A Stochastic-Gradient-based Interior-Point Algorithm for Solving Smooth Bound-Constrained Optimization Problems

TL;DR

This paper introduces a novel stochastic-gradient-based interior-point algorithm for smooth bound-constrained optimization, demonstrating convergence and superior performance over projection methods in experiments.

Contribution

It presents a new interior-point method using stochastic gradients and inner neighborhoods, with proven convergence guarantees in deterministic and stochastic contexts.

Findings

Algorithm outperforms projection-based methods in experiments.

Convergence guarantees established for both deterministic and stochastic cases.

Effective balance of barrier, step-size, and neighborhood sequences is crucial.

Abstract

A stochastic-gradient-based interior-point algorithm for minimizing a continuously differentiable objective function (that may be nonconvex) subject to bound constraints is presented, analyzed, and demonstrated through experimental results. The algorithm is unique from other interior-point methods for solving smooth nonconvex optimization problems since the search directions are computed using stochastic gradient estimates. It is also unique in its use of inner neighborhoods of the feasible region -- defined by a positive and vanishing neighborhood-parameter sequence -- in which the iterates are forced to remain. It is shown that with a careful balance between the barrier, step-size, and neighborhood sequences, the proposed algorithm satisfies convergence guarantees in both deterministic and stochastic settings. The results of numerical experiments show that in both settings the algorithm can outperform projection-based methods.