Characterizations, Dynamical Systems and Gradient Methods for Strongly Quasiconvex Functions

TL;DR

This paper explores properties of strongly quasiconvex functions and introduces gradient-based algorithms with proven exponential and linear convergence without requiring Lipschitz continuity, bridging dynamical systems and optimization.

Contribution

It provides new insights into strongly quasiconvex functions, establishing convergence results for gradient methods derived from dynamical systems without Lipschitz assumptions.

Findings

Exponential convergence of gradient systems for strongly quasiconvex functions.

Linear convergence of gradient descent and heavy ball methods.

Comparison with other nonconvex function classes in gradient descent.

Abstract

We study differentiable strongly quasiconvex functions for providing new properties for algorithmic and monotonicity purposes. Furthemore, we provide insights into the decreasing behaviour of strongly quasiconvex functions, applying this for establishing exponential convergence for first- and second-order gradient systems without relying on the usual Lipschitz continuity assumption on the gradient of the function. The explicit discretization of the first-order dynamical system leads to the gradient descent method while discretization of the second-order dynamical system with viscous damping recovers the heavy ball method. We establish the linear convergence of both methods under suitable conditions on the parameters as well as comparisons with other classes of nonconvex functions used in the gradient descent literature.