scholarly journals On Lipschitz Continuity of Projections onto Polyhedral Moving Sets

2020 ◽  
Author(s):  
Ewa M. Bednarczuk ◽  
Krzysztof E. Rutkowski
Keyword(s):  
Constraint Qualification ◽  
Lipschitz Continuity ◽  
Constant Rank ◽  
Locally Lipschitz Functions ◽  
Polyhedral Sets ◽  
Constant Rank Constraint Qualification ◽  
Rank Constraint ◽  
Locally Lipschitz ◽  
Systems Of Inequalities ◽  
Left And Right

Abstract In Hilbert space setting we prove local lipchitzness of projections onto parametric polyhedral sets represented as solutions to systems of inequalities and equations with parameters appearing both in left- and right-hand sides of the constraints. In deriving main results we assume that data are locally Lipschitz functions of parameter and the relaxed constant rank constraint qualification condition is satisfied.

scholarly journals R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization

2021 ◽  
Author(s):  
Patrick Mehlitz ◽  
Leonid I. Minchenko
Keyword(s):  
Linear Dependence ◽  
Constraint Qualification ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Bilevel Optimization ◽  
Regularity Property ◽  
Constant Rank Constraint Qualification ◽  
Existence Criterion ◽  
Set Valued Mappings

AbstractThe presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian–Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.

scholarly journals CONSTANT RANK CONSTRAINT QUALIFICATIONS: A GEOMETRIC INTRODUCTION

2014 ◽  
Vol 34 (3) ◽  
pp. 481-494 ◽  
Author(s):  
Roberto Andreani ◽  
Paulo J.S. Silva
Keyword(s):  
Constraint Qualifications ◽  
Constant Rank ◽  
Rank Constraint

Improved Convergence Properties of the Relaxation Schemes of Kadrani et al. and Kanzow and Schwartz for MPEC

2018 ◽  
Vol 35 (01) ◽  
pp. 1850008
Author(s):  
Na Xu ◽  
Xide Zhu ◽  
Li-Ping Pang ◽  
Jian Lv
Keyword(s):  
Linear Dependence ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Accumulation Point ◽  
Stationary Points ◽  
Nondegeneracy Condition ◽  
Equilibrium Constraints ◽  
Convergence Properties ◽  
Constant Rank Constraint Qualification ◽  
Relaxation Schemes

This paper concentrates on improving the convergence properties of the relaxation schemes introduced by Kadrani et al. and Kanzow and Schwartz for mathematical program with equilibrium constraints (MPEC) by weakening the original constraint qualifications. It has been known that MPEC relaxed constant positive-linear dependence (MPEC-RCPLD) is a class of extremely weak constraint qualifications for MPEC, which can be strictly implied by MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) and MPEC relaxed constant positive-linear dependence (MPEC-rCPLD), of course also by the MPEC constant positive-linear dependence (MPEC-CPLD). We show that any accumulation point of stationary points of these two approximation problems is M-stationarity under the MPEC-RCPLD constraint qualification, and further show that the accumulation point can even be S-stationarity coupled with the asymptotically weak nondegeneracy condition.

On quasi approximate solutions for nonsmooth robust semi-infinite optimization problems

2019 ◽  
Vol 35 (3) ◽  
pp. 417-426 ◽  
Author(s):  
CHANOKSUDA KHANTREE ◽  
RABIAN WANGKEEREE ◽  
◽  
Keyword(s):  
Constraint Qualification ◽  
Generalized Convexity ◽  
Optimization Problems ◽  
Approximate Solutions ◽  
Data Uncertainty ◽  
Necessary Condition ◽  
Duality Theorems ◽  
Approximate Form ◽  
Locally Lipschitz ◽  
Approximate Duality

This paper devotes to the quasi ε-solution for robust semi-infinite optimization problems (RSIP) involving a locally Lipschitz objective function and infinitely many locally Lipschitz constraint functions with data uncertainty. Under the fulfillment of robust type Guignard constraint qualification and robust type Kuhn-Tucker constraint qualification, a necessary condition for a quasi ε-solution to problem (RSIP). After introducing the generalized convexity, we give a sufficient optimality for such a quasi ε-solution to problem (RSIP). Finally, we also establish approximate duality theorems in term of Wolfe type which is formulated in approximate form.

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