Stability Analysis For Vector Optimization And Equilibrium Problems

It is the core task of optimization theory research to solve optimization prob-lems. However, most of the optimization problems are difcult to solve preciselyand directly. We can only construct approximate problems on the basis of orig-inal problems, and approximate the solutions of the original problems by usingthe solutions of the approximate problems. Therefore, the basic requirement ofthe algorithm design is that the solutions of the approximate problems shouldconverge to the solutions of the original problem when the parameters and func-tions in the approximate problems converge to the corresponding parameters andfunctions in the original problems. In the practical applications, the parametersand functions in the optimization problems are obtained from empirical data, andthere exists systematic error. In addition, the algorithm is usually achieved bycomputer programming. The rounding error is unavoidable. At the same time, inorder to improve the speed of operation, we will have to make some approxima-tions. Therefore, in order to meet the requirements of practical applications, thesolutions of the optimization problems must have certain stability. Defne the set-valued mapping called the solution mapping with the parameters and functionsof the optimization problems as independent variables, and their solution sets asdependent variable. Then the stability of the optimization problems is identifedwith the continuity and variational properties of this solution mapping.Most of the present literatures are devoted to the stability of the solutionswith respect to the parameters. The parametric approximate problems are con-structed by appending parameters to the original problems, and the continuity and variational properties of the solutions are analyzed with respect to the pa-rameters. However, the perturbations of the errors are not in accordance with thegiven parameters. Parametric stability can not ensure the stability of the solutionswith respect to the errors. In addition, as the range of applications of optimizationtheory continuously expands, non-parametric algorithms emerge. The basic ideaof non-parametric algorithms is to design algorithms by constructing approximateproblems in the function space.In this thesis, we study the non-parametric stability for semi-infnite vector op-timization problems, parametric vector optimization problems, vector equilibriumproblems and vector quasiequilibrium problems under functional perturbations.Establish a topological structure on the corresponding functional set, and discussthe upper semicontinuity, lower semicontinuity, closedness, Ho¨lder continuity ofthe solution mappings of the above-mentioned optimization problems and the es-sentiality of the solutions, respectively. In addition, we also analyze the density ofstable vector optimization and equilibrium problems. Obviously, the parametricstability is a special case of the non-parametric stability. Our results in this thesisgeneralize the related results in some literatures.In Chapter2, we devote to the continuity of solution mappings for semi-infnite vector optimization problems without compact constraint. The sufcientconditions for lower semicontinuity and upper semicontinuity of solution map-pings under functional perturbations of both objective functions and constraintfunctions are established. Some examples are given to analyze the assumptions inthe main results. We also show that every convex semi-infnite vector optimiza-tion problem can be arbitrarily approximated by stable convex semi-infnite vector optimization problems, i.e., the set of all stable convex semi-infnite vector opti-mization problems (that is, their the weak solution mappings are continuous orthe solution mappings are upper semicontinuous) is dense in the set of all convexsemi-infnite vector optimization problems with the given topology.In Chapter3, the concepts of essential solutions and essential solution sets forparametric vector optimization problems are introduced, and the relations amongessential solutions, essential solution sets and lower semicontinuity of solutionmappings are discussed. The characterizations of essential solutions are presented,and some sufcient conditions for the closedness of solution mappings are obtained.Finally, some corollaries of the main results are given as applications for somespecial optimization models.In Chapter4, the sensitivity analysis for vector equilibrium problems underfunctional perturbations is discussed. We show that the solution mapping is anupper semicontinuous set-valued mapping. The sufcient conditions for lowersemicontinuity and Ho¨lder continuity of the solution mapping are established.Finally, we derive some corollaries for special cases of vector equilibrium problemsas examples.In Chapter5, the continuity of solution mapping for vector quasiequilibriumproblems under mapping perturbations is investigated. We show that the solu-tion mapping is upper semicontinuous and Hausdorf upper semicontinuous. Thesufcient conditions for lower semicontinuity and Hausdorf lower semicontinuityof the solution mapping are established. Finally, we apply our results to trafcnetwork problems as example.