The adjoint state method for parametric definable optimization without smoothness or uniqueness

TL;DR

This paper introduces an adjoint state formula for nonconvex, possibly nonsmooth parametric optimization problems, enabling first-order analysis without smoothness or solution uniqueness.

Contribution

It develops a novel adjoint construction that works under minimal conditions, extending the applicability of first-order methods to broader classes of problems.

Findings

The adjoint state formula applies to nonconvex, nonsmooth problems.

It provides a conservative field for the value function without differentiating the solution map.

The method is relevant even when formal adjoint constructions fail in smooth cases.

Abstract

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.