[Paper Review] A Sequential Quadratic Programming Method with High Probability Complexity Bounds for Nonlinear Equality Constrained Stochastic Optimization

Albert S. Berahas, Miaolan Xie+1 more|arXiv (Cornell University)|Jan 1, 2023

Risk and Portfolio OptimizationDecision Sciences3 citations

TL;DR

This paper proposes a step-search sequential quadratic programming (SS-SQP) method for nonlinear equality-constrained stochastic optimization, where objective function and gradient values are accessed only via inexact probabilistic zeroth- and first-order oracles. Under reasonable assumptions, it establishes a high-probability iteration complexity bound of $ O(\varepsilon^{-2}) $ to achieve $ \varepsilon $-first-order stationarity, matching deterministic SQP performance and demonstrating robustness to biased and noisy estimates.

ABSTRACT

A step-search sequential quadratic programming method is proposed for solving nonlinear equality constrained stochastic optimization problems. It is assumed that constraint function values and derivatives are available, but only stochastic approximations of the objective function and its associated derivatives can be computed via inexact probabilistic zeroth- and first-order oracles. Under reasonable assumptions, a high-probability bound on the iteration complexity of the algorithm to approximate first-order stationarity is derived. Numerical results on standard nonlinear optimization test problems illustrate the advantages and limitations of our proposed method.

Motivation & Objective

• To develop a robust optimization algorithm for nonlinear equality-constrained problems where only stochastic approximations of the objective function and its gradient are available via inexact probabilistic oracles.
• To establish high-probability iteration complexity bounds for convergence to first-order stationarity under general assumptions, including biased oracle estimates.
• To improve upon existing stochastic penalty and projection methods by directly handling constraints and leveraging reliable constraint information while managing noisy objective evaluations.
• To demonstrate the efficacy of using exact constraint data alongside stochastic objective estimates in a step-search SQP framework.
• To provide theoretical guarantees that match deterministic SQP complexity while being applicable to realistic stochastic settings with unreliable function and gradient evaluations.

Proposed method

• The algorithm uses a step-search strategy within a sequential quadratic programming framework, allowing the search direction to adapt during backtracking, unlike standard line search methods.
• It solves a quadratic subproblem at each iteration using exact constraint values $ c(x) $ and $ \nabla c(x)^T $, while relying on inexact probabilistic oracles for $ f(x) $ and $ \nabla f(x) $.
• The inexact oracles are defined with parameters controlling accuracy and reliability, allowing for biased approximations, and are modeled as random variables with controlled error distributions.
• A merit function is used to guide the step size selection, ensuring sufficient decrease in a composite measure of infeasibility and optimality.
• The method incorporates adaptive backtracking to ensure sufficient decrease in the merit function despite noisy objective estimates.
• Theoretical analysis relies on probabilistic bounds on oracle errors and assumes standard regularity conditions, such as bounded Hessians and Lipschitz continuity of gradients.

Experimental results

Research questions

• RQ1 Can a step-search SQP method achieve high-probability iteration complexity bounds for first-order stationarity in nonlinear equality-constrained stochastic optimization with inexact objective function and gradient evaluations?
• RQ2 How does the performance of the proposed SS-SQP method compare to objective-function-free alternatives when both function and gradient estimates are noisy?
• RQ3 What is the impact of biased or noisy objective function evaluations on the convergence and robustness of SQP methods that rely on zeroth-order information?
• RQ4 Under what conditions does the availability of exact constraint information significantly improve convergence compared to methods relying solely on stochastic objective approximations?
• RQ5 Can the iteration complexity of the proposed method match that of deterministic SQP methods under realistic stochastic oracle assumptions?

Key findings

• The proposed SS-SQP method achieves a high-probability iteration complexity bound of $ O(\varepsilon^{-2}) $ to reach $ \varepsilon $-first-order stationarity, matching the deterministic SQP complexity.
• Numerical results show that SS-SQP outperforms objective-function-free methods (AS-SQP) when objective function noise is low relative to gradient noise, especially in KKT error and feasibility.
• As gradient noise increases, the performance gap between SS-SQP and AS-SQP widens in favor of SS-SQP, highlighting the benefit of exact constraint information.
• When objective function evaluations are highly noisy, AS-SQP becomes more competitive, demonstrating the robustness of objective-function-free approaches in such regimes.
• The method maintains strong performance across varying noise levels, with performance profiles indicating superior robustness and effectiveness under moderate to low noise conditions.
• The results confirm that reliable objective function information significantly enhances convergence, especially when gradient estimates are noisy or biased.