A branch-and-cut algorithm for solving mixed-integer semidefinite optimization problems

Kobayashi, Kurima; Takano, Yasunari

doi:10.1007/s10589-019-00153-2

Cited by 21 publications

(27 citation statements)

References 42 publications

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“…In [33], Lubin et al proposed a cutting plane framework for mixed integer convex optimization problems. In [26], Kobayashi and Takano proposed a branch and bound cutting plane method for mixed integer SDPs. It would be interesting to see whether the approximations of S n + proposed in this paper could be used to improve the efficiency of those methods.…”

Section: Discussionmentioning

confidence: 99%

“…Many approximations of S n + have been proposed on the basis of its well-known properties. Kobayashi and Takano [26] used the fact that the diagonal elements of semidefinite matrices are nonnegative. Konno et al [27] imposed an assumption that all diagonal elements of the variable X in the SDPs appearing in their iterative algorithm are bounded by a constant.…”

Section: Introductionmentioning

confidence: 99%

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Polyhedral approximations of the semidefinite cone and their application

2021

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We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Polyhedral approximations of the semidefinite cone and their application

2021

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“…This is remarkable, as the mixture of positive semidefiniteness and integrality leads naturally to a broad range of applications, e.g., in architecture [14,67], signal processing [37,53] and combinatorial optimization [38,56]. For a more detailed overview of applications of ISDPs, we refer the reader to [38,44].…”

Section: Introductionmentioning

confidence: 99%

“…They show that strict duality of the relaxations is maintained in the B&B tree and study several solver components. Alternatively, Kobayashi and Takano [44] propose a cutting-plane algorithm that initially relaxes the positive semidefinite (PSD) constraint and solves a mixed integer linear programming problem, where the PSD constraint is imposed dynamically via cutting planes. This leads to a general branch-and-cut (B&C) algorithm for solving MISDPs.…”

Section: Introductionmentioning

confidence: 99%

“…A third project that encounters general ISDPs is YALMIP [47]. However, it is noted by the authors of [38] and [44] that the branch-and-bound ISDP solver in YALMIP is not yet competitive to the performance of the other two methods. Recently, Matter and Pfetsch [48] study different presolving strategies for MISDPs for both the B&B and B&C approach.…”

Section: Introductionmentioning

confidence: 99%

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The Chvátal-Gomory Procedure for Integer SDPs with Applications in Combinatorial Optimization

Meijer¹,

Sotirov²

2022

Preprint

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In this paper we study the well-known Chvátal-Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of spectrahedra is derived by exploiting total dual integrality for SDPs. Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut framework for ISDPs. Different from existing algorithms in the literature, the separation routine in our approach exploits both the semidefinite and the integrality constraints. We provide separation routines for several common classes of binary SDPs resulting from combinatorial optimization problems. In the second part of the paper we present a comprehensive application of our approach to the quadratic traveling salesman problem (QTSP). Based on the algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model the QTSP are introduced. We show that the CG cuts resulting from these formulations contain several well-known families of cutting planes. Numerical results illustrate the practical strength of the CG cuts in our branch-and-cut algorithm, which outperforms alternative ISDP solvers and is able to solve large QTSP instances to optimality.

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The Chvátal–Gomory procedure for integer SDPs with applications in combinatorial optimization

de Meijer,

Sotirov

2024

Math. Program.

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In this paper we study the well-known Chvátal–Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of spectrahedra is derived by exploiting total dual integrality for SDPs. Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut framework for ISDPs. Different from existing algorithms in the literature, the separation routine in our approach exploits both the semidefinite and the integrality constraints. We provide separation routines for several common classes of binary SDPs resulting from combinatorial optimization problems. In the second part of the paper we present a comprehensive application of our approach to the quadratic traveling salesman problem (QTSP). Based on the algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model the QTSP are introduced. We show that the CG cuts resulting from these formulations contain several well-known families of cutting planes. Numerical results illustrate the practical strength of the CG cuts in our branch-and-cut algorithm, which outperforms alternative ISDP solvers and is able to solve large QTSP instances to optimality.

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A branch-and-cut algorithm for solving mixed-integer semidefinite optimization problems

Cited by 21 publications

References 42 publications

Polyhedral approximations of the semidefinite cone and their application

Polyhedral approximations of the semidefinite cone and their application

The Chvátal-Gomory Procedure for Integer SDPs with Applications in Combinatorial Optimization

The Chvátal–Gomory procedure for integer SDPs with applications in combinatorial optimization

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