Manifold Sampling for Nonconvex Optimization of Piecewise Linear Compositions

We develop a manifold sampling algorithm for the unconstrained minimization of 4 a nonsmooth composite function f , ψ + h ◦ F when ψ is smooth with known derivatives, h is a 5 nonsmooth, piecewise linear function, and F is smooth but expensive to evaluate. The trust-region 6 algorithm classifies points in the domain of h as belonging to different manifolds and uses this knowl7 edge when computing search directions. Since h is known, classifying objective manifolds using only 8 the values of F is simple. We prove that all cluster points of the sequence of the manifold sampling 9 algorithm iterates are Clarke stationary; this holds although points evaluated by the algorithm are 10 not assumed to be differentiable and when only approximate derivatives of F are available. Numer11 ical results show that manifold sampling using zero-order information is competitive with gradient 12 sampling algorithms that are given exact gradient values. 13