On density theorems, connectedness results and error bounds in vector optimization.

Yung Hon-wai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2001.Includes bibliographical references (leaves 133-139).Abstracts in English and Chinese.Chapter 0 --- Introduction --- p.1Chapter 1 --- Density Theorems in Vector Optimization --- p.7Chapter 1.1 --- Preliminary --- p.7Chapter 1.2 --- The Arrow-Barankin-Blackwell Theorem in Normed Spaces --- p.14Chapter 1.3 --- The Arrow-Barankin-Blackwell Theorem in Topolog- ical Vector Spaces --- p.27Chapter 1.4 --- Density Results in Dual Space Setting --- p.32Chapter 2 --- Density Theorem for Super Efficiency --- p.45Chapter 2.1 --- Definition and Criteria for Super Efficiency --- p.45Chapter 2.2 --- Henig Proper Efficiency --- p.53Chapter 2.3 --- Density Theorem for Super Efficiency --- p.58Chapter 3 --- Connectedness Results in Vector Optimization --- p.63Chapter 3.1 --- Set-valued Maps --- p.64Chapter 3.2 --- The Contractibility of the Efficient Point Sets --- p.67Chapter 3.3 --- Connectedness Results in Vector Optimization Prob- lems --- p.83Chapter 4 --- Error Bounds In Normed Spaces --- p.90Chapter 4.1 --- Error Bounds of Lower Semicontinuous Functionsin Normed Spaces --- p.91Chapter 4.2 --- Error Bounds of Lower Semicontinuous Convex Func- tions in Reflexive Banach Spaces --- p.100Chapter 4.3 --- Error Bounds with Fractional Exponents --- p.105Chapter 4.4 --- An Application to Quadratic Functions --- p.114Bibliography --- p.13

PC Points and their Application to Vector Optimization

AMS subject classification: 90C29, 90C48In this paper we present some results concerning stability under perturbations and some topological properties of minimal point sets in vector optimization. A common feature of all these results is that they exploit the notion of a point of continuity (PC point) of a subset of a topological vector space. The concept of a PC point plays an important role in investigation of the geometry of Banach spaces

Domination and Decomposition in Multiobjective Programming

During the last few decades, multiobjective programming has received much attention for both its numerous theoretical advances as well as its continued success in modeling and solving real-life decision problems in business and engineering. In extension of the traditionally adopted concept of Pareto optimality, this research investigates the more general notion of domination and establishes various theoretical results that lead to new optimization methods and support decision making. After a preparatory discussion of some preliminaries and a review of the relevant literature, several new findings are presented that characterize the nondominated set of a general vector optimization problem for which the underlying domination structure is defined in terms of different cones. Using concepts from linear algebra and convex analysis, a well known result relating nondominated points for polyhedral cones with Pareto solutions is generalized to nonpolyhedral cones that are induced by positively homogeneous functions, and to translated polyhedral cones that are used to describe a notion of approximate nondominance. Pareto-oriented scalarization methods are modified and several new solution approaches are proposed for these two classes of cones. In addition, necessary and sufficient conditions for nondominance with respect to a variable domination cone are developed, and some more specific results for the case of Bishop-Phelps cones are derived. Based on the above findings, a decomposition framework is proposed for the solution of multi-scenario and large-scale multiobjective programs and analyzed in terms of the efficiency relationships between the original and the decomposed subproblems. Using the concept of approximate nondominance, an interactive decision making procedure is formulated to coordinate tradeoffs between these subproblems and applied to selected problems from portfolio optimization and engineering design. Some introductory remarks and concluding comments together with ideas and research directions for possible future work complete this dissertation

Analyse post-Pareto en optimisation vectorielle stochastique et déterministe : étude théorique et algorithmes.

This thesis explore related aspects to post-Pareto analysis arising from Stochastic Vector Optimization Problem. A Stochastic Vector Optimization Problem is to optimize a random vector objective function defined on an arbitrary set, and taking values in a partially ordered set. Its solution set (called Pareto set) consists of the feasible solutions which ensure some sort of equilibrium amongst the objectives. That is to say, Pareto solutions are such that noneof the objectives values can be improved further without deterioring another. Technically speaking, each Pareto solution is acceptable. The natural question that arises is : how to choose one solution ? One possible answer is to optimize an other objective over the Pareto set. Considering the existence of a decision-maker with its own criteria, we deal with the post-Pareto Stochastic Optimization Problem of minimizing its real-valued criteria over the Pareto set.Cette thèse relate certains aspects liés à l'analyse post-Pareto issue de Problèmes d'Optimisation Vectorielle Stochastique. Un problème d'optimisation Vectorielle Stochastique consiste à optimiser l'espérance d'une fonction vectorielle aléatoire définie sur un ensemble arbitraire et à valeurs dans un espace sectoriel ordonné. L'ensemble des solutions de ce problème (appelé ensemble de Pareto) est composé des solutions admissibles qui assurent un certain équilibre entre les objectifs : il est impossible d'améliorer la valeur d'un objectif sans détériorer celle d'un autre. D'un point de vue technique, chaque solution de Pareto est acceptable. Nous nous posons alors le problème de la sélection de l'une d'entre elles : en supposant l'existence d'un décideur qui aurait son propre critère de décision, nous considérons le problème post-Pareto Stochastique qui vise à minimiser cette fonctionnelle sur l'ensemble de Pareto associé à un Problème d'Optimisation Vectorielle Stochastique

Multiobjective Problems of Mathematical Programming; Proceedings of an International Conference, Yalta, USSR, October 26 - November 2, 1988

IIASA's approach to research in Multiple Objective Decision Support, Multiple Criteria Optimization (MCO) and related topics assumes a high level of synergy between three main components: methodological and theoretical backgrounds, computer implementation and decision support systems and real life applications. This synergy is reflected in the subjects of papers presented at the Conference as well as in the structure of the Proceedings which is divided into three main sections. In the first section, "Theory and Methodology of Multiple Criteria Optimization," 21 papers discussing new theoretical developments in MCO are presented. The second section, "Applications of Multiple Criteria Optimization, " contains nine papers dealing with real-life applications of MCO. Five papers on the application of MCO in the development of Decision Support Systems are included in the final section, "Multiple Criteria Decision Support." Among the important outcomes of this Conference were conclusions regarding further directions of research for Multiple Criteria Optimization, in particular, in the context of cooperation between scientists from Eastern and Western countries