Minimizing Smooth Kurdyka-Łojasiewicz Functions via Generalized Descent Methods: Convergence Rate and Complexity

Abstract

This paper introduces a generalized descent algorithm (DEAL) for minimizing smooth nonconvex functions. If the objective function is nonsmooth, a smoothing technique (e.g., forward-backward and high-order Moreau envelopes) is applied to generate a smooth counterpart. The proposed framework unifies several methods, such as gradient-based methods with constant step-sizes and Armijo line search, and several proximal splitting methods. The method is built around a generalized descent inequality that adapts the amount of decrease to the geometry of the objective function. Under the Kurdyka-Łojasiewicz (KL) property, we establish global convergence of the generated sequence to critical points and provide a unified convergence rate analysis. In particular, we show that the convergence behavior depends jointly on the KL exponent and the descent order, and we identify a precise condition under which generalized descent methods achieve linear convergence. By choosing the order of high-order proximal regularization according to the KL exponent, our boosted high-order proximal-point method achieves linear convergence for arbitrary KL exponents. If the objective function satisfies a global KL inequality, we further strengthen the results by proving convergence to global minimizers and deriving explicit iteration-complexity bounds. Numerical experiments validate our theoretical foundation.