可能不连续函数和非紧决策集参数最优解的连续性
Continuity of Parametric Optima for Possibly Discontinuous Functions and Noncompact Decision Sets
摘要：
本文研究了值函数的连续性和参数最优化问题的解。这些问题在运筹学、控制和经济学中很重要，因为最优性方程是它们的特殊情况。经典的Berge极大值定理给出了值函数连续和解集值函数上连续的充分条件。Berge极大值定理假定目标函数是连续的，可行集的多功能是紧值的。这些假设在许多应用问题中并不满足，这在历史上限制了定理的相关性。本文从三个方面推广了Berge最大值定理：（i）目标函数不一定连续，（ii）可行集的多值性不一定是紧值的，（iii）给出了充要条件.为了说明主要定理，本文提供了应用程序的库存控制和鲁棒优化分析可能非紧的行动集和不连续的目标函数。
英文摘要：
This paper investigates continuity properties of value functions and solutions for parametric optimization problems. These problems are important in operations research, control, and economics because optimality equations are their particular cases. The classic fact, Berge's maximum theorem, gives sufficient conditions for continuity of value functions and upper semicontinuity of solution multifunctions. Berge's maximum theorem assumes that the objective function is continuous and the multifunction of feasible sets is compact-valued. These assumptions are not satisfied in many applied problems, which historically has limited the relevance of the theorem. This paper generalizes Berge's maximum theorem in three directions: (i) the objective function may not be continuous, (ii) the multifunction of feasible sets may not be compact-valued, and (iii) necessary and sufficient conditions are provided. To illustrate the main theorem, this paper provides applications to inventory control and to the analysis of robust optimization over possibly noncompact action sets and discontinuous objective functions.
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