Research On Optimization Research On Optimality For Robust Optimization Problems For Robust

Uncertainty is ubiquitous in the natural world,engineering systems,and our social lives.In the real world applications,the parameters in many optimization models are uncertain,so it is very important to explore uncertain optimization.Nowadays,as one of the most effective approaches to deal with uncertain optimization problems,robust optimization is becoming more and more popular.In this thesis,some robustness concepts and optimality conditions of robust optimization problems are investigated by virtue of nonlinear scalarizing,set optimization and image space analysis.For uncertain scalar optimization problems,two kinds of robust counterparts are defined,and various robustness characterizations are established.Three concepts of robust efficiency for uncertain multiobjective optimization problems are introduced,and different kinds of robustness on vectorization counterparts and set orderings are explored on the frame of image space analysis.The thesis is divided into seven chapters and organized as follows:In Chapter 1,the background and recent researches on the optimization problems and robust optimization are firstly recalled.Subsequently,two famous nonlinear scalarizing functions and their applications,the developments of set orderings and set optimization,the basic features of image space analysis,as well as its related topics are described.Lastly,the motivations and the main research works of this thesis are stated.In Chapter 2,some notations,basic definitions,two kinds of important nonlinear functions including Gerstewitz function and the oriented distance function,as well as their properties are recalled.Several set orderings,image problem and(regular)weak separation functions in image space analysis are introduced,which will be used in the following context.In Chapter 3,robust counterparts of scalar robust optimization problems and relations to multiobjective optimization are considered.By means of the Benson's scalarization method and elastic constraint method in multiobjective optimization,two kinds of robustness concepts which can be characterized as special cases of a general nonlinear scalarizing approach and applied to risk measures in investment decision problems.Moreover,both constrained and unconstrained multiobjective optimization problems are introduced,and their relations to scalar robust optimization are discussed.Particularly,optimal solutions of scalar robust optimization problems are weakly efficient solutions for the unconstrained multiobjective optimization problem,and these solutions are efficient under uniqueness assumptions.In Chapter 4,concepts of robust efficiency for uncertain multiobjective optimization problems in the context of set orderings are considered.First of all,three concepts of robust efficiency are proposed by replacing set order relations with the Minmax less order relation,the Minmax certainly less order relation and the Minmax certainly nondominated order relation,respectively.Afterwards,some interpretations for these concepts are given,and the relations between new concepts and the existent concepts of efficiency are revealed.At last,these concepts are used to analyze tourist's destination selection problem.In Chapter 5,by virtue of the image space analysis,various robustness characterizations of general scalar robust optimization problems are investigated.Under mild assumptions,several robust solutions for different kinds of robustness concepts are characterized by introducing the corrected images of the original uncertain problem or the images of its counterpart problems appropriately,which provides a unified approach to tackling with robustness for uncertain optimization problems.Furthermore,some robust optimality conditions,especially saddle point sufficient optimality conditions for scalar robust optimization problems are obtained by means of linear and nonlinear(regular)weak separation functions.Finally,an application for finding a shortest path is given to verify the validity of the results.In Chapter 6,based on the frame of image space analysis,some concepts of multiobjective robust efficiency under set orderings and vectorization models are characterized.By using linear and nonlinear scalarization results for several set order relations,some suitable subsets of scalarization image space are introduced to make equivalent characterizations for Upper set,Lower set,Set and Certainly less ordered robustness,respectively.Subsequently,by introducing corrected image of original uncertain problem or the selected and corrected images of its robust counterpart,an equivalent relation between multiobjective robustness and the separation of two sets in the image space is well established.Some Lagrangian-type sufficient robust optimality conditions are presented by means of linear vector and scalar separation functions.Especially,under suitable restriction assumptions,Lagrangian-type necessary robust optimality conditions in terms of nonlinear separation functions are derived.In Chapter 7,the main results of this thesis are summarized,and some problems which are worth exploration are put forward.Above all,some perspectives of robust optimization on neural network are described from an application point of view.