Structured Nonsmooth Optimization Using Functional Encoding and Branching Information

TL;DR

This paper introduces a novel gradient-based algorithm for nonsmooth nonconvex optimization that encodes active branches of nonsmooth operators, enabling efficient optimization with improved convergence and practical implementation benefits.

Contribution

The paper presents a new encoding-based gradient method for nonsmooth optimization that enhances branch tracking and convergence, integrating seamlessly with existing methods.

Findings

The BIGD method achieves local linear convergence under certain conditions.

Numerical experiments show improved performance over existing methods.

Functional encoding simplifies implementation and enhances efficiency.

Abstract

We develop a novel gradient-based algorithm for optimizing nonsmooth nonconvex functions where nonsmoothness arises from explicit nonsmooth operators in the objective's analytical form. Our key innovation involves encoding active smooth branches of these operators, enabling both branch function extraction at arbitrary points and transition detection through branch tracking. This approach yields a Branch-Information-Driven Gradient Descent (BIGD) method for encodable piecewise-differentiable functions, with an enhanced version achieving local linear convergence under appropriate conditions. The computationally efficient encoding mechanism is straightforward to implement. The power of using branch information has been proved via substantial numerical experiments compared to some existing nonsmooth optimization methods on standard test problems. Most importantly, for piecewise-smooth problems given analytical expressions, implementation of functional encoding can be integrated into a wide range of existing nonsmooth optimization methods to improve the bundle points management, reduce the complexity of the quadratic programming sub-problems, and improve the efficiency of line search.