Constraint Qualifications in Semi-Infinite Systems and Their Applications in Nonsmooth Semi-Infinite Problems with Mixed Constraints

In this paper, we present several constraint qualifications, and we show that conditions guarantee the nonvacuity of the Karush--Kuhn--Tucker multipliers set for nonsmooth semi-infinite programming problems with mixed constraints. The relationships with various constraint qualifications are investigated. All results are given in terms of the Michel--Penot subdifferential.

[1]  Hubert Halkin,et al.  Implicit Functions and Optimization Problems without Continuous Differentiability of the Data , 1974 .

[2]  Kenneth O. Kortanek,et al.  Semi-Infinite Programming: Theory, Methods, and Applications , 1993, SIAM Rev..

[3]  Jay S. Treiman.  Shrinking generalized gradients , 1988 .

[4]  S. Nobakhtian,et al.  Nonsmooth semi-infinite programming problems with mixed constraints , 2009 .

[5]  Marco A. López,et al.  Optimality conditions for nondifferentiable convex semi-infinite programming , 1983, Math. Program..

[6]  J. Borwein,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

[7]  Nader Kanzi,et al.  Optimality conditions for nonsmooth semi-infinite multiobjective programming , 2014, Optim. Lett..

[8]  S. Nobakhtian,et al.  Optimality conditions for non-smooth semi-infinite programming , 2010 .

[9]  D. W. Peterson.  A REVIEW OF CONSTRAINT QUALIFICATIONS IN FINITE-DIMENSIONAL SPACES* , 1973 .

[10]  Xiaoqi Yang,et al.  Lagrange Multipliers in Nonsmooth Semi-Infinite Optimization Problems , 2007, Math. Oper. Res..

[11]  Jane J. Ye,et al.  Nondifferentiable Multiplier Rules for Optimization and Bilevel Optimization Problems , 2004, SIAM J. Optim..

[12]  Boris S. Mordukhovich,et al.  Nonsmooth Cone-Constrained Optimization with Applications to Semi-Infinite Programming , 2014, Math. Oper. Res..

[13]  Marco A. López,et al.  Semi-infinite programming , 2007, Eur. J. Oper. Res..

[14]  Boris S. Mordukhovich,et al.  Constraint qualifications and optimality conditions for nonconvex semi-infinite and infinite programs , 2013, Math. Program..

[15]  G. Giorgi,et al.  Mathematics of Optimization: Smooth and Nonsmooth Case , 2004 .

[16]  Wu Li,et al.  Constraint Qualifications for Semi-Infinite Systems of Convex Inequalities , 2000, SIAM J. Optim..

[17]  Boris S. Mordukhovich,et al.  Subdifferentials of Nonconvex Supremum Functions and Their Applications to Semi-infinite and Infinite Programs with Lipschitzian Data , 2013, SIAM J. Optim..

[18]  Nader Kanzi.  Necessary optimality conditions for nonsmooth semi-infinite programming problems , 2011, J. Glob. Optim..

[19]  Philippe Michel,et al.  A generalized derivative for calm and stable functions , 1992, Differential and Integral Equations.

[20]  M. A. López-Cerdá,et al.  Linear Semi-Infinite Optimization , 1998 .

[21]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .