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We consider the minimization of an $L_0$-Lipschitz continuous andexpectation-valued function, denoted by $f$ and defined as $f(x)\triangleq\mathbb{E}[\tilde{f}(x,\omega)]$, over a Cartesian product of closed and convexsets with a view towards obtaining both asymptotics as well as rate andcomplexity guarantees for computing an approximate stationary point (in aClarke sense). We adopt a smoothing-based approach reliant on minimizing$f_{\eta}$ where $f_{\eta}(x) \triangleq \mathbb{E}_{u}[f(x+\eta u)]$, $u$ is arandom variable defined on a unit sphere, and $\eta > 0$. In fact, it isobserved that a stationary point of the $\eta$-smoothed problem is a$2\eta$-stationary point for the original problem in the Clarke sense. In sucha setting, we derive a suitable residual function that provides a metric forstationarity for the smoothed problem. By leveraging a zeroth-order frameworkreliant on utilizing sampled function evaluations implemented in ablock-structured regime, we make two sets of contributions for the sequencegenerated by the proposed scheme. (i) The residual function of the smoothedproblem tends to zero almost surely along the generated sequence; (ii) Tocompute an $x$ that ensures that the expected norm of the residual of the$\eta$-smoothed problem is within $\epsilon$ requires no greater than$\mathcal{O}(\tfrac{1}{\eta \epsilon^2})$ projection steps and$\mathcal{O}\left(\tfrac{1}{\eta^2 \epsilon^4}\right)$ function evaluations.These statements appear to be novel and there appear to be few results tocontend with general nonsmooth, nonconvex, and stochastic regimes viazeroth-order approaches.

Zeroth-order randomized block methods for constrained minimization of expectation-valued Lipschitz continuous functions