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

• URL: http://arxiv.org/abs/2211.15943v2
• Date: Mon, 29 Jan 2024 02:02:20 GMT
• Title: Fully Stochastic Trust-Region Sequential Quadratic Programming for Equality-Constrained Optimization Problems
• Authors: Yuchen Fang, Sen Na, Michael W. Mahoney, Mladen Kolar
• Abstract summary: We propose a sequential quadratic programming algorithm (TR-StoSQP) to solve nonlinear optimization problems with objectives and deterministic equality constraints. The algorithm adaptively selects the trust-region radius and, compared to the existing line-search StoSQP schemes, allows us to utilize indefinite Hessian matrices.
• Score: 62.83783246648714
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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