Semidefinite relaxations for quadratically constrained quadratic programming: A review and comparisons

At the intersection of nonlinear and combinatorial optimization, quadratic programming has attracted significant interest over the past several decades. A variety of relaxations for quadratically constrained quadratic programming (QCQP) can be formulated as semidefinite programs (SDPs). The primary...

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Bibliographic Details
Published in:Mathematical programming Vol. 129; no. 1; pp. 129 - 157
Main Authors: Bao, Xiaowei, Sahinidis, Nikolaos V., Tawarmalani, Mohit
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.09.2011
Springer Nature B.V
Subjects:
Calculus of Variations and Optimal Control; Optimization
Combinatorial analysis
Combinatorics
Computation
Constraints
Full Length Paper
Gaps
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical models
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Nonlinearity
Numerical Analysis
Optimization
Production planning
Quadratic programming
Semidefinite programming
Studies
Theoretical
Semidefinite programming
Nonconvex optimization
65K05
90C30
Lagrangian relaxation
90C22
49M29
90C26
Quadratic programming
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:At the intersection of nonlinear and combinatorial optimization, quadratic programming has attracted significant interest over the past several decades. A variety of relaxations for quadratically constrained quadratic programming (QCQP) can be formulated as semidefinite programs (SDPs). The primary purpose of this paper is to present a systematic comparison of SDP relaxations for QCQP. Using theoretical analysis, it is shown that the recently developed doubly nonnegative relaxation is equivalent to the Shor relaxation, when the latter is enhanced with a partial first-order relaxation-linearization technique. These two relaxations are shown to theoretically dominate six other SDP relaxations. A computational comparison reveals that the two dominant relaxations require three orders of magnitude more computational time than the weaker relaxations, while providing relaxation gaps averaging 3% as opposed to gaps of up to 19% for weaker relaxations, on 700 randomly generated problems with up to 60 variables. An SDP relaxation derived from Lagrangian relaxation, after the addition of redundant nonlinear constraints to the primal, achieves gaps averaging 13% in a few CPU seconds.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-011-0462-2