Manifold Sampling for Optimizing Nonsmooth Nonconvex Compositions

TL;DR

This paper introduces a manifold sampling algorithm for optimizing nonsmooth compositions, demonstrating its effectiveness and theoretical convergence properties without requiring Jacobian information.

Contribution

It develops a derivative-free manifold sampling method for nonsmooth compositions and proves convergence to Clarke stationary points.

Findings

Algorithm is competitive with state-of-the-art methods.

Convergence to Clarke stationary points is established.

Numerical results validate the approach's effectiveness.

Abstract

We propose a manifold sampling algorithm for minimizing a nonsmooth composition $f= h\circ F$, where we assume $h$ is nonsmooth and may be inexpensively computed in closed form and $F$ is smooth but its Jacobian may not be available. We additionally assume that the composition $h\circ F$ defines a continuous selection. Manifold sampling algorithms can be classified as model-based derivative-free methods, in that models of $F$ are combined with particularly sampled information about $h$ to yield local models for use within a trust-region framework. We demonstrate that cluster points of the sequence of iterates generated by the manifold sampling algorithm are Clarke stationary. We consider the tractability of three particular subproblems generated by the manifold sampling algorithm and the extent to which inexact solutions to these subproblems may be tolerated. Numerical results demonstrate that manifold sampling as a derivative-free algorithm is competitive with state-of-the-art algorithms for nonsmooth optimization that utilize first-order information about $f$.