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Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations

Authors:

Sunyoung Kim,

Masakazu KojimaAuthors Info & Claims

Computational Optimization and Applications, Volume 26, Issue 2

Pages 143 - 154

https://doi.org/10.1023/A:1025794313696

Published: 01 November 2003 Publication History

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Abstract

We show that SDP (semidefinite programming) and SOCP (second order cone programming) relaxations provide exact optimal solutions for a class of nonconvex quadratic optimization problems. It is a generalization of the results by S. Zhang for a subclass of quadratic maximization problems that have nonnegative off-diagonal coefficient matrices of quadratic objective functions and diagonal coefficient matrices of quadratic constraint functions. A new SOCP relaxation is proposed for the class of nonconvex quadratic optimization problems by extracting valid quadratic inequalities for positive semidefinite cones. Its effectiveness to obtain optimal values is shown to be the same as the SDP relaxation theoretically. Numerical results are presented to demonstrate that the SOCP relaxation is much more efficient than the SDP relaxation.

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Cited By

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Qiu YYıldırım E(2024)On exact and inexact RLT and SDP-RLT relaxations of quadratic programs with box constraintsJournal of Global Optimization10.1007/s10898-024-01407-y90:2(293-322)Online publication date: 1-Oct-2024
https://dl.acm.org/doi/10.1007/s10898-024-01407-y
Locatelli M(2024)A new technique to derive tight convex underestimators (sometimes envelopes)Computational Optimization and Applications10.1007/s10589-023-00534-887:2(475-499)Online publication date: 1-Mar-2024
https://dl.acm.org/doi/10.1007/s10589-023-00534-8
Suppakitpaisarn VHao J(2024)Utilizing Graph Sparsification for Pre-processing in Max Cut QUBO SolverMetaheuristics10.1007/978-3-031-62912-9_22(219-233)Online publication date: 4-Jun-2024
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Index Terms

Exact Solutions of Some Nonconvex Quadratic Optimization Problems via SDP and SOCP Relaxations
1. Mathematics of computing
  1. Mathematical analysis
    1. Mathematical optimization
      1. Continuous optimization
        Quadratic programming
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Continuous optimization
        Quadratic programming

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cover image Computational Optimization and Applications

Computational Optimization and Applications Volume 26, Issue 2

November 2003

102 pages

ISSN:0926-6003

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Kluwer Academic Publishers

United States

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Published: 01 November 2003

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Qiu YYıldırım E(2024)On exact and inexact RLT and SDP-RLT relaxations of quadratic programs with box constraintsJournal of Global Optimization10.1007/s10898-024-01407-y90:2(293-322)Online publication date: 1-Oct-2024
https://dl.acm.org/doi/10.1007/s10898-024-01407-y
Locatelli M(2024)A new technique to derive tight convex underestimators (sometimes envelopes)Computational Optimization and Applications10.1007/s10589-023-00534-887:2(475-499)Online publication date: 1-Mar-2024
https://dl.acm.org/doi/10.1007/s10589-023-00534-8
Suppakitpaisarn VHao J(2024)Utilizing Graph Sparsification for Pre-processing in Max Cut QUBO SolverMetaheuristics10.1007/978-3-031-62912-9_22(219-233)Online publication date: 4-Jun-2024
https://dl.acm.org/doi/10.1007/978-3-031-62912-9_22
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Azuma GFukuda MKim SYamashita M(2022)Exact SDP relaxations for quadratic programs with bipartite graph structuresJournal of Global Optimization10.1007/s10898-022-01268-386:3(671-691)Online publication date: 31-Dec-2022
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Zhang BGao YLiu XHuang X(2022)Outcome-space branch-and-bound outer approximation algorithm for a class of non-convex quadratic programming problemsJournal of Global Optimization10.1007/s10898-022-01255-886:1(61-92)Online publication date: 19-Dec-2022
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Azuma GFukuda MKim SYamashita M(2022)Exact SDP relaxations of quadratically constrained quadratic programs with forest structuresJournal of Global Optimization10.1007/s10898-021-01071-682:2(243-262)Online publication date: 1-Feb-2022
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Luo HChen SWu H(2021)A new branch-and-cut algorithm for non-convex quadratic programming via alternative direction method and semidefinite relaxationNumerical Algorithms10.1007/s11075-020-01065-788:2(993-1024)Online publication date: 1-Oct-2021
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Adjé A(2021)Quadratic Maximization of Reachable Values of Affine Systems with Diagonalizable MatrixJournal of Optimization Theory and Applications10.1007/s10957-021-01825-y189:1(136-163)Online publication date: 10-Feb-2021
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References

Cited By

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