M次微分下多目标优化问题的最优性条件

Optimality Conditions for Multiobjective Optimization Problems in Terms of the M Subdifferential

主要考虑了一类带有不等式约束的非光滑多目标优化模型。在多目标优化的研究过程中,解的最优性条件一直是众多学者关注的内容。而对于非光滑多目标优化问题,利用广义微分的概念对其进行研究是非常有意义的研究课题。经典的广义微分工具包括Clarke广义梯度、Mordukhovich次微分等等。利用Mordukhovich次微分的概念对多目标优化问题解的最优性条件进行推广,其中Mordukhovich次微分简记为M次微分。利用Mordukhovich伪凸的概念,建立了在M次微分意义下多目标优化问题弱有效解的必要充分最优性条件及其有效解的充分最优性条件。同时引入了M次微分意义下的线性化锥,并且利用该线性化锥构造了一个有效解的最优性条件的等价描述。最后对非光滑多目标优化的后续研究工作提出了一些可扩展的研究内容和问题。

Throughout this paper,we consider the nonsmooth multiobjective programming problem with inequality constraints.In the studying of the multiobjective programming problem,many researchers focus on the optimality of the solutions.What＇s more,for the nonsmooth multiobjective programming problem,it is meaningful to discuss in the term of the generalized differential.The classical generalized differential tools include the Clarke subdifferential and the Mordukhovich＇s subdifferential.In our paper,we mainly use the Mordukhovich＇s subdifferential to study the nonsmooth multiobjective programming problem.The M subdifferential is short of the Mordukhovich＇s subdifferential.In this paper,we introduce the definition of the Mordukhovich＇s pseudoconvex.Then the necessary and sufficient optimality condition of the weak efficiency solutions and the sufficient optimality condition of the efficiency solutions are given.What＇s more,we introduce the concept of the linearizing cones by using the Mordukhovich＇s subdifferential and propose an equivalent description of the efficient solutions＇optimality condition with it.Finally,we give some further research problems for following study.