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Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities

Imre Csiszár, František Matúš (2012)

Kybernetika

Integral functionals based on convex normal integrands are minimized subject to finitely many moment constraints. The integrands are finite on the positive and infinite on the negative numbers, strictly convex but not necessarily differentiable. The minimization is viewed as a primal problem and studied together with a dual one in the framework of convex duality. The effective domain of the value function is described by a conic core, a modification of the earlier concept of convex core. Minimizers...

Locally Lipschitz vector optimization with inequality and equality constraints

Ivan Ginchev, Angelo Guerraggio, Matteo Rocca (2010)

Applications of Mathematics

The present paper studies the following constrained vector optimization problem: min C f ( x ) , g ( x ) ∈ - K , h ( x ) = 0 , where f : ℝ n → ℝ m , g : ℝ n → ℝ p are locally Lipschitz functions, h : ℝ n → ℝ q is C 1 function, and C ⊂ ℝ m and K ⊂ ℝ p are closed convex cones. Two types of solutions are important for the consideration, namely w -minimizers (weakly efficient points) and i -minimizers (isolated minimizers of order 1). In terms of the Dini directional derivative first-order necessary conditions for a point x 0 to be a w -minimizer and first-order sufficient conditions for x 0 ...

Mixed complementarity problems for robust optimization equilibrium in bimatrix game

Guimei Luo (2012)

Applications of Mathematics

In this paper, we investigate the bimatrix game using the robust optimization approach, in which each player may neither exactly estimate his opponent’s strategies nor evaluate his own cost matrix accurately while he may estimate a bounded uncertain set. We obtain computationally tractable robust formulations which turn to be linear programming problems and then solving a robust optimization equilibrium can be converted to solving a mixed complementarity problem under the l 1 ∩ l ∞ -norm. Some numerical...

Multi-objective Optimization Problem with Bounded Parameters

Ajay Kumar Bhurjee, Geetanjali Panda (2014)

RAIRO - Operations Research - Recherche Opérationnelle

In this paper, we propose a nonlinear multi-objective optimization problem whose parameters in the objective functions and constraints vary in between some lower and upper bounds. Existence of the efficient solution of this model is studied and gradient based as well as gradient free optimality conditions are derived. The theoretical developments are illustrated through numerical examples.

New Farkas-type constraint qualifications in convex infinite programming

Nguyen Dinh, Miguel A. Goberna, Marco A. López, Ta Quang Son (2007)

ESAIM: Control, Optimisation and Calculus of Variations

This paper provides KKT and saddle point optimality conditions, duality theorems and stability theorems for consistent convex optimization problems posed in locally convex topological vector spaces. The feasible sets of these optimization problems are formed by those elements of a given closed convex set which satisfy a (possibly infinite) convex system. Moreover, all the involved functions are assumed to be convex, lower semicontinuous and proper (but not necessarily real-valued). The key result...

Nonlinear dynamic systems and optimal control problems on time scales

Yunfei Peng, Xiaoling Xiang, Yang Jiang (2011)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear dynamic systems on time scales. Introducing the reasonable weak solution of nonlinear dynamic systems, the existence of the weak solution for the nonlinear dynamic systems on time scales and its properties are presented. Discussing L1-strong-weak lower semicontinuity of integral functional, we give sufficient conditions for the existence of optimal controls. Using integration by parts formula...

Nonlinear dynamic systems and optimal control problems on time scales*

Yunfei Peng, Xiaoling Xiang, Yang Jiang (2011)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear dynamic systems on time scales. Introducing the reasonable weak solution of nonlinear dynamic systems, the existence of the weak solution for the nonlinear dynamic systems on time scales and its properties are presented. Discussing L1-strong-weak lower semicontinuity of integral functional, we give sufficient conditions for the existence of optimal controls. Using integration by parts formula...

Nonsmooth Problems of Calculus of Variations via Codifferentiation

Maxim Dolgopolik (2014)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper multidimensional nonsmooth, nonconvex problems of the calculus of variations with codifferentiable integrand are studied. Special classes of codifferentiable functions, that play an important role in the calculus of variations, are introduced and studied. The codifferentiability of the main functional of the calculus of variations is derived. Necessary conditions for the extremum of a codifferentiable function on a closed convex set and its applications to the nonsmooth problems of...

On constraint qualifications in directionally differentiable multiobjective optimization problems

Giorgio Giorgi, Bienvenido Jiménez, Vincente Novo (2004)

RAIRO - Operations Research - Recherche Opérationnelle

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints such that all functions are, at least, Dini differentiable (in some cases, Hadamard differentiable and sometimes, quasiconvex). Several constraint qualifications are given in such a way that generalize both the qualifications introduced by Maeda and the classical ones, when the functions are differentiable. The relationships between them are analyzed. Finally, we give several Kuhn-Tucker...

On constraint qualifications in directionally differentiable multiobjective optimization problems

Giorgio Giorgi, Bienvenido Jiménez, Vincente Novo (2010)

RAIRO - Operations Research

We consider a multiobjective optimization problem with a feasible set defined by inequality and equality constraints such that all functions are, at least, Dini differentiable (in some cases, Hadamard differentiable and sometimes, quasiconvex). Several constraint qualifications are given in such a way that generalize both the qualifications introduced by Maeda and the classical ones, when the functions are differentiable. The relationships between them are analyzed. Finally, we give several Kuhn-Tucker...

On dual vector optimization and shadow prices

Letizia Pellegrini (2004)

RAIRO - Operations Research - Recherche Opérationnelle

In this paper we present the image space analysis, based on a general separation scheme, with the aim of studying lagrangian duality and shadow prices in Vector Optimization. Two particular kinds of separation are considered; in the linear case, each of them is applied to the study of sensitivity analysis, and it is proved that the derivatives of the perturbation function can be expressed in terms of vector Lagrange multipliers or shadow prices.

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