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

Research output: Chapter in Book/Report/Conference proceedingConference contribution

## Abstract

We consider the minimization of an L-{0}-Lipschitz continuous and expectation-valued function, denoted by f and defined as f(\mathrm{x})\ {\buildrel \triangle\over=}\\mathbb{E}[\tilde{f}(\mathrm{x}, \omega)], over a Cartesian product of closed and convex sets with a view towards obtaining both asymptotics as well as rate and complexity guarantees for computing an approximate stationary point (in a Clarke sense). We adopt a smoothing-based approach reliant on minimizing f-{\eta} where f(\mathrm{x})\ {\buildrel \triangle\over=}\\mathbb{E}-{u}[f(\mathrm{x}+\eta u)],u is a random variable defined on a unit sphere, and \eta > 0. In fact, it is observed that a stationary point of the \eta-smoothed problem is a 2\eta-stationary point for the original problem in the Clarke sense. In such a setting, we derive a suitable residual function that provides a metric for stationarity for the smoothed problem. By leveraging a zeroth-order framework reliant on utilizing sampled function evaluations implemented in a block-structured regime, we make two sets of contributions for the sequence generated by the proposed scheme. (i) The residual function of the smoothed problem tends to zero almost surely along the generated sequence; (ii) To compute an \mathrm{x} that ensures that the expected norm of the residual of the \eta-smoothed problem is within \epsilon requires no greater than \mathcal{O}(\frac{1}{\eta\epsilon^{2}}) projection steps and \mathcal{O}\left(\frac{1}{\eta^{2}\epsilon^{4}}\right) function evaluations. These statements appear to be novel with few related results available to contend with general nonsmooth, nonconvex, and stochastic regimes via zeroth-order approaches.

Original language English (US) 2021 7th Indian Control Conference, ICC 2021 - Proceedings Institute of Electrical and Electronics Engineers Inc. 7-12 6 9781665409780 https://doi.org/10.1109/ICC54714.2021.9703135 Published - 2021 7th Indian Control Conference, ICC 2021 - Virtual, Online, IndiaDuration: Dec 20 2021 → Dec 22 2021

### Publication series

Name 2021 7th Indian Control Conference, ICC 2021 - Proceedings

### Conference

Conference 7th Indian Control Conference, ICC 2021 India Virtual, Online 12/20/21 → 12/22/21

## All Science Journal Classification (ASJC) codes

• Control and Systems Engineering
• Control and Optimization

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