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SDP_S: a Tool to formulate and solve Semidefinite relaxations for Bivalent Quadratic problems

G raud Delaporte, S bastien Jouteau, Frédéric Roupin

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

SDP_S is a stand-alone program which formulates semidefinite (SDP) relaxations for any 0-1 quadratic problem. Moreover, it can solve the SDP by using the Spectral Bundle Method of C.Helmberg. SDP_S is an implementation of the recipe proposed by F.Roupin in the Cedric technical report RC388-2002. One of the major advantages of SDP_S is that it requires no knowledge about semidefinite programming. Indeed, the input problem has just to be stated as a natural 0-1 quadratic program. The source code (C/C++) and documentation can be found athttp://semidef.free.fr.

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SDP_S is a stand-alone program which formulates semidefinite (SDP) relaxations for any 0-1 quadratic problem. Moreover, it can solve the SDP by using the Spectral Bundle Method of C.Helmberg. SDP_S is an implementation of the recipe proposed by F.Roupin in the Cedric technical report RC388-2002. One of the major advantages of SDP_S is that it requires no knowledge about semidefinite programming. Indeed, the input problem has just to be stated as a natural 0-1 quadratic program. The source code (C/C++) and documentation can be found athttp://semidef.free.fr.
Many computer vision problems can be formulated as binary quadratic programs (BQPs). Two classic relaxation methods are widely used for solving BQPs, namely, spec-tral methods and semidefinite programming (SDP), each with their own advantages and disadvantages. Spectral relaxation is simple and easy to implement, but its bound is loose. Semidefinite relaxation has a tighter bound, but its computational complexity is high for large scale prob-lems. We present a new SDP formulation for BQPs, with two desirable properties. First, it has a similar relaxation bound to conventional SDP formulations. Second, com-pared with conventional SDP methods, the new SDP for-mulation leads to a significantly more efficient and scalable dual optimization approach, which has the same degree of complexity as spectral methods. Extensive experiments on various applications including clustering, image segmen-tation, co-segmentation and registration demonstrate the usefulness of our SDP formulation for solving large-scale BQPs. 1.
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.
This repository contains the source code, computational experiments, analysis, and benchmarks accompanying the paper A Direct SDP-Based Framework for Certified Bounds for Constrained Quadratic Binary Optimization Problems. The repository implements a direct semidefinite programming (SDP) approach for constrained quadratic binary optimization, designed to obtain strong bounds for exact optimization. The SDP relaxation is strengthened using quadratized linear constraints and triangle (hypermetric) inequalities. To address the numerical infeasibility of SDP solutions obtained from numerical solvers such as MOSEK, the framework includes a correction procedure that converts approximate SDP solutions into mathematically certified bounds suitable for use in Branch-and-Bound algorithms. The repository also implements an adaptation of the Goemans–Williamson randomized hyperplane rounding scheme to constrained binary optimization, together with problem-specific feasibility restoration. Computational experiments are provided for the Sparsest k-Subgraph and Maximum Independent Set problems. The work presented in this repository was funded and developed under the QBIQ project (project no. 644), funded by the Slovenian Research and Innovation Agency (ARIS) through the Applied Projects programme. The code and accompanying data are intended to facilitate reproducibility of the computational results and provide a foundation for further research on exact SDP-based methods for constrained quadratic binary optimization.
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.
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
Abstract: We consider the constraint satisfaction problem (CSP), where the values must be assigned to variables which are subject to a set of constraints. This problem is naturally formulated as 0-1 quadratic knapsack problem subject to quadratic constraint. In this paper, we present a branch-and-bound algorithm for 0-1 quadratic programming, which is based on solving semidefinite relaxations. At each node of the enumeration tree, a lower bound is given naturally by the value of (SDP) problem and an upper bound is computed by satisfying the quadratic constraint. We show that this method is able to determine whether a (CSP) has a solution or not. Then we give some hints on how to reduce as much as possible the initial size of the (CSP). Some numerical examples assess the effectiveness of the theoretical results shown in this paper, and the advantage of the new modelization.