Multi-objective Optimization Problems And Its Application In Financial Investment

As a most important branch of optimization theory and its applications,multi-objective optimization’s theoretical research related to many branches such asconvex analysis, nonsmooth analysis, nonlinear analysis, etc. At the same time,multi-objective optimization also has important applications in many areas of nationallife. In multi-objective optimization problems, there are still some meaningful hotissues, such as the theoretical research of approximate solution, intelligentoptimization algorithm, applications in practical problems and so on. Many authorshave gained a lot of results. Inspired by these results, we discussed some questionsfocusing on-quasi weakly efficient solution in multi-objective optimization,non-dominated sorting genetic algorithm with elite strategy (NSGA-Ⅱ),multi-objective optimization in financial investment. Our main work was as follows:In chapter Ⅰ, we looked back on some research in multi-objective optimization andintroduced the preliminaries.In chapter Ⅱ, first of all, on the basis of, maintainning convexity assumptionsunchanged, we increased the equality constraint of the MP and extended it to the VPwith some general constraints. We defined a weak saddle point of VP,and introduced adual model. We get a lot of results, such as the existence of sufficient and necessaryconditions of-quasi weakly efficient solution and the saddle point in multiobjectiveoptimization of VP problem and we proved the corresponding duality theorem. Then,under the semi-convexity assumptions, necessary and sufficient conditions wereobtained for the existence of-quasi weakly efficient solution in multiobjectiveoptimization problem and the duality theorem was proved.In chapter Ⅲ, on the basis of the review of the existing multi-objectiveoptimization algorithm, we discussed a class of intelligent optimization algorithms ofmulti-objective optimization-NSGA-Ⅱ(non-dominated sorting genetic algorithmwith elite strategy). We gave the ideas of NSGA-Ⅱ and its processes,analyzed Paretoranking strategy used in the non-dominated sorting link’s defects, and proposed theimprovement of the cumulative sorting fitness assignment strategy algorithm, andmade a comparison between the improved algorithm and the original one in terms ofconvergence, and at last we improved that the former was more stable andefficient.In chapter Ⅳ, we considered two financial investment issues: the portfolioinvestment in friction market and the optimal allocation of bank capital. Using multi-objective optimization theory, when the transaction fee was a piecewisefunction, with opportunity costs, transaction costs constrained, we constructed amulti-objective non-linear portfolio optimization model.Under the restrictions ofexpected losses and rate of return in the actual expected, we built a mean-VaR modelof the optimal allocation of bank capital. We designed a model using the improvedNSGA-Ⅱ algorithm, and finally we stimulated the designed model using MATLAB,and solved both two in a better way.