Scalarization Methods Of Multi-objective Optimization Problems And Its Applications In Machine Learning

In recent years,as a very important direction in the optimization theory and its applications,multi-objective optimization has developed very rapidly.It has become a prevailing research focused in Optimization.Scalarization method is one of the ba-sic methods to deal with multi-objective optimization problems.The in-depth study of the scalarization methods and scalarization properties of various solutions for multi-objective optimization problems will not only provide important theoretical basis and technical support for the design of efficient algorithms for solving multi-objective opti-mization problems and the application of multi-objective optimization models and meth-ods to solve many practical problems in many fields,such as economic management,engineering design,transportation,ecological protection and optimal decision-making.In the meantime,it will also promote the development of multi-objective optimization theory and method itself.This thesis is mainly devoted to the scalarization property and application of multi-objective optimization problems,which mainly includes three aspects:the equivalent scalarization characterization for the exact solutions of multi-objective optimization problems,some sufficient conditions and necessary conditions for the approximate solutions of multi-objective optimization problems using scalariza-tion methods,and the reconstruction of the regularization models in machine learning based on the scalarization method and its applications.This thesis has five chapter and main points of each chapter are as follows:In Chapter 1,the background and significance about the research of the theory,methods and applications of multi-objective optimization are briefly described.The his-tory and status of the research directions closely related to this thesis are the definition of exact solutions and approximate solutions of multi-objective optimization problems,scalarization properties of various kinds of exact solutions and approximate solutions,multi-objective optimization problems and machine learning,especially the regulariza-tion methods and applications of sparse optimization problem are summarized,and then the main contents of this thesis are put forward.Chapter 2 is focused on some equivalent scalarization characterization results of weakly efficient solution,efficient solution,strictly efficient solution and properly effi-cient solution for multi-objective optimization problems.Based on the ideas of Tcheby-cheff norm and Epsilon-constraint scalarization,the generalized Tchebycheff norm scalarization model,the generalized Tchebycheff norm scalarization model with surplus variables and the generalized Tchebycheff norm scalarization model with slack variables and surplus variables are proposed for multi-objective optimization problems,and then based on these three kinds of scalarization models respectively,the parameters range in the scalarization models is adjusted to establish some nonlinear equivalent scalarization characterization results of weakly efficient solutions,efficient solutions,strictly efficient solutions and properly efficient solutions for multi-objective optimization problems,and some concrete examples are given to analyze and explain the main results.In addition,some relations and basic properties among efficient solution,weakly efficient solution and properly efficient solution for a class of multi-objective optimization problems with special structure are also studied in this chapter.Besides,some scalarization results of the solution are established,and applications of the multi-objective optimization model in the location problem and data fitting problem are given.Chapter 3 mainly includes some scalarization results of approximate solutions of multi-objective optimization problems,such as ?-weakly efficient solution,?-efficient solution and ?-properly efficient solution.Firstly,the necessary conditions and sufficient conditions of ?-weakly efficient solution,the sufficient conditions of ?-efficient solution and the necessary conditions and sufficient conditions of ?-properly efficient solution are established by using the generalized Tchebycheff scalarization model with surplus variables and the generalized Tchebycheff scalarization model with slack variables and surplus variables respectively.Furthermore,some sufficient conditions and necessary conditions of several kinds of approximate solutions for multi-objective optimization problems are established by using the Flexible Pascoletti-Serafini scalarization model and Modified Pascoletti-Serafini scalarization model proposed by Akbari,Ghaznavi and Khorram.Finally,some quasi interior properties of improvement sets are studied,and a linear scalarization result of weakly efficient solutions defined by quasi interior is es-tablished,and the linear scalarization result is pointed out through specific examples that the generalized convexity condition in the results can not be weaken to the classical near-subconvexlikeness.In Chapter 4,the reconstruction model of regularization model in machine learning and its application are studied.A kind of reconstruction form of regularization model in machine learning is proposed by using the classical Tchebycheff norm scalarization model and method of multi-objective optimization problems.Then the application of reconstruction model in the signal recovery is studied.Chapter 5 summarizes the research work of the whole thesis and puts forward some ideas and prospects for the future research work.