Trust-Region Sequential Quadratic Programming for Stochastic Optimization with Random Models

TL;DR

This work proposes a Trust-Region Sequential Quadratic Programming method to find both first- and second-order stationary points, utilizing a random model to represent the objective function and establishing global almost sure first- and second-order convergence guarantees for this method.

Abstract

In this work, we consider solving optimization problems with a stochastic objective and deterministic equality constraints. We propose a Trust-Region Sequential Quadratic Programming method to find both first- and second-order stationary points. Our method utilizes a random model to represent the objective function, which is constructed from stochastic observations of the objective and is designed to satisfy proper adaptive accuracy conditions with a high but fixed probability. To converge to first-order stationary points, our method computes a gradient step in each iteration defined by minimizing a quadratic approximation of the objective subject to a (relaxed) linear approximation of the problem constraints and a trust-region constraint. To converge to second-order stationary points, our method additionally computes an eigen step to explore the negative curvature of the reduced Hessian matrix, as well as a second-order correction step to address the potential Maratos effect, which arises due to the nonlinearity of the problem constraints. Such an effect may impede the method from moving away from saddle points. Both gradient and eigen step computations leverage a novel parameter-free decomposition of the step and the trust-region radius, accounting for the proportions among the feasibility residual, optimality residual, and negative curvature. We establish global almost sure first- and second-order convergence guarantees for our method, and present computational results on CUTEst problems, regression problems, and saddle-point problems to demonstrate its superiority over existing line-search-based stochastic methods.

Figures (3)

• Figure 1: KKT residual box plots over 47 CUTEst problems with given initialization (left) and random initialization (right). Each panel has four different noise levels. For each noise level, the first four boxes correspond to TR-SQP-STORM with different types of ${\bar{H}}_k$; the fifth box corresponds to TR-SQP-STORM2; and the last two boxes correspond to $\ell_2$-SSQP and AL-SSQP, respectively.
• Figure 2: Trajectories of KKT residuals of four datasets. Each panel corresponds to a dataset and includes seven lines representing the seven algorithms.
• Figure 3: Trajectories of the KKT residuals and the smallest eigenvalue of the reduced Lagrangian Hessians under four noise levels. The top four figures show the trajectories of the KKT residuals, while the bottom four figures show the trajectories of the smallest eigenvalues. Each figure corresponds to a noise level and includes seven lines representing the seven algorithms.

Theorems & Definitions (22)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 3.1
• Remark 3.2
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5
• Lemma 4.6