Exact Second-Order Cone Programming Relaxations for Some Nonconvex Minimax Quadratic Optimization Problems

V. Jeyakumar University of New South Wales Guoyin Li University of New South Wales

Optimization and Control mathscidoc:2108.27001

SIAM Journal on Optimization, 28, (1), 760–787, 2018.12
In this paper, we study, for the first time, nonconvex minimax separable quadratic optimization problems with multiple separable quadratic constraints and their second-order cone programming (SOCP) relaxations. Under suitable conditions, we establish exact SOCP relaxation for minimax nonconvex separable quadratic programs. We show that various important classes of specially structured minimax quadratic optimization problems admit exact SOCP relaxations under easily verifiable conditions. These classes include some minimax extended trust-region problems, minimax uniform quadratic optimization problems, max dispersion problems, and some robust quadratic optimization problems under bounded data uncertainty. The present work shows that nonconvex minimax separable quadratic problems with quadratic constraints, which contain a hidden closed and convex epigraphical set, exhibit exact SOCP relaxations.
nonconvex quadratic programs, robust optimization, second-order cone programming, minimax quadratic programs
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@inproceedings{v.2018exact,
  title={Exact Second-Order Cone Programming Relaxations for Some Nonconvex Minimax Quadratic Optimization Problems},
  author={V. Jeyakumar, and Guoyin Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210823155151652952864},
  booktitle={SIAM Journal on Optimization},
  volume={28},
  number={1},
  pages={760–787},
  year={2018},
}
V. Jeyakumar, and Guoyin Li. Exact Second-Order Cone Programming Relaxations for Some Nonconvex Minimax Quadratic Optimization Problems. 2018. Vol. 28. In SIAM Journal on Optimization. pp.760–787. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210823155151652952864.
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