FlexCone: An Always-Feasible First-Order Solver for Conic Optimization

Conic subproblems produced by linearization or convexification, as in sequential convex programming (SCP) and related methods, are frequently infeasible, even when the original nonlinear constraints are satisfiable. Existing solvers certify such infeasibility but return nothing to act on, so controllers fall back on stale solutions that violate constraints. We propose FlexCone, an always-feasible first-order conic solver that handles feasible and infeasible problems within a single framework. FlexCone applies ADMM to an exact relaxation of the conic program that provably recovers the original solution when the problem is feasible and converges to a minimum-violation point when it is not. Because every instance runs the same operations, FlexCone batches efficiently on a GPU and can be trained using deep unfolding. On standard convex benchmarks, learned FlexCone matches or improves upon learned projection-based solvers. In closed-loop model predictive control, it is competitive on feasible subproblems and recovers from infeasible ones, eliminating the constraint violations and collisions of the standard fallback. On neural network robustness certification, where infeasibility of an LP relaxation certifies robustness, our batched solver returns certificates orders of magnitude faster than baseline solvers on the same formulations.