Optimality Conditions in Vector Optimization


by

Manuel A. Jiménez

DOI: 10.2174/97816080511061100101
eISBN: 978-1-60805-110-6, 2010
ISBN: 978-1-60805-368-1



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Indexed in: Scopus

Vector optimization is continuously needed in several science fields, particularly in economy, business, engineering, physics and math...[view complete introduction]

Table of Contents

Foreword

- Pp. i-ii (2)

Fernando Lobo Pereira

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Preface

- Pp. iii

Manuel Arana Jim´enez, Gabriel x Gabriel Ruiz Garz´on and Antonio Rufi´an Lizana

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Contributors

- Pp. iv-v (2)

Manuel Arana Jim´enez, Gabriel x Gabriel Ruiz Garz´on and Antonio Rufi´an Lizana

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Pseudoinvexity: A good condition for efficiency and weak efficiency in multiobjective mathematical programming. Characterization

- Pp. 1-16 (16)

M. Arana-Jim´enez, G. Ruiz-Garz´on and A. Rufi´an-Lizana

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Optimality and constraint qualifications in vector optimization

- Pp. 17-34 (18)

Carosi Laura and Martein Laura

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Second order optimality conditions in vector optimization problems.

- Pp. 35-60 (26)

M. Hachimi and B. Aghezzaf

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Invex functions and existence of weakly efficient solutions for nonsmooth vector optimization

- Pp. 61-74 (14)

Lucelina Batista Santos, Marko Rojas-Medar, Gabriel Ruiz-Garz´on and Antonio Rufi´an-Lizana

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Proper efficiency and duality for differentiable multiobjective programming problems with B-(p,r)-invex functions

- Pp. 75-96 (22)

Tadeusz Antczak

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On nonsmooth constrained optimization involving generalized type-I conditions.

- Pp. 97-104 (8)

S. K. Mishra, J. S. Rautelay and Sanjay Oli

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Duality theory for the multiobjective nonlinear programming involving generalized convex functions

- Pp. 105-118 (14)

R. Osuna-Gomezy, M. B. Hernnandez-Jimenez and L. L. Salles Neto

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Mixed type duality for multiobjective optimization problems with set constraints

- Pp. 119-142 (24)

Riccardo Cambini and Laura Carosi

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Necessary and sufficient optimality conditions for continuous-time multiobjective optimization problems

- Pp. 143-163 (21)

Adilson J. V. Brand~ao, Valeriano Antunes de Oliveira, Marko Antonio Rojas-Medarx and Lucelina Batista Santos

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Optimality conditions and duality for nonsmooth multiobjective continuous-time problems

- Pp. 164-182 (19)

S. Nobakhtian and M. R. Pouryayevali

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Index

- Pp. 183-184 (2)

Manuel Arana Jimenez, Gabriel Ruiz Garzon and Antonio Rufian Lizana

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Foreword

Fuelled by the wealth of applications and by the beauty of the scientific building, optimization has a very long tradition of developments that currently translate into a complex and a very sophisticated body of results. Progress beyond the state-of-the-art requires high quality contributions. This is precisely what this book achieves in the specific area of vector-valued optimization. Here, nonsmoothness, various types of generalized convexity, higher order conditions, duality, non-degeneracy, infinite dimensionality of underlying spaces, and nonlinearity are compounded with the elaborated solution concepts considered in the area of optimization of vector-valued objective functions.

Let us be more specific.

While in chapter one, it is shown that invexity and pseudo-invexity are appropriate properties for vector-valued functions in that every vector critical point is an efficient or a weakly efficient solution of a Multi-objective Programming Problem, thus generalizing analogue results for scalar case, in chapter three, generalized convexity and variational-like inequalities are used to derive the existence of weakly efficient solutions for some nonsmooth and nonconvex vector optimization problems defined on infinite and finite-dimensional Banach spaces.

Optimality conditions of optimality for nonsmooth continuous-time multiobjective optimization problems under generalized convexity assumptions are derived in both chapters nine and ten. While the former also encompasses problems with smooth data, the later also establishes weak and strong duality theorems for two dual models.

In chapter five, a concept of proper efficiency is presented to develop the optimality conditions and duality results for differentiable non-convex multi-objective programming problems. A relaxation of the proper efficiency assumption enabled the extension of known results to a wider class of nonconvex vector optimization problems. A combination of duality results and necessary and sufficient optimality conditions, now of the Kuhn-Tucker type are obtained in chapter six for non-smooth constrained optimization problems involving generalized type-I functions on the objective and constraint functions involved in the problem.

A constraint qualification condition weaker than those currently available in the literature is derived in chapter two via a unifying approach based on a separation theorem in the vector optimization context.

Chapter three discusses various types of second-order optimality conditions for scalar and vector valued abstract optimization problems. Moreover, second-order sufficient optimality conditions exhibiting a very small gap with the necessary optimality conditions are also considered. The application of these results to derive optimality conditions for optimization problems with equality and inequality constraints is also considered.

Different forms of weak, strong and converse duality theorems are obtained in chapter seven for the Wolfe and Mond-Weir dual problems associated with the vector optimization problem with constraints whose data satisfies assumptions of invexity, strictly invexity or quasi-invexity. Duality results are also derived in chapter eight for pairs of dual programs where the primal is a vector optimization problem with a feasible region defined by a set constraint, equality and inequality constraints, and the duals are of "mixed type" and satisfy suitably generalized concavity properties.

Clearly, this book contains a significant set of contributions which constitute a step forward in shaping the today's Optimization Theory.

Fernando Lobo Pereira
The Institute for Systems and Robotics, Porto
University of Porto
Portugal


Preface

Vector optimization is continuously needed in several science fields, particularly in economy and engineering. The evolution of these fields depends, in part, on the improvements in vector optimization in mathematical programming.

The search for solutions to vector or multiobjective mathematical programming problems has been carried out through the study of optimality conditions and of the properties of the functions that are involved, as well as through the study of the respective dual problems. In the case of optimality conditions, it is customary to use critical points of the Kuhn-Tucker or Fritz John type. In the case of the classes of functions employed in mathematical programming problems, the tendency has been to substitute convex functions with more general ones, with the objective of obtaining a solution through an optimality condition. Meanwhile the inverse result has also been sought. At the same time, optimality conditions, vector functions and optimality results are being generalized from the differentiable case to the non-differentiable one, and otherwise, extended to other kind of problems, such as continuous time problems.

The aim of this book is to present the last developments in vector optimization. We deeply appreciate the contribution of many of the most relevant researchers in the area that have made possible the existence of this book.

Manuel Arana Jiménez
Gabriel Ruiz Garzón
and
Antonio Rufián Lizana


List of Contributors

Editor(s):
Manuel A. Jiménez
University of Cádiz
Spain




Co-Editor(s):
Gabriel Ruiz Garzón
University of Cádiz
Spain


Antonio Rufián Lizana
University of Sevilla
Spain




Contributor(s):
B. Aghezzaf
Département de Mathématiques et d'Informatique, Facultédes Sciences
Université Hassan II Aïn chock, B.P.5366
Maârif Casablanca
Morocco


Tadeusz Antczak
Faculty of Mathematics and Computer Science
University of Lodz, Banacha 22
Lodz, 90-238
Poland


Valeriano Antunes de Oliveira
Universidade Federal de Uberlândia
Faculdade de Ciências Integradas do Pontal. Av. Jose Joãao Dib, 2545 Progresso 38302-000
Ituiutaba
MG
Brazil


M. Arana-Jiménez
Departamento de Estadística e Investigación Operativa
Escuela Superior de Ingeniería, Universidad de Cádiz
Cádiz , 11002
Spain


Lucelina Batista Santos
Departamento de Matematica
Universidade Federal do Paraná
CP 19081 CEP, 81531-990 Curitiba
Paraná
Brazil


Adilson J. V. Brandão
Departamento de Matemática
Universidade Federal de São Carlos, Campus Sorocaba
Rodovia João Leme dos Santos, Km 110 - SP-264 Itinga
Sorocaba
SP , 18052-780
Brazil


Riccardo Cambini
Department of Statistics and Applied Mathematics,
Faculty of Economics, University of Pisa,
Via Cosimo Ridolfi, 10
Pisa, 56124
Italy


Laura Carosi
Department of Statistics and Applied Mathematics
University of Pisa, Via Ridolfi, 10
Pisa, 56124
Italy


M. Hachimi
Université Ibn Zohr, Faculté des Sciences Juridiques Economiques et Sociales
B.P. 8658 Hay Dakhla
Agadir , 80000
Morocco


M. B. Hernández-Jiménez
Departamento de Economía, Métodos Cuantitativos e Ha Económica
Area de iv Estadística e Investigación Operativa. Universidad Pablo de Olavide
Spain


Laura Martein
Department of Statistics and Applied Mathematics
University of Pisa
Via Ridolfi, 10,
Pisa, 56124
Italy


S. K. Mishra
Department of Mathematics, Faculty of Science
Banaras Hindu University
Varanasi, 221005
India


S. Nobakhtian
Department of Mathematics
University of Isfahan
P.O. Box 81745-163
Isfahan
Iran


Sanjay Oli
Department of Mathematics
Asia Pacific Institute of Information Technology SD
India


R. Osuna-G´omez
Departamento de Estadística e Investigación Operativa
Facultad de Matemáticas, Universidad de Sevilla
Spain


M.R. Pouryayevali
Department of Mathematics
University of Isfahan
P.O. Box 81745-163
Isfahan
Iran


J. S. Rautela
Department of Mathematics
Faculty of Applied Science and Humanities, Echelon Institute of Technology
Faridabad , 121001
India


Marko Rojas-Medar
Departamento de Ciencias Básicas
Universidad Del Bío-Bío. Facultad de Ciencias
Casilla: 447 Av. Andrés Bello S/N
Chillán
Chile


A. Rufián-Lizana
Departamento de Estadística e Investigacíon Operativa, Facultad de Matemáticas
Universidad de Sevilla
Spain


G. Ruiz-Garzón
Departamento de Estadística e Investigación Operativa, Facultad de Ciencias Sociales y de la Comunicación, ,
Universidad de Cádiz
Spain


L. L. Salles Neto
Departamento de Ciência e Tecnologia
Universidade Federal de São Paulo
São Paulo
Brazil




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