论文标题
统一的研究针对最小值和Chebyshev问题的必要和足够的最佳条件和圆锥形约束的问题
A Unified Study of Necessary and Sufficient Optimality Conditions for Minimax and Chebyshev Problems with Cone Constraints
论文作者
论文摘要
我们提出了一项统一的研究，对最小值和Chebyshev优化问题的一阶和二阶和二阶的最佳条件与锥体约束。可以以几种不同的形式提出此类问题的一阶最佳条件：就拉格朗日乘数（KKT点）而言，就线性乘数（KKT点）而言，就次异性和正常锥体而言，就非平滑惩罚函数而言，就积极的Cadre Multipliers以及替代形式而言。我们描述了所有这些形式的必要形式和充分的最佳条件之间的互连，并证明看似不同的条件实际上是等效的。我们还展示了如何以更方便的形式重新重新制定锥体限制优化问题，并扩展经典的二阶优化条件，以使平滑锥体约束问题的平滑锥体限制性问题限制为具有锥体约束的Minimax和Chebyshev问题。本文获得的最佳条件为开发新的有效结构探索方法开发了解决锥体约束的最小值和Chebyshev问题的方法。
We present a unified study of first and second order necessary and sufficient optimality conditions for minimax and Chebyshev optimisation problems with cone constraints. First order optimality conditions for such problems can be formulated in several different forms: in terms of a linearised problem, in terms of Lagrange multipliers (KKT-points), in terms of subdifferentials and normal cones, in terms of a nonsmooth penalty function, in terms of cadres with positive cadre multipliers, and in an alternance form. We describe interconnections between all these forms of necessary and sufficient optimality conditions and prove that seemingly different conditions are in fact equivalent. We also demonstrate how first order optimality conditions can be reformulated in a more convenient form for particular classes of cone constrained optimisation problems and extend classical second order optimality condition for smooth cone constrained problems to the case of minimax and Chebyshev problems with cone constraints. The optimality conditions obtained in this article open a way for a development of new efficient structure-exploiting methods for solving cone constrained minimax and Chebyshev problems.
