参数可定义优化的伴随状态法无需光滑性或唯一性
The adjoint state method for parametric definable optimization without smoothness or uniqueness
AI总结：
本文提出一种无需光滑性或解唯一性的参数可定义优化方法，通过伴随状态公式构造保守场，为价值函数提供可计算的一阶对象，适用于非凸问题及非唯一解的情况。
摘要：
我们证明了非凸的参数可定义优化问题，即使目标函数非光滑、不等式约束、锥约束系统以及非唯一原解和对偶解，只要满足一个资格条件，就存在伴随状态公式。伴随构造为价值函数提供保守场的选择，无需求解映射的微分。通过例子表明，即使在光滑问题中，正式的伴随构造在缺乏保守性或可定义性时也会失效，这凸显了这些概念在理解该方法理论方面的重要性。本文提供了一种可以直接与现有原对偶求解器结合的工具，适用于广泛参数优化问题。
英文摘要：
We establish that nonconvex definable parametric optimization problems with possibly nonsmooth objectives, inequality constraints, conic constraint systems, and non-unique primal and dual solutions admit an adjoint state formula under a mere qualification condition. The adjoint construction yields a selection of a conservative field for the value function, providing a computable first-order object without requiring differentiation of the solution mapping. Through examples, we show that even in smooth problems, the formal adjoint construction fails without conservativity or definability, illustrating the relevance of these concepts to grasp theoretical aspects of the method. This work provides a tool which can be directly combined with existing primal-dual solvers for a wide range of parametric optimization problems.
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