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Journal of Applied Mathematics

A Rank-Two Feasible Direction Algorithm for the Binary Quadratic Programming

Xuewen Mu, Yaling Zhang

Published Jan 1, 2013• 1 citation
Publisher

Abstract

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.

DOI
10.1155/2013/963563

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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.
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
The Goemans--Williamson randomized algorithm guarantees a high-quality approximation to the MAX-CUT problem, but the cost associated with such an approximation can be excessively high for large-scale problems due to the need for solving an expensive semidefinite relaxation. In order to achieve better practical performance, we propose an alternative, rank-two relaxation and develop a specialized version of the Goemans--Williamson technique. The proposed approach leads to continuous optimization heuristics applicable to MAX-CUT as well as other binary quadratic programs, for example the MAX-BISECTION problem. A computer code based on the rank-two relaxation heuristics is compared with two state-of-the-art semidefinite programming codes that implement the Goemans--Williamson randomized algorithm, as well as with a purely heuristic code for effectively solving a particular MAX-CUT problem arising in physics. Computational results show that the proposed approach is fast and scalable and, more importantly, attains a higher approximation quality in practice than that of the Goemans--Williamson randomized algorithm. An extension to MAX-BISECTION is also discussed, as is an important difference between the proposed approach and the Goemans--Williamson algorithm; namely, that the new approach does not guarantee an upper bound on the MAX-CUT optimal value.
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.
-Let (QP) be a binary quadratic program that consists in minimizing a quadratic function subject to linear constraints. To solve (QP) we reformulate it into an equivalent program with a convex objective function. Our reformulation, that we call EQCR (Extended Quadratic Convex Reformulation), is optimal from the continuous relaxation bound point of view. We show that this best reformulation can be deduced from the solution of a semidefinite relaxation of (QP) and that EQCR outperforms QCR. We carry out computational experiments on Max-Cut and on the k-cluster problem.
In computer vision, many problems can be formulated as binary quadratic programs (BQPs), which are in general NP hard. Finding a solution when the problem is of large size to be of practical interest typically requires relaxation. Semidefinite relaxation usually yields tight bounds, but its computational complexity is high. In this work, we present a semidefinite programming (SDP) formulation for BQPs, with two desirable properties. First, it produces similar bounds to the standard SDP formulation. Second, compared with the conventional SDP formulation, the proposed SDP formulation leads to a considerably more efficient and scalable dual optimization approach. We then propose two solvers, namely, quasi-Newton and smoothing Newton methods, for the simplified dual problem. Both of them are significantly more efficient than standard interior-point methods. Empirically the smoothing Newton solver is faster than the quasi-Newton solver for dense or medium-sized problems, while the quasi-Newton solver is preferable for large sparse/structured problems.