[Paper Review] Worst-Case Complexity of an SQP Method for Nonlinear Equality Constrained Stochastic Optimization

Frank E. Curtis, Michael J. O’Neill+1 more|arXiv (Cornell University)|Dec 29, 2021

Stochastic Gradient Optimization Techniques4 citations

TL;DR

This paper presents a worst-case complexity analysis of a stochastic sequential quadratic programming (SQP) method for nonlinear equality-constrained optimization with a stochastic objective function. It establishes an ${\cal O}(\varepsilon^{-4})$ iteration complexity bound with high probability, comparable to stochastic gradient methods in the unconstrained case, even when the merit parameter is adaptively updated.

ABSTRACT

A worst-case complexity bound is proved for a sequential quadratic optimization (commonly known as SQP) algorithm that has been designed for solving optimization problems involving a stochastic objective function and deterministic nonlinear equality constraints. Barring additional terms that arise due to the adaptivity of the monotonically nonincreasing merit parameter sequence, the proved complexity bound is comparable to that known for the stochastic gradient algorithm for unconstrained nonconvex optimization. The overall complexity bound, which accounts for the adaptivity of the merit parameter sequence, shows that a result comparable to the unconstrained setting (with additional logarithmic factors) holds with high probability.

Motivation & Objective

• To analyze the worst-case iteration complexity of an SQP method for solving stochastic optimization problems with deterministic nonlinear equality constraints.
• To address the theoretical challenge introduced by adaptive merit parameter updates, which dynamically alter the merit function during optimization.
• To extend existing complexity results—previously limited to unconstrained or simple-constrained problems—into the realm of stochastic, equality-constrained optimization.
• To demonstrate that the convergence rate under adaptive merit parameter selection remains comparable to the unconstrained stochastic gradient method, up to logarithmic factors.
• To provide a high-probability complexity bound that accounts for the stochasticity in gradient estimates and the adaptivity of the merit parameter sequence.

Proposed method

• The algorithm uses a line search on a merit function to enforce sufficient decrease, adapting the merit parameter sequence monotonically nonincreasingly.
• Stochastic gradient estimates are used in place of exact gradients, with bounded variance assumed, enabling application to large-scale or simulation-based problems.
• The method solves subproblems that approximate the primal-dual necessary conditions for optimality via sequential quadratic programming.
• A merit function is employed to balance objective improvement and constraint violation, with the parameter updated adaptively based on progress.
• The analysis leverages a stochastic approximation framework with conditional expectations and tail probability bounds to control the impact of adaptive parameter updates.
• Key technical tools include modified lemmas for handling three distinct events related to the sign of the gradient step and the merit parameter update, combined with union bounds to derive high-probability results.

Experimental results

Research questions

• RQ1 What is the worst-case iteration complexity of an SQP method when applied to stochastic, equality-constrained optimization with adaptive merit parameter updates?
• RQ2 How does the complexity bound change when the merit parameter is unknown and adaptively updated, compared to the idealized case with known threshold?
• RQ3 Can the convergence rate of the stochastic SQP method be bounded with high probability despite the stochasticity in gradient estimates and dynamic merit function?
• RQ4 To what extent does the adaptivity of the merit parameter sequence affect the convergence rate, and can the bound still match that of unconstrained stochastic gradient methods?
• RQ5 Are the analytical techniques developed in this work generalizable to other adaptive methods in constrained stochastic optimization, such as augmented Lagrangian or interior point methods?

Key findings

• The worst-case complexity bound for the adaptive SQP method is ${\cal O}(\varepsilon^{-4})$ iterations to achieve a point where the expected norm of the gradient of the Lagrangian and the constraint violation are both $\mathcal{O}(\varepsilon)$, with high probability.
• The complexity bound matches that of the stochastic gradient method in the unconstrained nonconvex setting, up to logarithmic factors arising from the adaptive merit parameter sequence.
• The high-probability guarantee stems solely from the uncertainty in the adaptive merit parameter updates, not from the stochastic gradient estimates themselves.
• The analysis establishes that the method achieves a primal-dual stationarity measure decaying at a rate of $k^{-4}$, ignoring logarithmic terms.
• The result holds under standard assumptions, including bounded variance of stochastic gradients and Lipschitz continuity of the objective and constraint functions.
• The theoretical framework is extendable to other adaptive methods in constrained stochastic optimization, such as penalty, augmented Lagrangian, and interior point methods, due to its general treatment of dynamic parameter sequences.