Nonsmooth Mathematical Programs with Vanishing Constraints in Banach Spaces

Vivek Laha (), Vinay Singh (), Yogendra Pandey () and S. K. Mishra ()
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Vivek Laha: Institute of Science, Banaras Hindu University
Vinay Singh: National Institute of Technology Mizoram
Yogendra Pandey: Satish Chandra College
S. K. Mishra: Institute of Science, Banaras Hindu University

A chapter in High-Dimensional Optimization and Probability, 2022, pp 395-417 from Springer

Abstract: Abstract In this chapter, we study the optimization problems with equality, inequality, and vanishing constraints in a Banach space where the objective function and the binding constraints are either differentiable at the optimal solution or Lipschitz near the optimal solution. We derive nonsmooth Karush–Kuhn–Tucker (KKT) type necessary optimality conditions for the above problem where Fréchet (or Gâteaux or Hadamard) derivatives are used for the differentiable functions and the Michel-Penot (M-P) subdifferentials are used for the Lipschitz continuous functions. We also introduce several modifications of some known constraint qualifications like Abadie constraint qualification, Cottle constraint qualification, Slater constraint qualification, Mangasarian–Fromovitz constraint qualification, and linear independence constraint qualification for the above mentioned problem which is called as the nonsmooth mathematical programs with vanishing constraints (NMPVC) in terms of the M-P subdifferentials and establish relationships among them.

Keywords: KKT optimality conditions; Constraint qualifications; Michel-Penot subdifferential; Fréchet differentiability; Nonsmooth mathematical programs; Vanishing constraints; Lipschitz continuity; 49J50; 49J52; 90C46; 90C26; 90C30 (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-031-00832-0_13

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DOI: 10.1007/978-3-031-00832-0_13

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