approach. We derive new theoretical insights in rank-one ternary positive semidefinite matrices, which lead to a basic SDP relaxation that is further strengthened by valid triangle, RLT, split and kkk-gonal inequalities. These are embedded in a tailored branch-and-bound algorithm that iteratively solves strengthened SDPs, separates violated inequalities, applies a ternary branching strategy and computes high-quality feasible solutions. We test our algorithm on TQP variations motivated by practice, including unconstrained, linearly constrained and quadratic ratio problems. Computational results on these instances demonstrate the effectiveness of the proposed algorithm.
Following the seminal work of Goemans and Williamson [1], there has been an outgrowth in the use of semidefinite programming (SDP) for designing approximation algorithms. Recall that anα-approximation algorithm for a problem P is a polynomial-time algorithm such that for every instance I of P , it delivers a solution that is within a factor of α of the optimum value [2]. It is well known that SDPs can be solved in polynomial time (up to any prescribed accuracy) via interior-point algorithms (see, e.g., Refs. [3,4]), and they have been used very successfully in the design of approximation algorithms for a host of NP-hard problems, e.g., graph partitioning, graph coloring, and quadratic optimization [1-9], just to name a few.