Structured Nonsmooth Optimization Using Functional Encoding and Branching Information

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.