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On Computing the Nonlinearity Interval in Parametric Semidefinite Optimization

Author

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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)

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.

Suggested Citation

  • Jonathan D. Hauenstein & Ali Mohammad-Nezhad & Tingting Tang & Tamás Terlaky, 2022. "On Computing the Nonlinearity Interval in Parametric Semidefinite Optimization," Mathematics of Operations Research, INFORMS, vol. 47(4), pages 2989-3009, November.
    • Handle: RePEc:inm:ormoor:v:47:y:2022:i:4:p:2989-3009
    DOI: 10.1287/moor.2021.1234
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