General Semi-Infinite Programming: Critical Point Theory
(2011) Optimization : a journal of mathematical programming and operations research — Vol. 60, n° 7, p. 859-873 (2011)
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- Abstract
- We study General Semi-Infinite Programming (GSIP) from a topological point of view. Under the Symmetric Mangasarian-Fromovitz Constraint Qualification (Sym-MFCQ) two basic theorems from Morse theory (deformation theorem and cell-attachment theorem) are proved. Outside the set of Karush-Kuhn-Tucker (KKT) points, 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 KKT level, the topology of the lower level set changes via the attachment of a q-dimensional cell. The dimension q equals the so-called GSIP-index of the (nondegenerate) KKT-point. Here, the Nonsmooth Symmetric Reduction Ansatz (NSRA) allows to perform a local reduction of GSIP to a Disjunctive Optimization Problem. The GSIP-index then coincides with the stationary index from the corresponding Disjunctive Optimization Problem.
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Jongen, H. Th., & Shikhman, V. (2011). General Semi-Infinite Programming: Critical Point Theory. Optimization : a journal of mathematical programming and operations research, 60(7), 859-873. https://doi.org/10.1080/02331934.2010.543134 (Original work published 2011)