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

arXiv: https://arxiv.org/abs/2603.26503 · HTML: https://mlsanity.com/paper/2603.26503/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Mar 2026.

## Built on by (most cited first, 1 of 1)

- [dOPT: Differentiating Conic Optimization via Geometric Reduction](https://mlsanity.com/paper/2609.33828/) (2026)

## Abstract (arXiv)

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.