Faster, but weaker, relaxations for quadratically constrained quadratic programs

We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve m...

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Published in:Computational optimization and applications Vol. 59; no. 1-2; pp. 27 - 45
Main Authors: Burer, Samuel, Kim, Sunyoung, Kojima, Masakazu
Format: Journal Article
Language:English
Published: Boston Springer US 01.10.2014
Springer Nature B.V
Subjects:
Analysis
Blocking
Computation
Computer science
Constraints
Convex and Discrete Geometry
Linear programming
Management Science
Mathematical analysis
Mathematical programming
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Programming
Quadratic programming
Semidefinite programming
Statistics
Studies
Semidefinite programming
Nonconvex quadratic programming
Second-order cone programming
Difference of convex
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Abstract We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than those gotten by SDP. The framework allows one to choose a block-diagonal structure for the mixed SOCP-SDP, which in turn allows one to control the speed and bound quality. For a fixed block-diagonal structure, we also introduce a procedure to improve the bound quality without increasing computation time significantly. The effectiveness of our framework is illustrated on a large sample of QCQPs from various sources.
AbstractList We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than those gotten by SDP. The framework allows one to choose a block-diagonal structure for the mixed SOCP-SDP, which in turn allows one to control the speed and bound quality. For a fixed block-diagonal structure, we also introduce a procedure to improve the bound quality without increasing computation time significantly. The effectiveness of our framework is illustrated on a large sample of QCQPs from various sources.
Issue Title: Methods for Optimization and Equilibrium Problems We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than those gotten by SDP. The framework allows one to choose a block-diagonal structure for the mixed SOCP-SDP, which in turn allows one to control the speed and bound quality. For a fixed block-diagonal structure, we also introduce a procedure to improve the bound quality without increasing computation time significantly. The effectiveness of our framework is illustrated on a large sample of QCQPs from various sources.[PUBLICATION ABSTRACT]
Author Kim, Sunyoung
Kojima, Masakazu
Burer, Samuel
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  surname: Burer
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  givenname: Sunyoung
  surname: Kim
  fullname: Kim, Sunyoung
  organization: Department of Mathematics, Ewha W. University
– sequence: 3
  givenname: Masakazu
  surname: Kojima
  fullname: Kojima, Masakazu
  organization: Research and Development Initiative & JST CREST, Chuo University
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Difference of convex
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Snippet We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on...
Issue Title: Methods for Optimization and Equilibrium Problems We introduce a new relaxation framework for nonconvex quadratically constrained quadratic...
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SubjectTerms Analysis
Blocking
Computation
Computer science
Constraints
Convex and Discrete Geometry
Linear programming
Management Science
Mathematical analysis
Mathematical programming
Mathematics
Mathematics and Statistics
Operations Research
Operations Research/Decision Theory
Optimization
Programming
Quadratic programming
Semidefinite programming
Statistics
Studies
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Title Faster, but weaker, relaxations for quadratically constrained quadratic programs
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