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Semidefinite programming for discrete optimization and matrix completion problems

Semidefinite programming for discrete optimization and matrix completion problems
Semidefinite programming for discrete optimization and matrix completion problems
Semidefinite programming (SDP) is currently one of the most active areas of research in optimization. SDP has attracted researchers from a wide variety of areas because of its theoretical and numerical elegance as well as its wide applicability. In this paper we present a survey of two major areas of application for SDP, namely discrete optimization and matrix completion problems. In the first part of this paper we present a recipe for finding SDP relaxations based on adding redundant constraints and using Lagrangian relaxation.
We illustrate this with several examples. We first show that many relaxations for the max-cut problem (MC) are equivalent to both the Lagrangian and the well-known SDP relaxation. We then apply the recipe to obtain new strengthened SDP relaxations for MC as well as known SDP relaxations for several other hard discrete optimization problems. In the second part of this paper we discuss two completion problems, the positive semidefinite matrix completion problem and the Euclidean distance matrix completion problem. We present some theoretical results on the existence of such completions and then proceed to the application of SDP to find approximate completions.
We conclude this paper with a new application of SDP to find approximate matrix completions for large and sparse instances of Euclidean distance matrices.
semidefinite programming, discrete optimization, lagrangian relaxation, max-cut problem, euclidean distance matrix, matrix completion problem
0166-218X
513-577
Wolkowicz, Henry
3ef6bc24-0956-4eb2-a542-802d82454cf7
Anjos, Miguel F.
0103b24d-cab5-4109-9924-dc9eebd04257
Wolkowicz, Henry
3ef6bc24-0956-4eb2-a542-802d82454cf7
Anjos, Miguel F.
0103b24d-cab5-4109-9924-dc9eebd04257

Wolkowicz, Henry and Anjos, Miguel F. (2002) Semidefinite programming for discrete optimization and matrix completion problems. Discrete Applied Mathematics, 123 (1-3), 513-577. (doi:10.1016/S0166-218X(01)00352-3).

Record type: Article

Abstract

Semidefinite programming (SDP) is currently one of the most active areas of research in optimization. SDP has attracted researchers from a wide variety of areas because of its theoretical and numerical elegance as well as its wide applicability. In this paper we present a survey of two major areas of application for SDP, namely discrete optimization and matrix completion problems. In the first part of this paper we present a recipe for finding SDP relaxations based on adding redundant constraints and using Lagrangian relaxation.
We illustrate this with several examples. We first show that many relaxations for the max-cut problem (MC) are equivalent to both the Lagrangian and the well-known SDP relaxation. We then apply the recipe to obtain new strengthened SDP relaxations for MC as well as known SDP relaxations for several other hard discrete optimization problems. In the second part of this paper we discuss two completion problems, the positive semidefinite matrix completion problem and the Euclidean distance matrix completion problem. We present some theoretical results on the existence of such completions and then proceed to the application of SDP to find approximate completions.
We conclude this paper with a new application of SDP to find approximate matrix completions for large and sparse instances of Euclidean distance matrices.

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Published date: 2002
Keywords: semidefinite programming, discrete optimization, lagrangian relaxation, max-cut problem, euclidean distance matrix, matrix completion problem
Organisations: Operational Research

Identifiers

Local EPrints ID: 29673
URI: http://eprints.soton.ac.uk/id/eprint/29673
ISSN: 0166-218X
PURE UUID: 9c303ffb-45d5-4fcc-ad81-790310cb0e3e

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Date deposited: 15 May 2006
Last modified: 15 Mar 2024 07:33

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Contributors

Author: Henry Wolkowicz
Author: Miguel F. Anjos

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