scholarly journals Improved enhanced Fritz John condition and constraints qualifications using convexificators

2019 ◽  
Author(s):  
Abeka Khare ◽  
Triloki Nath
Keyword(s):  
Nonsmooth Optimization ◽  
Optimality Condition ◽  
Optimization Problems ◽  
Extremal Point ◽  
Necessary Optimality Condition ◽  
Kkt Condition

In this paper, using convexificators we derive enhanced Fritz John optimality condition for nonsmooth optimization problems having equality, inequality and abstract set constraint. This necessary optimality condition provides some more information about the extremal point in terms of converging sequences towards it. Then we employ this optimality condition to study enhanced KKT condition and to define associated ∂^*- pseudonormality and ∂^*-quasinormality concepts in terms of convexificators. Later, sufficiency for ∂^*-pseudonormality and some more results based on these concepts are investigated.

scholarly journals Optimality conditions for non-Lipschitz vector problems with inclusion constraints

2019 ◽  
Vol 128 (1D) ◽  
Author(s):  
Phan Nhật Tĩnh
Keyword(s):  
Optimality Conditions ◽  
Optimality Condition ◽  
Efficient Solutions ◽  
Necessary Optimality Condition ◽  
Font Size ◽  
Kkt Condition ◽  
Generalized Derivative ◽  
Numerical Examples

<span class="fontstyle0"><span style="font-size: small;">We use the concept of approximation introduced by D.T Luc et al.  as<br />generalized derivative for non-Lipschitz vector functions to consider vector problems<br />with non-Lipschitz data under inclusion constraints. Some calculus of approximations<br />are presented. A necessary optimality condition, type of KKT condition, for local<br />efficient solutions of the problems is established under an assumption on regularity.<br />Applications and numerical examples are also given.</span></span>

scholarly journals A Novel Approach for Solving Nonsmooth Optimization Problems with Application to Nonsmooth Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Hamid Reza Erfanian ◽  
M. H. Noori Skandari ◽  
A. V. Kamyad
Keyword(s):  
Nonsmooth Optimization ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Linear Optimization ◽  
Optimal Solution ◽  
Piecewise Linear Function ◽  
Nonsmooth Equations ◽  
Smooth Functions ◽  
Generalized Derivative ◽  
Novel Approach

We present a new approach for solving nonsmooth optimization problems and a system of nonsmooth equations which is based on generalized derivative. For this purpose, we introduce the first order of generalized Taylor expansion of nonsmooth functions and replace it with smooth functions. In other words, nonsmooth function is approximated by a piecewise linear function based on generalized derivative. In the next step, we solve smooth linear optimization problem whose optimal solution is an approximate solution of main problem. Then, we apply the results for solving system of nonsmooth equations. Finally, for efficiency of our approach some numerical examples have been presented.

Optimality of Approximate Quasi-Weakly Efficient Solutions for Vector Equilibrium Problems via Convexificators

2021 ◽  
Author(s):  
Guolin Yu ◽  
Siqi Li ◽  
Xiao Pan ◽  
Wenyan Han
Keyword(s):  
Optimality Condition ◽  
Generalized Convexity ◽  
Equilibrium Problems ◽  
Efficient Solutions ◽  
Sufficient Optimality Condition ◽  
Necessary Optimality Condition ◽  
Weakly Efficient Solutions ◽  
Convexity Assumption ◽  
Pseudoconvex Function ◽  
Vector Equilibrium

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.

scholarly journals A note on the paper "Optimality conditions for nonsmooth interval-valued and multiobjective semi-infinite programming"

2020 ◽  
Author(s):  
Nazih Abderrazzak Gadhi ◽  
Aissam Ichatouhane
Keyword(s):  
Optimality Conditions ◽  
Programming Problem ◽  
Optimality Condition ◽  
Short Proof ◽  
Necessary Optimality Condition ◽  
Main Result Theorem ◽  
Correct Formulation ◽  
Result Theorem ◽  
Infinite Programming ◽  
Interval Valued

A nonsmooth semi-infinite interval-valued vector programming problem is solved in the paper by Jennane et all. (RAIRO-Oper. Res. doi: 10.1051/ro/2020066, 2020). The necessary optimality condition obtained by the authors, as well as its proof, is false. Some counterexamples are given to refute some results on which the main result (Theorem 4.5) is based. For the convinience of the reader, we correct the faulty in those results, propose a correct formulation of Theorem 4.5 and give also a short proof.

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