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Disjunctive Optimization: Critical Point Theory

Author

Listed:
  • H. T. Jongen

    (RWTH Aachen)

  • J. J. Rückmann

    (RWTH Aachen)

  • O. Stein

    (University of Trier)

Abstract

In this paper, we introduce the concepts of (nondegenerate) stationary points and stationary index for disjunctive optimization problems. Two basic theorems from Morse theory, which imply the validity of the (standard) Morse relations, are proved. The first one is a deformation theorem which applies outside the stationary point set. The second one is a cell-attachment theorem which applies at nondegenerate stationary points. The dimension of the cell to be attached equals the stationary index. Here, the stationary index depends on both the restricted Hessian of the Lagrangian and the set of active inequality constraints. In standard optimization problems, the latter contribution vanishes.

Suggested Citation

  • H. T. Jongen & J. J. Rückmann & O. Stein, 1997. "Disjunctive Optimization: Critical Point Theory," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 321-336, May.
    • Handle: RePEc:spr:joptap:v:93:y:1997:i:2:d:10.1023_a:1022650006477
    DOI: 10.1023/A:1022650006477
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    Citations

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    Cited by:

    1. D. Aussel & J. J. Ye, 2008. "Quasiconvex Minimization on a Locally Finite Union of Convex Sets," Journal of Optimization Theory and Applications, Springer, vol. 139(1), pages 1-16, October.
    2. Peter Kirst & Fabian Rigterink & Oliver Stein, 2017. "Global optimization of disjunctive programs," Journal of Global Optimization, Springer, vol. 69(2), pages 283-307, October.
    3. Peter Kirst & Oliver Stein, 2016. "Solving Disjunctive Optimization Problems by Generalized Semi-infinite Optimization Techniques," Journal of Optimization Theory and Applications, Springer, vol. 169(3), pages 1079-1109, June.
    4. J. J. Rückmann & A. Shapiro, 1999. "First-Order Optimality Conditions in Generalized Semi-Infinite Programming," Journal of Optimization Theory and Applications, Springer, vol. 101(3), pages 677-691, June.
    5. Jan Rückmann & Juan Alfredo Gómez, 2006. "On generalized semi-infinite programming," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(1), pages 1-32, June.

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