MPVC: Critical Point Theory

Dorsch, Dominik;Shikhman, Vladimir;Stein, Olivier
(2012) Journal of Global Optimization : an international journal dealing with theoretical and computational aspects of seeking global optima and their applications in science, management, and engineering — Vol. 52, n° 3, p. 591-605 (2012)

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Authors
  • Dorsch, DominikRWTH Aachen University
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  • Shikhman, VladimirUCLouvain
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  • Stein, OlivierKarsruhe Institute of Technology
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Abstract
We study mathematical programs with vanishing constraints (MPVCs) from a topological point of view. We introduce the new concept of a T-stationary point for MPVC. Under the Linear Independence Constraint Qualification we derive an equivariant Morse Lemma at nondegenerate T-stationary points. Then, two basic theorems from Morse Theory (deformation theorem and cell-attachment theorem) are proved. Outside the T-stationary point set, continuous deformation of lower level sets can be performed. As a consequence, the topological data (such as the number of connected components) then remain invariant. However, when passing a T-stationary level, the topology of the lower level set changes via the attachment of a q-dimensional cell. The dimension q equals the stationary T-index of the (nondegenerate) T-stationary point. The stationary T-index depends on both the restricted Hessian of the Lagrangian and the number of bi-active vanishing constraints. Further, we prove that all T-stationary points are generically nondegenerate. The latter property is shown to be stable under C^2-perturbations of the defining functions. Finally, some relations with other stationarity concepts, such as strong, weak, and M-stationarity, are discussed.
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Dorsch, D., Shikhman, V., & Stein, O. (2012). MPVC: Critical Point Theory. Journal of Global Optimization : an international journal dealing with theoretical and computational aspects of seeking global optima and their applications in science, management, and engineering, 52(3), 591-605. https://doi.org/10.1007/s10898-011-9805-z (Original work published 2012)