A MINRES-based Linesearch Algorithm for Nonconvex Optimization with Non-positive Curvature Detection

Abstract: We propose a MINRES-based Newton-type algorithm for solving unconstrained nonconvex optimization problems. Our approach uses the minimal residual method (MINRES), a well-known solver for indefinite symmetric linear systems, to compute descent directions that leverage second-order and non-positive curvature (NPC) information. Comprehensive asymptotic convergence properties are derived under standard assumptions. In particular, under the Kurdyka-Łojasiewicz inequality and a mild NPC-detectability condition, we prove that our algorithm can avoid strict saddle points and converge to second-order critical points. This is primarily achieved by integrating proper regularization techniques and forward linesearch mechanisms along NPC directions. Furthermore, fast local superlinear convergence to potentially non-isolated minima is established, when the local Polyak-Łojasiewicz condition is satisfied. Numerical experiments on the CUTEst test collection and on a deep auto-encoder problem illustrate the efficiency of the proposed method.