【作者】 赵丹；

【导师】 李军；

【作者基本信息】 西华师范大学 ， 应用数学， 2022， 硕士

【摘要】 研究解的最优性条件在完善最优化理论中起着不可或缺的作用,为设计优化问题的求解算法奠定了关键的理论基础。本文主要利用Suzuki^[1]研究的GP次微分的性质,讨论可行集在解处的法锥后,得到拟凸半无限优化中解的最优性条件;类似地方法用来讨论目标函数的下水平集,将Kanzi和Soleimani-damaneh^[2]中连续强拟凸函数的假设弱化为上半连续强拟凸后,得到拟凸多目标优化问题弱有效解的最优性条件。文章主要由以下五个章节组成:第一章,介绍了拟凸优化问题中解的最优性条件研究背景与意义,以及该领域研究现状和本文结构内容。第二章,给出了本文研究所需要的基础知识以及介绍了部分重要的性质定理。第三章,考虑在Slater约束规范和适当假设下,利用Suzuki^[1]介绍的下水平集在点处的法锥与GP次微分之间的关系和性质,将文献[1]在带不等式约束的拟凸优化问题中刻画可行集在解处的法锥的方法推广到有限维Euclidean空间中的拟凸半无限优化问题,得到解的最优性条件。进一步讨论退化为有限个约束函数以及凸半无限优化的结果。第四章,考虑有限维Euclidean空间中的拟凸多目标优化问题,借鉴Suzuki^[1]在带不等式约束的拟凸优化问题中刻画可行集在解处的法锥的方法,对所有目标函数的下水平集的交集在解处的法锥进行刻画,将文献[2]给出的目标函数连续强拟凸假设弱化为上半连续强拟凸后,得到弱有效解的最优性条件。同时,讨论了退化为单目标情形以及可行集的变化。第五章,对全文进行了总结,阐述了本文的主要成果以及未来的研究展望。更多还原

【Abstract】 It is indispensable to explore the optimality condition of solution when improving the optimization theory,which lays a key theoretical foundation for the designing algorithm to solve the optimization problems.In this dissertation,the optimality condition of the solution in quasiconvex semi-infinite optimization is obtained by using the properties of GP subdifferential studied by Suzuki[1],after discussing the normal cone of the feasible set at the solution;the optimality condition of weakly efficient solution of quasiconvex multiobjective optimization problem is obtained after weakening the assumption of continuous strongly quasiconvex function in Kanzi and soleimani damaneh[2] to upper semicontinuous strongly quasiconvex by using similar method to discuss the lower level set of the objective function.This dissertation consists of five chapters as follows:In the first chapter,the research background and the meaning of the optimality conditions of the solutions in the quasiconvex optimization problems is introduced,as well as the current situation of the research in this field and the structure and content of this dissertation.In the second chapter,the necessary basic knowledge for the research discussed in this thesis and some significant property theorems are introduced.In the third chapter,considering the relationship and properties between the normal cone at the lower level set and the GP subdifferential introduced by Suzuki[1] under the Slater constraint specification and appropriate assumptions,the optimality conditions of solution is obtained by extending the characterization of the normal cone of the feasible set at the solution of the quasiconvex optimization problem with inequality constraints by Suzuki[1] to the quasiconvex semi-infinite problem in finite dimensional Euclidean space.The results of reduction to finite constraint function and convex semi-infinite optimization are further discussed.In the fourth chapter,considering the quasiconvex multiobjective optimization problem in a finite dimensional Euclidean space,the optimality condition of weakly efficient solution is obtained after weakening the assumption of continuous strongly quasiconvex objective function given by reference[2] to upper semicontinuous strongly quasiconvex by using Suzuki’s method of describing the normal cone of the feasible set at the solution of the quasiconvex optimization problem with inequality constraints to describe the intersection of the lower level sets of all objective functions.At the same time,the degradation to a single objective and the change of feasible set are discussed.In the fifth chapter,the summarization of the whole dissertation is given.The main achievements and the prospects of the research in the future are also be expounded.更多还原