Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets

Kojima, M.; Tunçel, Levent

doi:10.1137/s1052623498336450

Cited by 65 publications

(65 citation statements)

References 16 publications

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“…In our case we add quadratic constraints. The idea of quadratic valid inequalities has been used in [10]; and closing the duality gap has been discussed in [20].…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

On Lagrangian Relaxation of Quadratic Matrix Constraints

Anstreicher¹,

Wolkowicz²

2000

SIAM J. Matrix Anal. & Appl.

115

107

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Abstract. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations.For several special cases of QQP, e.g., convex programs and trust region subproblems, the Lagrangian relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective.In this paper we consider a certain QQP where the quadratic constraints correspond to the matrix orthogonality condition XX T = I. For this problem we show that the Lagrangian dual based on relaxing the constraints XX T = I and the seemingly redundant constraints X T X = I has a zero duality gap. This result has natural applications to quadratic assignment and graph partitioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix. We also show that the technique of relaxing quadratic matrix constraints can be used to obtain a strengthened semidefinite relaxation for the max-cut problem.Key words. Lagrangian relaxations, quadratically constrained quadratic programs, semidefinite programming, quadratic assignment, graph partitioning, max-cut problems AMS subject classifications. 49M40, 52A41, 90C20, 90C27 PII. S0895479898340299 1. Introduction. Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. They often provide a much improved model compared to the simpler linear relaxation of a problem. However, very large linear models can be solved efficiently, whereas QQPs are in general NP-hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Moreover these relaxations can be shown to be equivalent to semidefinite programming (SDP) relaxations, and SDP problems can be solved efficiently, i.e., they are polynomial time problems; see, e.g., [31].SDP relaxations provide a tractable approach for finding good bounds for many hard combinatorial problems. The best example is the application of SDP to the max-cut problem, where a 87% performance guarantee exists [11,12]. Other examples include matrix completion problems [23,22], as well as graph partitioning problems and the quadratic assignment problem (references given below).In this paper we consider several quadratically constrained quadratic (nonconvex) programs arising from hard combinatorial problems. In particular, we look at the orthogonal relaxations of the quadratic assignment and graph partitioning problems. We show that the resulting well-known eigenvalue bounds for these problems can be obtained from the Lagrangian dual of the orthogonally constrained relaxations,

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“…In our case we add quadratic constraints. The idea of quadratic valid inequalities has been used in [10]; and closing the duality gap has been discussed in [20].…”

Section: Discussionmentioning

confidence: 99%

“…One can relate the geometry of the original feasible set of QQP with the feasible set of the SDP relaxation. The connection is through valid quadratic inequalities, i.e., nonnegative (convex) combinations of the quadratic functions; see [10,20].…”

Section: General Qqpmentioning

confidence: 99%

On Lagrangian Relaxation of Quadratic Matrix Constraints

Anstreicher¹,

Wolkowicz²

2000

SIAM J. Matrix Anal. & Appl.

115

107

View full text Add to dashboard Cite

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“…See also Balas [5] and Sherali and Shetti [35] on disjunctive programming methods. Kojima and Tunçel [18] give successive semidefinite relaxations converging to the convex hull of a nonconvex set defined by quadratic functions. Lasserre [19] describes a hierarchy of semidefinite relaxations of nonlinear 0-1 programs.…”

Section: Conic Integer Programmingmentioning

confidence: 99%

Conic mixed-integer rounding cuts

2008

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Abstract. A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints.Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve either secondorder conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree.Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming.We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean-variance capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability.

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“…The SCRM (Successive Convex Relaxation Method) proposed by Kojima-Tuncel [7] is a powerful numerical method to compute upper bounds of general QOPs by repeated applications of SDP (semidefinite programming) relaxations. The SCRM generates and solves a large number of SDP problems at each iteration.…”

Section: Parallel Successive Convex Relaxation Methods Formentioning

confidence: 99%

High performance grid and cluster computing for some optimization problems

Kojima

Yamashita

2004

2004 International Symposium on Applications and the Internet Workshops. 2004 Workshops.

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Abstract.Solving large scale optimization problems requires a huge amount of computational power. The size of optimization problems that can be solved on a few CPUs has been limited due to a lack of computational power. Grid and cluster computing has received much attention as a powerful and inexpensive way of solving large scale optimization problems that an existing single-unit CPU cannot process. The aim of this paper is to show that grid and cluster computing provides tremendous power to optimization methods. The methods that this article picks up are a successive convex relaxation method for quadratic optimization problems, a polyhedral homotopy method for polynomial systems of equations and a primal-dual interior-point method for semidefinite programs. Their parallel implementations on grids and clusters together with numerical results are reported. The article also mentions a grid portal system for optimization problems briefly.

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Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets

Cited by 65 publications

References 16 publications

On Lagrangian Relaxation of Quadratic Matrix Constraints

On Lagrangian Relaxation of Quadratic Matrix Constraints

Conic mixed-integer rounding cuts

High performance grid and cluster computing for some optimization problems

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