Moving higher-order Taylor approximations method for smooth constrained minimization problems

Abstract

In this paper we develop a higher-order method for solving composite (non)convex minimization problems with smooth (non)convex functional constraints. At each iteration our method approximates the smooth part of the objective function and of the constraints by higher-order Taylor approximations, leading to a moving Taylor approximation method (MTA). We present convergence guarantees for MTA algorithm for both, nonconvex and convex problems. In particular, when the objective and the constraints are nonconvex functions, we prove that the sequence generated by MTA algorithm converges globally to a KKT point. Moreover, we derive convergence rates in the iterates when the problem data satisfy the Kurdyka-Lojasiewicz (KL) property. Further, when the objective function is (uniformly) convex and the constraints are also convex, we provide (linear/superlinear) sublinear convergence rates for our algorithm. Finally, we present an efficient implementation of the proposed algorithm and compare it with existing methods from the literature.