On Computing the Nonlinearity Interval in Parametric Semidefinite Optimization

Jonathan D. Hauenstein (), Ali Mohammad-Nezhad (), Tingting Tang () and Tamás Terlaky ()
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Jonathan D. Hauenstein: Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, Indiana 46556
Ali Mohammad-Nezhad: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Tingting Tang: Department of Mathematics and Statistics, San Diego State University Imperial Valley, Calexico, California 92231
Tamás Terlaky: Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, Pennsylvania 18015

Mathematics of Operations Research, 2022, vol. 47, issue 4, 2989-3009

Abstract: This paper revisits the parametric analysis of semidefinite optimization problems with respect to the perturbation of the objective function along a fixed direction. We review the notions of invariancy set, nonlinearity interval, and transition point of the optimal partition, and we investigate their characterizations. We show that the set of transition points is finite and the continuity of the optimal set mapping, on the basis of Painlevé–Kuratowski set convergence, might fail on a nonlinearity interval. Under a local nonsingularity condition, we then develop a methodology, stemming from numerical algebraic geometry, to efficiently compute nonlinearity intervals and transition points of the optimal partition. Finally, we support the theoretical results by applying our procedure to some numerical examples.

Keywords: Primary: 90C22; secondary: 90C31; 90C51; parametric semidefinite optimization; optimal partition; nonlinearity interval; numerical algebraic geometry (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:47:y:2022:i:4:p:2989-3009

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