New optimization method achieves O(K^{-1/3}) convergence for nonconvex constraints

Linglingzhi Zhu and Jiajin Li propose a new first-order augmented Lagrangian method for solving nonconvex optimization problems where both the objective and inequality constraints have a convex-composite structure (i.e., convex outer functions composed with smooth nonlinear inner mappings). The key challenge addressed is that in nonconvex functional inequality systems, constraint violation and unbounded multipliers complicate analysis. The authors restrict the dual variable to an auxiliary compact set and reformulate the problem as a nonsmooth nonconvex-concave minimax problem.

Their main theoretical contribution is a finite-time mechanism that converts stationarity of the truncated minimax problem into a KKT certificate for the original constrained problem. They show that for a sufficiently large penalty parameter, most iterates enter a near-feasible region where a local conic regularity condition uniformly bounds prox-linear multipliers. This yields explicit convergence rates: with dual regularization, they achieve a global O(K^{-1/3}) rate; under additional assumptions like piecewise linear outer functions, a local dual error bound gives the sharper O(K^{-1/2}) rate. This work advances the theory of primal-dual methods for nonconvex optimization and has implications for machine learning and control.