摘要
广义凸性在数学规划与最优化理论中具有十分重要的作用。它们在一定程度上保留了凸函数的一些优秀性质,是凸函数的拓广与发展。目前,许多学者已经研究了各类广义凸性的条件下各类优化问题的最优性条件,鞍点,对偶理论等。
本文主要研究了两类广义凸性即r -半预不变凸性和非光滑的B - ( p,r)-不变凸性。以及在这两种广义凸性假设条件下多目标优化问题的最优性条件、对偶理论等。主要内容包括:
第一章介绍了研究的理论意义,应用意义及广义凸性的一些研究进展。
第二章在r -半预不变凸性假设条件下利用弧式方向可微研究了一类多目标优化问题(MP)的最优性条件及对偶理论。在r -半预不变凸性条件建立了多目标优化问题的KKT必要条件和KKT充分条件;同时,考虑了多目标优化问题的Mond-Weir对偶模型并在r -半预不变凸性条件证明了弱对偶定理、强对偶定理、逆对偶定理等对偶结果。
第三章主要研究了多目标优化问题的混合对偶。在r -半预不变凸性假设条件下证明了多目标优化问题的混合对偶模型的弱对偶定理、强对偶定理和逆对偶定理等;在非光滑B - ( p,r)-不变凸性条件假设下研究了一类带等式和不等式约束的非光滑多目标规划问题(NMOP)的混合对偶的弱对偶定理、强对偶定理、逆对偶定理等对偶结果。
第四章对全文进行了总结,并提出了一些可以进一步开展研究工作的思路。
全文的创新之处主要体现在第二章和第三章。
Convexity is an important mathematical concept. For the need of solving practical problems, People has generalized the convexity from different point of view.Invexity is an important kind of generalized forms. So,studying the generalized forms of convexity and applications in optimization theory is very important and interesting.This thesis study two kinds of generalized convexity--r-semipreinvex and B-(p,r)-invexity. We consider the optimality conditions and duality for multiobjective under these generalize- d convexity assumptions.
Chapter 1 acts as the general introduction to the significance and current situation in the study of the invexity.
Chapter 2 consider the optimality conditions and duality for multiobjective progra- mming (MP) under the arcwise directionally differentiable conditions in the r-semipreinvex assumption.Under the assumption of r-semipreinvex, we establish the KKT necessary conditions and the KKT sufficient conditions.Meanwhile,we study the Mond-Weir weak duality, strong duality, deverse duality etc for multiobjective programming.
Chapter 3 consider a class of nonsmooth multiobjective programming (NMOP) with equality and inequality constraints, and establish the mixed dual model for this programming; and so discuss the duality theories between the dual problem and the primal problem in terms of Clarke subdifferential, under the assumption of the nonsmoooth B-(p,r)-invxity. And we study the mixed weak duality, strong duality, deverse duality etc for multiobjective programming (MP) under the assumption of r-semipreinvex.
Chapter 4 comes to a conclusion.In the meanwhile, it put forward some problems for further study.
In this thesis, main results gather in the 2-th,3-th chapter.
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