Second-Order Optimality Conditions for Constrained Optimization Problems with $$C^1$$ Data Via Regular and Limiting Subdifferentials

2021 ◽  
Author(s):  
Mohammad Taghi Nadi ◽  
Jafar Zafarani
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Optimization Problems ◽  
Second Order ◽  
Constrained Optimization Problems ◽  
Second Order Optimality Conditions

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.

scholarly journals Constructive generalization of classical sufficient second-order optimality conditions

2019 ◽  
Vol 487 (5) ◽  
pp. 493-495
Author(s):  
Yu. G. Evtushenko ◽  
A. A. Tret’yakov
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Optimization Problems ◽  
Sufficient Conditions ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Constrained Optimization Problems ◽  
Constrained Problems ◽  
Slack Variables ◽  
Equality Constrained Optimization

In this paper, we consider new sufficient conditions of optimality of the second-order for equality constrained optimization problems, which essentially enhance and complement the classical ones and are constructive. For example, they establish equivalence between sufficient conditions in the equality constrained optimization problems and sufficient conditions for optimality in equality constrained problems by reducing the latter to equalities with the help of introducing slack variables. Previously, when using the classical sufficient optimality conditions, this fact was not considered to be true, that is, the existing classical sufficient conditions were not complete, so the proposed optimality conditions complement the classical ones and close the question of the equivalence of the problems with inequalities and the problems with equalities when reducing the first to the second by introducing slack variables.

Convergence of Subdifferentials of Convexly Composite Functions

1999 ◽  
Vol 51 (2) ◽  
pp. 250-265 ◽  
Author(s):  
C. Combari ◽  
R. Poliquin ◽  
L. Thibault
Keyword(s):  
Banach Space ◽  
Constrained Optimization ◽  
Optimization Problems ◽  
Second Order ◽  
Constrained Optimization Problems ◽  
Composite Functions ◽  
Kuratowski Convergence ◽  
General Banach Space

AbstractIn this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlevé-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions.

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