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Calmness modulus of linear semi-infinite programs

M.J. Canovas (canovas***at***umh.es)
A.Y. Kruger (a.kruger***at***ballarat.edu.au)
M.A. Lopez (marco.antonio***at***ua.es)
J. Parra (parra***at***umh.es)
M.A. Thera (michel.thera***at***unilim.fr)

Abstract: Our main goal is to compute or estimate the calmness modulus of the argmin mapping of linear semi-infinite optimization problems under canonical perturbations, i.e., perturbations of the objective function together with continuous perturbations of the right-hand side of the constraint system (with respect to an index ranging in a compact Hausdorff space). Specifically, we provide a lower bound on the calmness modulus for semi-infinite programs with unique optimal solution which turns out to be the exact modulus when the problem is finitely constrained. The relationship between the calmness of the argmin mapping and the same property for the (sub)level set mapping (with respect to the objective function), for semi-infinite programs and without requiring the uniqueness of the nominal solution, is explored too, providing an upper bound on the calmness modulus of the argmin mapping. When confined to finitely constrained problems, we also provide a computable upper bound as it only relies on the nominal data and parameters, not involving elements in a neighborhood. Illustrative examples are provided.

Keywords: Isolated calmness, calmness modulus, variational analysis, linear programming, semi-infinite programming

Category 1: Infinite Dimensional Optimization (Semi-infinite Programming )

Category 2: Linear, Cone and Semidefinite Programming (Linear Programming )

Citation: Published in SIAM Journal on Optimization 24 (2014) no. 1, 29-48

Download: [Postscript][PDF]

Entry Submitted: 01/23/2013
Entry Accepted: 01/31/2013
Entry Last Modified: 01/13/2014

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