Zhichao Jia – Optimization Online

Goldstein Stationarity in Lipschitz Constrained Optimization

Published: 2023/10/05, Updated: 2023/10/12

Constrained Nonlinear Optimization, Nonlinear Optimization constraint qualifications, convergence rate, Goldstein, nonconvex nonsmooth optimization, subgradient method

We prove the first convergence guarantees for a subgradient method minimizing a generic Lipschitz function over generic Lipschitz inequality constraints. No smoothness or convexity (or weak convexity) assumptions are made. Instead, we utilize a sequence of recent advances in Lipschitz unconstrained minimization, which showed convergence rates of $O(1/\delta\epsilon^3)$ towards reaching a “Goldstein” stationary point, that … Read more

First-Order Methods for Nonsmooth Nonconvex Functional Constrained Optimization with or without Slater Points

Published: 2023/04/28

Zhichao Jia

Benjamin Grimmer

Constrained Nonlinear Optimization, Nonsmooth Optimization constrained optimization, first-order method, Nonconvexity, Nonsmoothness, sparsity

Constrained optimization problems where both the objective and constraints may be nonsmooth and nonconvex arise across many learning and data science settings. In this paper, we show a simple first-order method finds a feasible, ϵ-stationary point at a convergence rate of O(ϵ−4) without relying on compactness or Constraint Qualification (CQ). When CQ holds, this convergence is measured by … Read more