First-Order Optimality Conditions for Degenerate Index Sets in Generalized Semi-Infinite Optimization

This paper gives the formal definition of a class of optimization problems, that is, problems of finding conditional extrema of given set-measurable functions. It also formulates the generalization of Lyapunov convexity theorem which is used in the proof of first-order optimality conditions for the mentioned class of optimization problems.

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Quasidifferential calculus and first-order optimality conditions in nonsmooth optimization

Mathematical Programming Studies - Quasidifferential Calculus ◽

10.1007/bfb0121136 ◽

1986 ◽

pp. 56-68 ◽

Cited By ~ 12

Author(s):

Alexander Shapiro

Keyword(s):

Optimality Conditions ◽

Nonsmooth Optimization ◽

First Order ◽

Quasidifferential Calculus ◽

First Order Optimality Conditions

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First order optimality conditions in vector optimization involving stable functions

Optimization ◽

10.1080/02331930601120516 ◽

2008 ◽

Vol 57 (3) ◽

pp. 449-471 ◽

Cited By ~ 23

Author(s):

B. Jiménez ◽

V. Novo

Keyword(s):

Optimality Conditions ◽

Vector Optimization ◽

First Order ◽

First Order Optimality Conditions

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Second- and First-Order Optimality Conditions in Vector Optimization

International Journal of Information Technology & Decision Making ◽

10.1142/s0219622014500540 ◽

2015 ◽

Vol 14 (04) ◽

pp. 747-767 ◽

Cited By ~ 3

Author(s):

Vsevolod I. Ivanov

Keyword(s):

Objective Function ◽

Optimality Conditions ◽

Stationary Point ◽

Lagrange Function ◽

Necessary Optimality Conditions ◽

Second Order ◽

Weak Efficiency ◽

First Order ◽

Vector Problem ◽

First Order Optimality Conditions

In this paper, we obtain second- and first-order optimality conditions of Kuhn–Tucker type and Fritz John one for weak efficiency in the vector problem with inequality constraints. In the necessary conditions, we suppose that the objective function and the active constraints are continuously differentiable. We introduce notions of KTSP-invex problem and second-order KTSP-invex one. We obtain that the vector problem is (second-order) KTSP-invex if and only if for every triple [Formula: see text] with Lagrange multipliers [Formula: see text] and [Formula: see text] for the objective function and constraints, respectively, which satisfies the (second-order) necessary optimality conditions, the pair [Formula: see text] is a saddle point of the scalar Lagrange function with a fixed multiplier [Formula: see text]. We introduce notions second-order KT-pseudoinvex-I, second-order KT-pseudoinvex-II, second-order KT-invex problems. We prove that every second-order Kuhn–Tucker stationary point is a weak global Pareto minimizer (global Pareto minimizer) if and only if the problem is second-order KT-pseudoinvex-I (KT-pseudoinvex-II). It is derived that every second-order Kuhn–Tucker stationary point is a global solution of the weighting problem if and only if the vector problem is second-order KT-invex.

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