Manifold Sampling for Optimizing Nonsmooth Nonconvex Compositions

Abstract

We propose a manifold sampling algorithm for minimizing a nonsmooth composition f = h • 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 • 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 .

Error Analysis of Surrogate Models Constructed through Operations on Sub-models

“…Khan et al [16] develop an algorithm for minimizing F = φ+h•f where φ is smooth with known derivatives, h is known, nonsmooth piecewise linear function and f is smooth but expensive to evaluate. Later in [18], Larson et al aim to minimize F = h • f where h is nonsmooth but inexpensive to compute and f is smooth but its Jacobian is not available.…”