The adjoint state method for parametric definable optimization without smoothness or uniqueness
This work offers a theoretical tool for handling nonsmooth and non-unique parametric optimization, potentially enhancing primal-dual solvers, but it appears incremental as it builds on existing adjoint methods with added conditions like definability.
The paper tackles parametric optimization problems with nonsmooth objectives and constraints, deriving an adjoint state formula under a qualification condition that yields a conservative field for the value function without differentiating the solution mapping. This provides a computable first-order object applicable to a wide range of problems, as demonstrated through examples showing failure without conservativity or definability.
Definable parametric optimization problems with possibly nonsmooth objectives, inequality constraints, 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.