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

TL;DR

A global almost sure convergence guarantee for TR-StoSQP is established, and 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 is illustrated.

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.

Figures (3)

• Figure 1: KKT residual boxplots for CUTEst problems. For each $\sigma^2$, there are five boxes. The first four boxes correspond to the proposed TR-StoSQP method with four different choices of $B_k$, while the last box corresponds to the $\ell_1$-StoSQP method.
• Figure 2: KKT residual boxplots for CUTEst problems with different relaxation techniques. The Hessian approximation $B_k$ is set as identity matrix. For each $\sigma^2$, there are three boxes. The first box corresponds to the proposed adaptive relaxation technique. The second box corresponds to the adaptive technique in Remark \ref{['rem:3']} (i). The last box corresponds to the nonadaptive technique in Remark \ref{['rem:3']} (ii).
• Figure 3: KKT residual boxplots for constrained logistic regression problems. For each setup of $\beta_k$, there are five boxes. The first four boxes correspond to the proposed TR-StoSQP method with four different choices of $B_k$, while the last box corresponds to the $\ell_1$-StoSQP method.

Theorems & Definitions (17)

• Lemma 2.1
• Remark 2.2
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4
• Lemma 4.4
• Lemma 4.5
• Lemma 4.6
• Lemma 4.7