Fully Stochastic Trust-Region Sequential Quadratic Programming for Equality-Constrained Optimization Problems

Abstract

We propose a trust-region stochastic sequential quadratic programming algorithm (TR-StoSQP) to solve nonlinear optimization problems with stochastic objectives and deterministic equality constraints. We consider a fully stochastic setting, where at each step a single sample is generated to estimate the objective gradient. The algorithm adaptively selects the trust-region radius and, compared to the existing line-search StoSQP schemes, allows us to utilize indefinite Hessian matrices (i.e., Hessians without modification) in SQP subproblems. As a trust-region method for constrained optimization, our algorithm must address an infeasibility issue — the linearized equality constraints and trust-region constraints may lead to infeasible SQP subproblems. In this regard, we propose an adaptive relaxation technique to compute the trial step, consisting of a normal step and a tangential step. To control the lengths of these two steps while ensuring a scale-invariant property, we adaptively decompose the trust-region radius into two segments, based on the proportions of the rescaled feasibility and optimality residuals to the rescaled full KKT residual. The normal step has a closed form, while the tangential step is obtained by solving a trust-region subproblem, to which a solution ensuring the Cauchy reduction is sufficient for our study. We establish a global almost sure convergence guarantee for TR-StoSQP, and illustrate its empirical performance on both a subset of problems in the CUTEst test set and constrained logistic regression problems using data from the LIBSVM collection.

Yuchen Fang

MS in Computational and Applied Math (2021-2023)

Yuchen Fang is a master student in the Computational and Applied Math program at UChicago, working with Mladen Kolar and Sen Na on stochastic nonlinear optimization. He is also interested in numerical linear algebra, high-dimensional statistics, statistical learning, as well as mathematical finance.

Sen Na

Postdoc in Statistics and ICSI

Sen Na is a postdoctoral scholar in the Statistics department and ICSI at UC Berkeley. His research interests broadly lie in the mathematical foundations of data science, with topics including high-dimensional statistics, graphical models, semiparametric models, optimal control, and large-scale and stochastic nonlinear optimization. He is also interested in applying machine learning methods in biology, neuroscience, and engineering.

Michael W. Mahoney

Professor in Statistics and ICSI, Amazon Scholar

Michael Mahoney is a Professor in the Statistics department and ICSI at UC Berkeley. He is also the director of the NSF/TRIPODS-funded Foundations of Data Analysis (FODA) Institute at UC Berkeley. He works on the algorithmic and statistical aspects of modern large-scale data analysis. Much of his recent research has focused on large-scale machine learning, including randomized matrix algorithms and randomized numerical linear algebra.

Mladen Kolar

Associate Professor of Econometrics and Statistics

Mladen Kolar is an Associate Professor of Econometrics and Statistics at the University of Chicago Booth School of Business. His research is focused on high-dimensional statistical methods, graphical models, varying-coefficient models and data mining, driven by the need to uncover interesting and scientifically meaningful structures from observational data.