scholarly journals Error Bound property in mathematical programming

2019 ◽  
Vol 55 (3) ◽  
pp. 309-318
Author(s):  
D. E. Berezhnov ◽  
L. I. Minchenko
Keyword(s):  
Mathematical Programming ◽  
Large Class ◽  
Optimality Conditions ◽  
Error Bound ◽  
Numerical Optimization ◽  
Constraint Qualification ◽  
Necessary Conditions ◽  
Optimization Algorithms ◽  
Sufficient Conditions ◽  
Mathematical Programming Problems

This article is devoted to the Error Bound property (also named R-regularity) in mathematical programming problems. This property plays an important role in analyzing the convergence of numerical optimization algorithms, a topic covered by multiple publications, and at the same time it is a relatively generic constraint qualification that guarantees the satisfaction of the necessary Kuhn – Tucker optimality conditions in mathematical programming problems. In the article, new sufficient conditions for the error bound property are described, and it’s also shown that several known necessary conditions are insufficient. The sufficient conditions obtained can be used to prove the regularity of a large class of sets including sets that cannot be proven regular by other known constraints.

scholarly journals Exact Penalization in Stochastic Programming—Calmness and Constraint Qualification

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Martin Branda
Keyword(s):  
Stochastic Programming ◽  
Error Bound ◽  
Constraint Qualification ◽  
Sufficient Conditions ◽  
Discrete Distributions ◽  
Exact Penalty ◽  
Asymptotic Equivalence ◽  
Exact Penalization ◽  
Local Minimizers ◽  
Constrained Problems

We deal with the conditions which ensure exact penalization in stochastic programming problems under finite discrete distributions. We give several sufficient conditions for problem calmness including graph calmness, existence of an error bound, and generalized Mangasarian-Fromowitz constraint qualification. We propose a new version of the theorem on asymptotic equivalence of local minimizers of chance constrained problems and problems with exact penalty objective. We apply the theory to a problem with a stochastic vanishing constraint.

scholarly journals Optimality conditions via scalarization for approximate quasi efficiency in multiobjective optimization

Filomat ◽  
2017 ◽  
Vol 31 (3) ◽  
pp. 671-680 ◽  
Author(s):  
Mehrdad Ghaznavi
Keyword(s):  
Multiobjective Optimization ◽  
Optimality Conditions ◽  
Optimization Problem ◽  
Original Problem ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Multiobjective Optimization Problem ◽  
Alternative Theorem ◽  
Convexity Assumptions ◽  
Interesting Area

Approximate problems that scalarize and approximate a given multiobjective optimization problem (MOP) became an important and interesting area of research, given that, in general, are simpler and have weaker existence requirements than the original problem. Recently, necessary conditions for approximation of several types of efficiency for MOPs have been obtained through the use of an alternative theorem. In this paper, we use these results in order to extend them to sufficient conditions for approximate quasi (weak, proper) efficiency. For this, we use two scalarization techniques of Tchebycheff type. All the provided results are established without convexity assumptions.

scholarly journals Optimality Conditions for a Nonsmooth Uncertain Multiobjective Programming Problem

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Wenyan Han ◽  
Guolin Yu ◽  
Tiantian Gong
Keyword(s):  
Optimality Conditions ◽  
Generalized Convexity ◽  
Multiobjective Programming ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Efficient Solutions ◽  
Type I ◽  
Generalized Convex Functions ◽  
Alternative Theorem ◽  
Multiobjective Programming Problem

This note is devoted to the investigation of optimality conditions for robust approximate quasi weak efficient solutions to a nonsmooth uncertain multiobjective programming problem (NUMP). Firstly, under the extended nonsmooth Mangasarian–Fromovitz constrained qualification assumption, the optimality necessary conditions of robust approximate quasi weak efficient solutions are given by using an alternative theorem. Secondly, a class of generalized convex functions is introduced to the problem (NUMP), which is called the pseudoquasi-type-I function, and its existence is illustrated by a concrete example. Finally, under the pseudopseudo-type-I generalized convexity hypothesis, the optimality sufficient conditions for robust approximate quasi weak efficient solutions to the problem (NUMP) are established.

A Boundary Tracking Optimization Algorithm for Constrained Nonlinear Problems

1978 ◽  
Vol 100 (2) ◽  
pp. 292-296 ◽  
Author(s):  
J. Y. Moradi ◽  
M. Pappas
Keyword(s):  
Mathematical Programming ◽  
Optimization Algorithm ◽  
Numerical Optimization ◽  
Nonlinear Problems ◽  
Nonlinear Constraints ◽  
Comparison Study ◽  
Boundary Tracking ◽  
Nonlinear Mathematical Programming ◽  
Mathematical Programming Problems ◽  
Tracking Strategy

A new procedure for numerical optimization of constrained nonlinear problems is described. The method makes use of an efficient “Boundary Tracking” strategy to move on the constraint surfaces. In a comparison study it was found to be an effective method for treating nonlinear mathematical programming problems particularly those with difficult nonlinear constraints.

scholarly journals Optimality and duality without a constraint qualification for minimax programming

2003 ◽  
Vol 67 (1) ◽  
pp. 121-130 ◽  
Author(s):  
Houchun Zhou ◽  
Wenyu Sun
Keyword(s):  
Optimality Conditions ◽  
Constraint Qualification ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dual Model ◽  
Duality Theorems ◽  
Minimax Programming ◽  
Optimality And Duality ◽  
Necessary And Sufficient

Without the need of a constraint qualification, we establish the optimality necessary and sufficient conditions for generalised minimax programming. Using these optimality conditions, we construct a parametric dual model and a parameter-free mixed dual model. Several duality theorems are established.

scholarly journals Necessary and Sufficient Conditions for Set-Valued Maps with Set Optimization

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Abdessamad Oussarhan ◽  
Ikram Daidai
Keyword(s):  
Optimality Conditions ◽  
Existence Of Solutions ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Set Optimization ◽  
Contingent Derivative ◽  
Necessary And Sufficient

Optimality conditions are studied for set-valued maps with set optimization. Necessary conditions are given in terms of S-derivative and contingent derivative. Sufficient conditions for the existence of solutions are shown for set-valued maps under generalized quasiconvexity assumptions.

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