利用切向次微分研究双层规划的必要最优性条件
Necessary Optimality Conditions for Bilevel Programming using Tangential Subdifferentials
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中文总结 AI 辅助
本文利用切向次微分推导非光滑双层规划的必要最优性条件，提出序贯增强切向 Fritz--John 条件及相应 KKT 系统，并研究切向约束规范以保证其有效性，最后应用于值函数重构。
中文摘要 AI 辅助
双层规划是一类约束条件涉及参数优化问题解的优化问题。众所周知，值函数重构可提供等价的单层优化问题，但这会得到一个非光滑优化问题，且该问题通常不满足常规约束规范，如 Mangasarian--Fromovitz 约束规范（MFCQ）。本文利用切向次微分研究非光滑双层规划的必要最优性条件。我们针对一类非光滑非线性规划推导出序贯增强切向 Fritz--John 型条件，并通过对序贯条件取极限，得到相应的增强切向 Fritz--John 系统和增强切向 Karush--Kuhn--Tucker（KKT）系统。我们进一步研究了切向约束规范条件，包括 T-拟正规性和切向锥-连续性性质（T--CCP），并分析了这些条件如何保证切向 KKT 型条件的有效性。最后，将这些结果应用于双层规划问题的值函数重构。
英文摘要
The bilevel program is an optimization problem in which the constraint involves solutions to a parametric optimization problem. It is well known that the value function reformulation provides an equivalent single-level optimization problem, but it results in a nonsmooth optimization problem that never satisfies the usual constraint qualification, such as the Mangasarian--Fromovitz constraint qualification (MFCQ). In this paper, we study necessary optimality conditions for nonsmooth bilevel programming using tangential subdifferentials. We derive a sequential enhanced tangential Fritz--John type condition for a class of nonsmooth nonlinear programs and then obtain corresponding enhanced tangential Fritz--John and enhanced tangential Karush--Kuhn--Tucker (KKT) systems by taking limits of the sequential condition. We further investigate tangential constraint qualification conditions, including T-quasinormality and a tangential cone--continuity property (T--CCP), and analyze how they guarantee the validity of tangential KKT type conditions. Finally, these results are applied to the value-function reformulation of a bilevel programming problem.
发表机构
- School of Mathematics, Hunan University（湖南大学数学学院）
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