Zhang, semidefinite relaxation of quadratic optimization problems, in ieee sp magazine, special issue on convex. Semidefinite programming sdp is a subfield of convex optimization concerned with the optimization of a linear objective function a userspecified function that the user wants to minimize or maximize over the intersection of the cone of positive semidefinite matrices with an affine space, i. Optimization online centering admm for the semidefinite. An effective algorithm for quadratic optimization with non. Two classic relaxation methods are widely used for solving bqps, namely. Polynomial optimization, quadratic optimization, semidefinite programming relaxation, sparsity. These include semidefinite programs, parametric trust region problems and concave quadratic maximization. They appear in a wide variety of applications in computer vision, such as image segmentationpixel labelling, image registrationmatching, image denoisingrestoration.
The final optimization problem 12 constitutes a convex quadratic program qp in terms of the transmitter currents ct. In particular, we study a vector lifting relaxation and compare it to a signi. Optimization online a new relaxation framework for. We study quadratically constrained quadratic programming qcqp, i. A geometric interpretation is the optimization of a linear functional, over the intersection of an af. Quadratic maximization and semidefinite relaxation. The worst case running time of these methods is exponential in n, making it impractical for problems of large size. Problems of the form qp are natural models that arise in a variety of settings. In this paper we study the quality of semidefinite relaxation for a global quadratic optimization problem with diagonal quadratic consraints. In this paper we study the approximation algorithms for a class of discrete quadratic. Therefore, relaxations such as the lagrangian relaxation are used. A fast semidefinite approach to solving binary quadratic problems. For sp, may 2010 1 semidefinite relaxation of quadratic optimization problems zhiquan luo, wingkin ma, anthony mancho so, yinyu ye, and shuzhong zhang.
Semidefinite programming relaxations of nonconvex quadratic. Department of mathematical and computing sciences, tokyo. A semidefinite relaxation scheme for multivariate quartic. Semidefinite programming relaxations for semialgebraic. Semidefinite relaxation and nonconvex quadratic optimization. One apparent diculty in analyzing sdp relaxationbased algorithms for problems 1 and 2 is that the usual goemanswilliamson analysis 6,7 and its variants thereof only provides a termbyterm estimate of the objective. This is a fairly broad class, including problems with a combination of continuous and discrete variables, and easily seen to be nphard in the general case. Sdp relaxations for quadratic optimization problems derived from.
Semidefinite programming relaxation for nonconvex quadratic programsdiscrete and continuous structures in optimization. Apr 15, 2010 semidefinite relaxation of quadratic optimization problems abstract. N2 we present a general semidefinite relaxation scheme for general nvariate quartic polynomial optimization under homogeneous quadratic constraints. In this paper we study a class of quadratic maximization problems and their semidefinite programming sdp relaxation. Torr2 abstractin computer vision, many problems can be formulated as binary quadratic programs bqps, which are in general np hard. Semidefinite relaxation of quadratic optimization problems article pdf available in ieee signal processing magazine 273. Quadratic assignment problems qaps are among the hardest discrete optimization problems. Request pdf incorporating prior information into semidefinite relaxation of quadratic optimization problems this paper focuses on equalization as a representative of the large class of. Semidefinite relaxation bounds for indefinite homogeneous. Semidefinite relaxation approximation for multivariate bi. Pdf semidefinite relaxation of quadratic optimization problems. A special case of the problem that we study corresponds to the max3cut model used in a recent paper of goemans and williamson j. For a special subclass of the problems we show that the sdp relaxation provides an exact optimal solution.
A special case of the problem that we study corresponds to. Parrilo in this lecture, we will discuss one of the most important applications of semide. In particular, we study the limit of that model, in. Recent study shows that even obtaining a strong lower bound for qaps is a computational challenge. Largescale binary quadratic optimization using semide. Quadratically constrained quadratic programs, denoted q 2p, are an important modelling tool, e. T1 a semidefinite relaxation scheme for multivariate quartic polynomial optimization with quadratic constraints. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
This paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its semidefinite programming sdp relaxation. Qcqps subsume a wide variety of nphard optimization problems, and. Citeseerx semidefinite relaxation bounds for indefinite. A special case of the problem that we study corresponds to the max3cut model used. Semidefinite relaxationbased optimization of multipleinput. It is shown that the proposed polynomial time convex con ditions are at least as strong as the standard ones, and usually better, but at a higher computational cost. Using lagrangian duality, we prove equivalence of the relaxations in a unified and. Article pdf available in ieee signal processing magazine 273. Quadratic functions, optimization, and quadratic forms. Convex and more specifically semidefinite programming relaxations of combinatorial optimization problems have attracted the attention of many researchers see, e. A recipe for semidefinite relaxation for 0,1quadratic programming article pdf available in journal of global optimization 71. Quadratic maximization and semidefinite relaxation springerlink. Roughly speaking, sdr is a powerful, computationally efficient.
A key technique in signal processing and communications wingkin ken ma anthony mancho so dept. Semidefinite relaxation of quadratic optimization problems ieee. Semidefinite programming relaxation for nonconvex quadratic programsdiscrete and continuous structures in. Furthermore, if the objective matrix is realvalued positive semidefinite with nonpositive offdiagonal elements, then the performance ratio improves to 0. Recently, there were several results on solving different forms of quadratic problems. Semidefinite relaxation of quadratic optimization problems abstract. For example, consider the problem of approximately solving. Semidefinite symplexmethod for solving the quadratic. Centering admm for the semidefinite relaxation of the qap. Semidefinite programming relaxation for nonconvex title. All relaxations that we consider lead to efficiently solvable problems. A new relaxation framework for quadratic assignment problems based on matrix splitting. As extensions, we apply the new approach to the relaxation problem of the sparse principal component analysis and the nonconvex. Keywords convex optimization integer quadratic programming mixedinteger.
Convex quadratic and semidefinite programming relaxations. Semidefinite relaxation based branchandbound method for. Semidefinite relaxation of quadratic optimization problems 2010. These relaxations result in the interesting, special, case where only the dual problem of the sdp relaxation has strict interior, i. Semidefinite symplexmethod for solving the quadratic optimization problems we propose a new semidefinite simplexmethod for solving the semidefinite optimization problems. Therefore, relaxations suc h as the lagrangian relaxation are used. In this paper, a general quadratic problem is transformed to a linear semidefinite one using a semidefinite relaxation.
In this article, we have provided general, comprehensive coverage of the sdr technique, from its practical deployments and scope of applicability to key theoretical results. Convex quadratic and semidefinite programming relaxations in. This paper studies the relationship between the socalled bi quadratic optimization problem and its semidefinite programming sdp relaxation. Pdf complex quadratic optimization and semidefinite. It dominates the state of theart sdpbased bound for qpl1. Semidefinite programming sdp relaxations for the quadratic assignment problem qap are derived using the dual of the homogenized lagrangian dual of appropriate equivalent representations of qap. Largescale binary quadratic optimization using semidefinite. A disadvantage of the sdp semidefinite programming relaxation method for quadratic andor combinatorial optimization problems lies in its expensive computational cost.
However, it does lead to provably approximation solutions for certain type of quadratic optimization problems, see 1,12,20, which motivates us to extend the existing methods for quadratic optimization problems to bi quadratic optimization. Abstract not available bibtex entry for this abstract preferred format for this abstract see preferences. We propose a new method for solving the semidefinite sd relaxation of the quadratic assignment problem qap, called the centering admm. A new semidefinite programming relaxation scheme for a class of quadratic matrix problems amir beck, yoel drori, marc teboulle computer science, mathematics.
Quadratically constrained quadratic programs, denoted q 2 p, are an imp ortan t mo delling to ol, e. A tight bound for the boolean quadratic optimization problem and its use in a branch and bound algorithm. We take a systematic look at various conic relaxations of q a p. Copositive and semidefinite relaxations of the quadratic. We then present several specific applications in section 4. In this article, we have provided general, comprehensive coverage of the. Second, to ensure that the obtained solution is related to an ellipse, we introduce a special quadratic equality constraint into the aforementioned cls model, which results in the nonconvex quadratically constrained quadratic programming problem. Management the chinese university of hong kong the chinese university of hong kong hong kong hong kong eusipco 2011 tutorial, aug 29, 2011. This paper proposes a socp secondordercone programming relaxation method.
Semidefinite programming relaxations of nonconvex quadratic optimization. Semidefinite relaxation bounds for biquadratic optimization. Finally, we derive the semidefinite relaxation version of the aforementioned problem in terms of. Quadratically constrained quadratic programs, denoted q 2 p, are an important modelling tool, e. Sdp relaxation of homogeneous quadratic optimization. The following nonconvex problem is then a relaxation of 1. Because sdp is solvable via interior point methods, most of these applications can usually be solved very e. Optimization online complex quadratic optimization and. A fast semidefinite approach to solving binary quadratic. Complex quadratic optimization and semidefinite programming. A semidefinite programming method for integer convex. In this paper we introduce a new approach to the formulation of computable relax. This provides a recipe for obtaining a quadratic relaxation, as well as the semidefinite relaxation, for hard combinatorial problems.
Nov 27, 2014 binary quadratic programs bqps are a class of combinatorial optimization problems with binary variables, quadratic objective function and linear quadratic constraints. Robust ellipse fitting via halfquadratic and semidefinite. We present a convex conic relaxation for a problem of maximizing an inde nite quadratic form over a set of convex constraints on. Semidefinite relaxation based branchandbound method for nonconvex quadratic programming by sha hu submitted to the school of engineering on august 11, 2006, in partial fulfillment of the requirements for the degree of master of science in computation for design and optimization abstract in this thesis, we use a semidefinite relaxation based.
We prove that such relaxation approximates the exact solution of the problem with relative accuracy. In the sdp relaxation of quadratic optimization, x ax is rewritten as a x with x xx,x 0, and then discard the rank restriction. By a similar technique to that used in quadratic optimization, the biquadratic optimization problems 1and2 are relaxed to the following bilinear sdp. The centering admm is an alternating direction method of multipliers admm combining the centering. Semidefinite relaxation of quadratic optimization problems. Semidefinite relaxation bounds for indefinite homogeneous quadratic optimization. The quadratic assign ment problem, qap, is treated in section 4. Roughly speaking, sdr is a powerful, computationally efficient approximation technique. Many hard combinatorial problems can be formulated as a quadratically constrained quadratic program qcqp with matrix variables. By simai he, zhiquan luo, jiawang nie and shuzhong zhang.
A recipe for semidefinite relaxation for 0,1quadratic. K0rner, e 1992, remarks on a difficult test problem for quadratic boolean programming. Ab in this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the hermitian complex form. Exact semidefinite formulations for a class of random and non. Semidefinite relaxation of quadratic optimization problems and. Semidefinite programming relaxations for semialgebraic problems. A recipe for semidefinite relaxation for 0,1 quadratic programming in memory of svata poljak. In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the hermitian complex form. The main contributions of the paper are the following. Since it is hard to optimize over this cone, we also. We first show that q a p can equivalently be formulated as a linear program over the cone of completely positive matrices. On approximating complex quadratic optimization problems via. This chapter introduces a semidefinite programming sdp relaxation procedure for this class of quadratic optimization problems which can generate a provably approximately optimal solution with a. It dominates the stateoftheart sdpbased bound for qpl1.