摘要
复合优化问题是一类非常重要的优化问题,因为它不仅涵盖了一般意义下的优化问题,同时也为研究许多算法的收敛性提供了统一的结构框架。而凸复合优化问题是一类十分重要且基础的优化问题,但绝大多数实际的优化问题均是非凸且是多目标的。为满足实际问题的需要,许多学者对凸性作了多种形式的推广,其中锥广义不变凸性是一类重要的推广形式。因此,在锥广义不变凸性下研究复合非光滑多目标优化问题具有十分重要的理论意义和较强的应用价值。本文在锥广义不变凸性下主要研究一类复合非光滑多目标优化问题的最优性条件、(弱)鞍点定理及对偶。
第一章介绍复合优化问题及广义不变凸性的研究现状。
第二章介绍全文所需要的一些预备知识。
第三章主要研究复合非光滑多目标优化问题的最优性条件,其目标函数和约束函数均是局部Lipschitz和Gateaux可导的局部Lipschitz函数的复合。首先,利用广义择一定理和半无限Gordon定理,建立复合非光滑优化问题的最优性必要条件。其次,引入η?广义零空间条件概念,在此基础上给出复合非光滑多目标优化问题的最优性充分条件。最后,通过具体例子解释最优性充分条件。
第四章在锥广义不变凸性下建立复合非光滑多目标优化问题的(弱)鞍点定理。
第五章在锥广义不变凸性下,利用η?广义零空间条件,分别建立复合非光滑多目标优化问题的Mond-Weir型、Wolfe型以及混合型对偶结果。
第六章对全文作了简单总结并提出了一些有待进一步研究的问题。
本文的创新之处主要体现在第三章、第四章、第五章。
Composite optimization problem is a great important class of optimization problem. This is not only because it covers various common multiobjective problems, but it provides a unified framework for studying the convergence behaviour of many algorithms. Moreover, convex composite programming problem is an important and basic class of optimization problem. However, many programs in our real life are nonconvex and multiobjective. So as to meet the demands of solving practical problems, many generalizations to convexity have been made, among which the cone-invexity is an important form. Therefore, it is significant to study the composite nonsmmoth multiobjective programming under such generalized invexity. This paper is concerned with the optimality conditions, (weak) saddle point theorem and duality results for a class of composite nonsmooth multiobjective programs under the cone generalized invexity.
The outline of the thesis is as follows.
In chapter 1, we introduce the current situation of composite optimization problems and generalized invexity.
In chapter 2, we present some preliminaries for the full article.
In chapter 3, we focus on the optimality conditions of composite nonsmooth multiobjective programs, where objective function and the constraints are compositions of locally Lipschitz, and Gateaux differentiable and locally Lipschitz functions. First of all, the necessary optimality condition of composite nonsmooth programs is established by generalized alternative theorem and semi-infinite Gordon theorem. Furthermore, a new class of conditions namedη? generalized null space condition is formulated to discuss the sufficient optimality conditions. At last, some examples are given to show the sufficient optimality condition.
In chapter 4, the weak saddle point theorem and saddle point theorem are derived for the composite nonsmooth multiobjective programming under the cone generalized invexity.
In chapter 5, Mond-Weir、Wolfe and Mixed duality results are established for composite nonsmmoth multiobjective programs by the cone generalized invexity and η?generalized null space condition, which are weak dual theorems、strong dual theorems and restricted converse dual theorems.
In chapter 6, we give a summary of this paper and put forward some problems for further study.
The innovation of this thesis lies in the 3-th,4-th,5-th chapter.
引文
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