Lucelina Batista Santos

Existence of weakly efficient solutions in nonsmooth vector optimization

In this paper we study the existence of weakly efficient solutions for some nonsmooth and nonconvex vector optimization problems. We consider problems whose objective functions are defined between infinite and finite-dimensional Banach spaces. Our results are stated under hypotheses of generalized convexity and make use of variational-like inequalities.

Some relations between variational-like inequality problems and vectorial optimization problems in Banach spaces

In this work, we will establish some relations between variational-like inequality problems and vectorial optimization problems in Banach spaces under invexity hypotheses. This paper extends the earlier work of Ruiz-Garzon et al. [G. Ruiz-Garzon, R. Osuna-Gomez, A. Rufian-Lizana, Relationships between vector variational-like inequality and optimization problems, European J. Oper. Res. 157 (2004) 113-119].

Necessary and sufficient conditions for weak efficiency in non-smooth vectorial optimization problems

By using the concepts of local cone approximations and K-directional derivatives, we obtain necessary conditions of optimality for the weak efficiency of the vectorial optimization problems with the inequalities and abstract constraints. We introduce the notion of stationary point (weak and strong) and we prove that, under suitable hypotheses of K-invexity, the stationary points are weakly efficient solutions (global).

Optimality conditions for nonlinear infinite programming problems

An infinite programming problem consists in minimizing a functional defined on a real Banach space under an infinite number of constraints. The main purpose of this article is to provide sufficient conditions of optimality under generalized convexity assumptions. Such conditions are necessarily satisfied when the problem possesses the property that every stationary point is a global minimizer.

Some relations between variational-like inequalities and efficient solutions of certain nonsmooth optimization problems.

We study the equivalence between the solutions of the variational-like inequality problem and the solutions of certain nonsmooth and nonconvex vectorial optimization problem.

Condições de otimalidade para problemas de controle ótimo discreto multiobjetivos

Neste trabalho, obtemos condicoes necessarias de otimalidade para a eficiencia fraca para problemas de controle otimo discreto. Abordamos o problema com restricoes de igualdade e de desigualdade sobre o estado e o controle e uma condicao de contorno. Assumimos ainda, que as funcoes envolvidas no problema sao continuamente diferenciaveis. As condicoes necessarias (Principio do Maximo Discreto) foram obtidas atraves do formalismo de Dubovitskii-Milyutin (DM).

Convexidade generalizada e condições de otimalidade para problemas multiobjectivos

Apresentamos um estudo sobre o papel das funcoes convexas generalizadas no estudo de problemas multiobjetivos. Inicialmente, caracterizaremos a eficiencia fraca em termos de escolarizacao. E sabido que, sob hipotese de convexidade, toda solucao fracamente eficiente e solucao do problema ponderado. Mais geralmente, um problema multiobjetivo e KT-invexo se e somente se toda solucao fracamente eficiente e solucao de um problema ponderado. Consideramos tambem as condicoes necessarias de otimalidade de 1a. e de 2a. ordem. E bastante conhecido que, sob hipoteses de convexidade, as condicoes de Kuhn-Tucker sao suficientes para a otimalidade. Consideraremos a classe dos problemas KT-pseudoinvexos, os quais possuem a seguinte propriedade: um problema e KT-invexo se, e somente se, todo ponto Kuhn-Tucker e solucao fracamente eficiente do problema multiobjetivo. Al_em disto, vale um resultado semelhante a este, utilizando-se as condicoes necessarias de 2a. ordem. Veremos que um problema multiobjetivo e KT-pseudoinvexo de segunda ordem se, e somente se, todo ponto Kuhn-Tucker de segunda ordem e solucao fracamente eficiente. Na obtencao destes resultados, os teoremas de alternativa foram amplamente utilizados.

Optimality and constraint qualifications in vector optimization

In this appealing and well-written text, Richard BronsonВ starts with the concrete and computational, and leads the reader to a choice of major applications. The first three chapters address the basics: matrices, vector spaces, and linear transformations. The next three cover eigenvalues, Euclidean inner products, and Jordan canonical forms, offering possibilities that can be tailored to the instructors taste and to the length of the course. Bronsons approach to computation is modern and algorithmic, and his theory is clean and straightforward. Throughout, the views of the theory presented are broad and balanced andВ key material is highlighted in the text and summarized at the end of each chapter. The book also includes ample exercises with answers and hints. Prerequisite: One year of calculus is recommended. Introduces deductive reasoning and helps the reader develop a facility with mathematical proofsProvides a balanced approach to computation and theory by offering computational algorithms for finding eigenvalues and eigenvectors Offers excellent exercise sets, ranging from drill to theoretical/challeging along with useful and interesting applications not found in other introductory linear algebra texts