scholarly journals First-order and second-order optimality conditions for nonsmooth constrained problems via convolution smoothing

Optimization ◽  
2011 ◽  
Vol 60 (1-2) ◽  
pp. 253-275 ◽  
Author(s):  
Andrew C. Eberhard ◽  
Boris S. Mordukhovich
Keyword(s):  
Optimality Conditions ◽  
Second Order ◽  
First Order ◽  
Second Order Optimality Conditions ◽  
Constrained Problems

scholarly journals Notes on Constraint Qualifications for Second-Order Optimality Conditions

2019 ◽  
Vol 11 (5) ◽  
pp. 16
Author(s):  
Giorgio Giorgi
Keyword(s):  
Optimality Conditions ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Second Order ◽  
Necessary Optimality Condition ◽  
First Order ◽  
Second Order Optimality Conditions ◽  
Order Constraint ◽  
Local Approximations ◽  
Abadie Constraint Qualification

In the first part of this paper we point out some basic properties of the critical cones used in second-order optimality conditions and give a simple proof of a strong second-order necessary optimality condition by assuming a “modified” first-order Abadie constraint qualification. In the second part we give some insights on second-order constraint qualifications related to second-order local approximations of the feasible set.

scholarly journals Second-Order Optimality Conditions for Nonconvex Set-Constrained Optimization Problems

2022 ◽  
Author(s):  
Helmut Gfrerer ◽  
Jane J. Ye ◽  
Jinchuan Zhou
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Support Function ◽  
Optimization Problems ◽  
Second Order ◽  
Constrained Optimization Problems ◽  
Second Order Optimality Conditions ◽  
Constrained Problems ◽  
Second Order Tangent Set ◽  
Order Tangent

In this paper, we study second-order optimality conditions for nonconvex set-constrained optimization problems. For a convex set-constrained optimization problem, it is well known that second-order optimality conditions involve the support function of the second-order tangent set. In this paper, we propose two approaches for establishing second-order optimality conditions for the nonconvex case. In the first approach, we extend the concept of the support function so that it is applicable to general nonconvex set-constrained problems, whereas in the second approach, we introduce the notion of the directional regular tangent cone and apply classical results of convex duality theory. Besides the second-order optimality conditions, the novelty of our approach lies in the systematic introduction and use, respectively, of directional versions of well-known concepts from variational analysis.

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