Research On Robust Optimization And Related Problems

Robust optimization is one branch of optimization under uncertainty,which deals with opti-mization problems subject to uncertainty.In robust optimization,uncertainty takes the form of uncontrollable parameters on which all information available is summarized in an un-certainty set.Owing to the practical requirement and its effective implementation,robust optimization has become of importance and wide use in optimization under uncertainty.On the basis of the concept of robustness,this dissertation is concerned with robust optimization and some related vital topics from infinitely many objective optimization problems,vector optimization problems under uncertainty and complementarity problems under uncertainty.Main contributions are listed as follows.1.After analyzing high robustness,unanimous robustness and strict robustness elaborately,we introduce a new robustness called relaxed robustness.In terms of adjustable variables,relaxed robustness contains many models in the literature which are intended to relieve the over-conservatism of strict robustness,such as deviation robustness,reliable robustness,soft robustness,and expected-value formulation and risk-aversion formulation in stochastic op-timization.This unified framework shows that the model to be chosen depends on the infor-mation Decision-Makers have,their attitudes to this information and available methods in mathematics.Besides,the robustness measure is extracted,and its basic properties are stud-ied including translation equivariance,monotonity,positive homogeneity and convexity.2.With respect to the order relation built on component-wise comparisons,Pareto efficien-cy and Geoffrion proper efficiency are introduced for optimization problems with infinitely many objectives.The intimate relationships between infinitely-many-objective optimization and uncertain/robust optimization are revealed.For general optimization problems under uncertainty,we analyze the Pareto efficiency of robust solutions and obtain methods of gen-erating Pareto robust solutions via a generalized ?-constraint method.In terms of a family of cones,we define the proper efficiency which maintains Geoffrion's structure,and uncover the essential difference:Pareto efficiency requires bounded tradeoffs for any other elements while Geoffrion proper efficiency needs uniformly bounded tradeoffs for others.It is of in-terest that Hurwicz decision rule can be derived by applying Geoffrion proper efficiency to the robust counterpart,thus two famous concepts from different fields are connected.3.Following the line in robust scalar optimization,we establish the first robust counterpart in the hard sense for the vector optimization problems under uncertainty.Out of overcoming its deficiency,we relax it in the light of Pareto efficiency to the second robust counterpart in the tight sense.These two robust models belong to vector methods of dealing with uncertain multiobjective/vector optimization,while in the literature set methods are more prevail.Comparisons between robust models from these two categories confirm that vector methods have their particularity and potential for further development.4.For the complementarity problem with fuzzy parameters,using possibility measure and necessity measure from possibility theory,we propose two deterministic models called possibility-satisficing counterpart and necessity-satisficing counterpart.Next,we analyze these two models elaborately with an emphasis on the latter,and present characterizations of their solutions from different aspects.Then,we compare this methodology with others of dealing with different uncertainty in complementarity problems,such as fuzzy comple-mentarity problems for fuzzy mappings,robust complementarity problems for ambiguity and stochastic complementarity problems for risk.Finally,we extend the methodology to fuzzy optimization,fuzzy game and mathematical programs with fuzzy complementarity constraints.