保守参数最优性与温和极小极大问题的岭方法
Conservative parametric optimality and the ridge method for tame min-max problems
AI总结：
本文研究无凸性、可微性等假设下极小极大问题岭方法的收敛性，证明在可定义语境中参数最优性公式提供保守场，使岭方法收敛至均衡集，并指出可定义性是关键。
摘要：
我们研究用于极小极大问题的岭方法，并在无任何凸性、可微性或限定条件假设下考察其收敛性。核心问题在于确定“参数最优性公式”是否提供了一个保守场，这是一种非常适合优化的广义导数概念。在半代数及更一般的可定义语境中，该问题的答案是肯定的。证明涉及对可定义保守场的一种新刻画，这具有独立意义。由此，应用于可定义目标函数的岭方法被证明具有极小化行为，并收敛至满足最优性条件的均衡集。可定义性对我们的证明至关重要：我们表明，对于更一般的非光滑函数类，参数最优性公式的保守性可能失效，从而导致岭方法的荒谬行为。
英文摘要：
We study the ridge method for min-max problems, and investigate its convergence without any convexity, differentiability or qualification assumption. The central issue is to determine whether the ''parametric optimality formula'' provides a conservative field, a notion of generalized derivative well suited for optimization. The answer to this question is positive in a semi-algebraic, and more generally definable, context. The proof involves a new characterization of definable conservative fields which is of independent interest. As a consequence, the ridge method applied to definable objectives is proved to have a minimizing behavior and to converge to a set of equilibria which satisfy an optimality condition. Definability is key to our proof: we show that for a more general class of nonsmooth functions, conservativity of the parametric optimality formula may fail, resulting in an absurd behavior of the ridge method.
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