多目标优化问题近似解的标量化研究

发布时间：2018-01-08 00:01

本文关键词：多目标优化问题近似解的标量化研究　出处：《重庆师范大学》2016年硕士论文　论文类型：学位论文

【摘要】：多目标优化是数学规划学科中的重要分支学科,是具有重要应用价值和多学科交叉的研究领域.这一问题的理论研究涉及到凸分析、非光滑分析等多门学科.而且,在经济规划、环境保护、金融管理以及社会可持续发展等管理决策中,都存在大量的多目标优化问题.本文主要研究了多目标优化问题拟近似解的非线性标量化和向量变分不等式问题近似解的最优性条件及其与向量优化问题近似解的关系.本文的主要内容安排如下：1.第一章简要叙述了多目标优化问题的研究意义,并对多目标优化及其与本文相关的研究方向的研究现状进行了综述,继而提出了本文主要研究内容.2.第二章研究了多目标优化问题拟近似解的非线性标量化.首先,我们在一般锥序下,利用非线性标量化方法,在没有任何凸性条件下,给出了多目标优化问题拟近似(弱)有效解的充分和必要条件.然后,我们在自然锥序下,利用两种非线性标量化方法,在没有任何凸性条件下,给出了多目标优化问题拟近似(弱、真)有效解的的充分和必要条件,并利用范数对多目标优化问题的拟近似(弱)有效解进行了非线性标量化刻画.3.第三章研究了向量变分不等式问题近似解的最优性条件及其与向量优化问题近似解的关系.首先,我们在Banach空间中,利用co-radiant集定义了向量变分不等式问题一类新的近似解,并研究了这种近似解的最优性条件.然后,我们在Rn空间中研究了这类向量变分不等式问题的近似解与向量优化问题近似解的关系.最后,我们在一定的广义凸性条件下研究了一类向量变分不等式问题的解与向量优化问题的拟近似解之间的关系,并在拟近似伪凸的条件下,证明了向量优化问题的临界点和拟近似弱有效解与向量变分不等式问题的解之间的等价性.
[Abstract]:Multi-objective optimization is an important branch of mathematics planning, and it is a research field with important application value and multi-disciplinary intersection. The theoretical study of this problem involves convex analysis. Non-smooth analysis and other disciplines. And, in economic planning, environmental protection, financial management and social sustainable development and other management decisions. In this paper, we mainly study the optimality conditions of the approximate solutions of quasi approximate solutions of multiobjective optimization problems and vector variational inequalities and their approximate solutions to vector optimization problems. The main contents of this paper are as follows:. 1. In the first chapter, the research significance of multi-objective optimization problem is briefly described. The research status of multi-objective optimization and its related research directions are summarized. In chapter 2, we study the nonlinear scalarization of the quasi approximate solution of the multiobjective optimization problem. Firstly, we use the nonlinear scalarization method under the general cone order. In this paper, the necessary and sufficient conditions for quasi-approximate (weak) efficient solutions of multi-objective optimization problems are given without any convexity. Then, we use two nonlinear scalar quantization methods under natural cone order. In this paper, the necessary and sufficient conditions for quasi-approximate (weak, true) efficient solutions of multi-objective optimization problems are given without any convexity. Quasi approximation (weak) for multiobjective optimization problem using norm. In chapter 3, we study the optimality condition of the approximate solution of vector variational inequality problem and its relation with the approximate solution of vector optimization problem. In Banach spaces, we define a class of new approximate solutions of vector variational inequality problems by using co-radiant sets, and study the optimality conditions of such approximate solutions. We study the relation between the approximate solution of the vector variational inequality problem and the approximate solution of the vector optimization problem in rn space. In this paper, we study the relationship between the solutions of a class of vector variational inequality problems and the quasi approximate solutions of vector optimization problems under certain generalized convexity conditions. The equivalence between the critical point and quasi-approximate weak efficient solution of vector optimization problem and the solution of vector variational inequality problem is proved.
【学位授予单位】：重庆师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O221.6

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