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

arXiv: https://arxiv.org/abs/2409.15734 · HTML: https://mlsanity.com/paper/2409.15734/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 54 references that are in the map.
- First posted on arXiv 24 Sep 2024.

## Builds on (most cited first, 30 of 30)

- [Constrained Policy Optimization](https://mlsanity.com/paper/1705.10528/) (2017)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Identifying and attacking the saddle point problem in high-dimensional non-convex optimization](https://mlsanity.com/paper/1406.2572/) (2014)
- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [Sequential Quadratic Optimization for Nonlinear Equality Constrained Stochastic Optimization](https://mlsanity.com/paper/2007.10525/) (2020)
- [Stochastic Optimization Using a Trust-Region Method and Random Models](https://mlsanity.com/paper/1504.04231/) (2015)
- [An Adaptive Stochastic Sequential Quadratic Programming with Differentiable Exact Augmented Lagrangians](https://mlsanity.com/paper/2102.05320/) (2021)
- [A Stochastic Sequential Quadratic Optimization Algorithm for Nonlinear Equality Constrained Optimization with Rank-Deficient Jacobians](https://mlsanity.com/paper/2106.13015/) (2021)
- [Fully Stochastic Trust-Region Sequential Quadratic Programming for Equality-Constrained Optimization Problems](https://mlsanity.com/paper/2211.15943/) (2022)
- [Worst-Case Complexity of an SQP Method for Nonlinear Equality Constrained Stochastic Optimization](https://mlsanity.com/paper/2112.14799/) (2021)
- [Inequality Constrained Stochastic Nonlinear Optimization via Active-Set Sequential Quadratic Programming](https://mlsanity.com/paper/2109.11502/) (2021)
- [Hessian Averaging in Stochastic Newton Methods Achieves Superlinear Convergence](https://mlsanity.com/paper/2204.09266/) (2022)
- [Accelerating Stochastic Sequential Quadratic Programming for Equality Constrained Optimization using Predictive Variance Reduction](https://mlsanity.com/paper/2204.04161/) (2022)
- [An Adaptive Sampling Sequential Quadratic Programming Method for Equality Constrained Stochastic Optimization](https://mlsanity.com/paper/2206.00712/) (2022)
- [Variance-reduced first-order methods for deterministically constrained stochastic nonconvex optimization with strong convergence guarantees](https://mlsanity.com/paper/2409.09906/) (2024)
- [Convergence of trust-region methods based on probabilistic models](https://mlsanity.com/paper/1304.2808/) (2013)
- [Statistical Inference of Constrained Stochastic Optimization via Sketched Sequential Quadratic Programming](https://mlsanity.com/paper/2205.13687/) (2022)
- [Sequential Quadratic Optimization for Stochastic Optimization with Deterministic Nonlinear Inequality and Equality Constraints](https://mlsanity.com/paper/2302.14790/) (2023)
- [First- and Second-Order High Probability Complexity Bounds for Trust-Region Methods with Noisy Oracles](https://mlsanity.com/paper/2205.03667/) (2022)
- [A Sequential Quadratic Programming Method with High Probability Complexity Bounds for Nonlinear Equality Constrained Stochastic Optimization](https://mlsanity.com/paper/2301.00477/) (2023)
- [A Sequential Quadratic Programming Method for Optimization with Stochastic Objective Functions, Deterministic Inequality Constraints and Robust Subproblems](https://mlsanity.com/paper/2302.07947/) (2023)
- [A Stochastic-Gradient-based Interior-Point Algorithm for Solving Smooth Bound-Constrained Optimization Problems](https://mlsanity.com/paper/2304.14907/) (2023)
- [Constrained Optimization in the Presence of Noise](https://mlsanity.com/paper/2110.04355/) (2021)
- [Sample Complexity Analysis for Adaptive Optimization Algorithms with Stochastic Oracles](https://mlsanity.com/paper/2303.06838/) (2023)
- [An Adaptive Sampling Augmented Lagrangian Method for Stochastic Optimization with Deterministic Constraints](https://mlsanity.com/paper/2305.01018/) (2023)
- [Almost-sure convergence of iterates and multipliers in stochastic sequential quadratic optimization](https://mlsanity.com/paper/2308.03687/) (2023)
- [Adaptive sampling strategies for risk-averse stochastic optimization with constraints](https://mlsanity.com/paper/2012.03844/) (2020)
- [Design Guidelines for Noise-Tolerant Optimization with Applications in Robust Design](https://mlsanity.com/paper/2401.15007/) (2024)
- [Computation of sparse low degree interpolating polynomials and their application to derivative-free optimization](https://mlsanity.com/paper/1306.5729/) (2013)
- [Modified Line Search Sequential Quadratic Methods for Equality-Constrained Optimization with Unified Global and Local Convergence Guarantees](https://mlsanity.com/paper/2406.11144/) (2024)

## Built on by (most cited first, 5 of 5)

- [High Probability Complexity Bounds of Trust-Region Stochastic Sequential Quadratic Programming with Heavy-Tailed Noise](https://mlsanity.com/paper/2503.19091/) (2025)
- [Why is Normalization Preferred? A Worst-Case Complexity Theory for Stochastically Preconditioned SGD under Heavy-Tailed Noise](https://mlsanity.com/paper/2602.13413/) (2026)
- [A Trust-Region Interior-Point Stochastic Sequential Quadratic Programming Method](https://mlsanity.com/paper/2603.10230/) (2026)
- [TRSVR: An Adaptive Stochastic Trust-Region Method with Variance Reduction](https://mlsanity.com/paper/2601.14647/) (2026)
- [TR-SSQP: A Trust-Region Method for Constrained Stochastic Optimization under Heavy-Tailed Noise](https://mlsanity.com/paper/2609.30732/) (2026)

## Abstract (arXiv)

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.