# Almost-sure convergence of iterates and multipliers in stochastic sequential quadratic optimization

arXiv: https://arxiv.org/abs/2308.03687 · HTML: https://mlsanity.com/paper/2308.03687/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 39 references that are in the map.
- First posted on arXiv 7 Aug 2023.

## Builds on (most cited first, 16 of 16)

- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Sequential Quadratic Optimization for Nonlinear Equality Constrained Stochastic Optimization](https://mlsanity.com/paper/2007.10525/) (2020)
- [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)
- [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)
- [Sequential Quadratic Optimization for Stochastic Optimization with Deterministic Nonlinear Inequality and Equality Constraints](https://mlsanity.com/paper/2302.14790/) (2023)
- [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)
- [Constrained Optimization in the Presence of Noise](https://mlsanity.com/paper/2110.04355/) (2021)
- [Revisiting the central limit theorems for the SGD-type methods](https://mlsanity.com/paper/2207.11755/) (2022)
- [Bregman three-operator splitting methods](https://mlsanity.com/paper/2203.00252/) (2022)

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

- [Variance-reduced first-order methods for deterministically constrained stochastic nonconvex optimization with strong convergence guarantees](https://mlsanity.com/paper/2409.09906/) (2024)
- [Trust-Region Sequential Quadratic Programming for Stochastic Optimization with Random Models](https://mlsanity.com/paper/2409.15734/) (2024)
- [High Probability Complexity Bounds of Trust-Region Stochastic Sequential Quadratic Programming with Heavy-Tailed Noise](https://mlsanity.com/paper/2503.19091/) (2025)
- [MoSSP: A Momentum-Based Single-Loop Stochastic Penalty Method for Nonconvex Constrained DC-Regularized Optimization](https://mlsanity.com/paper/2605.29635/) (2026)
- [TR-SSQP: A Trust-Region Method for Constrained Stochastic Optimization under Heavy-Tailed Noise](https://mlsanity.com/paper/2609.30732/) (2026)

## Abstract (arXiv)

Stochastic sequential quadratic optimization (SQP) methods for solving continuous optimization problems with nonlinear equality constraints have attracted attention recently, such as for solving large-scale data-fitting problems subject to nonconvex constraints. However, for a recently proposed subclass of such methods that is built on the popular stochastic-gradient methodology from the unconstrained setting, convergence guarantees have been limited to the asymptotic convergence of the expected value of a stationarity measure to zero. This is in contrast to the unconstrained setting in which almost-sure convergence guarantees (of the gradient of the objective to zero) can be proved for stochastic-gradient-based methods. In this paper, new almost-sure convergence guarantees for the primal iterates, Lagrange multipliers, and stationarity measures generated by a stochastic SQP algorithm in this subclass of methods are proved. It is shown that the error in the Lagrange multipliers can be bounded by the distance of the primal iterate to a primal stationary point plus the error in the latest stochastic gradient estimate. It is further shown that, subject to certain assumptions, this latter error can be made to vanish by employing a running average of the Lagrange multipliers that are computed during the run of the algorithm. The results of numerical experiments are provided to demonstrate the proved theoretical guarantees.