An objective-function-free algorithm for nonconvex stochastic optimization with deterministic equality and inequality constraints

TL;DR

This paper introduces a novel stochastic optimization algorithm that does not evaluate the objective function, relying solely on gradients, and achieves convergence rates comparable to unconstrained methods.

Contribution

It presents an objective-function-free algorithm for constrained stochastic optimization with proven convergence rates and adaptive step size selection.

Findings

Achieves O(1/√k) convergence rate for full-rank Jacobian problems.

Uses an adaptive Adagrad-type stepsize for iteration selection.

Operates solely with gradient information, not function evaluations.

Abstract

An algorithm is proposed for solving optimization problems with stochastic objective and deterministic equality and inequality constraints. This algorithm is objective-function-free in the sense that it only uses the objective's gradient and never evaluates the function value. It is based on an adaptive selection of function-decreasing and constraint-improving iterations, the first ones using an Adagrad-type stepsize. When applied to problems with full-rank Jacobian, the combined primal-dual optimality measure is shown to decrease at the rate of O(1/sqrt{k}), which is identical to the convergence rate of first-order methods in the unconstrained case.