Single-Loop Deterministic and Stochastic Interior-Point Algorithms for Nonlinearly Constrained Optimization

TL;DR

This paper introduces a novel single-loop interior-point algorithm framework for solving nonlinearly constrained optimization problems, capable of handling stochastic gradient estimates and providing convergence guarantees.

Contribution

It presents a new single-loop interior-point method suitable for stochastic and deterministic settings, with proven convergence and practical effectiveness.

Findings

Good performance on a large set of test problems

Convergence guarantees for both deterministic and stochastic cases

Effective handling of nonconvex, nonlinear constraints

Abstract

An interior-point algorithm framework is proposed, analyzed, and tested for solving nonlinearly constrained continuous optimization problems. The main setting of interest is when the objective and constraint functions may be nonlinear and/or nonconvex, and when constraint values and derivatives are tractable to compute, but objective function values and derivatives can only be estimated. The algorithm is intended primarily for a setting that is similar for stochastic-gradient methods for unconstrained optimization, namely, the setting when stochastic-gradient estimates are available and employed in place of gradients of the objective, and when no objective function values (nor estimates of them) are employed. This is achieved by the interior-point framework having a single-loop structure rather than the nested-loop structure that is typical of contemporary interior-point methods. For completeness, convergence guarantees for the framework are provided both for deterministic and stochastic settings. Numerical experiments show that the algorithm yields good performance on a large set of test problems.