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Sequential Quadratic Optimization for Solving Expectation Equality Constrained Stochastic Optimization Problems

Haoming Shen, Yang Zeng, Baoyu Zhou

Abstract

A sequential quadratic programming method is designed for solving general smooth nonlinear stochastic optimization problems subject to expectation equality constraints. We consider the setting where the objective and constraint function values, as well as their derivatives, are not directly available. The algorithm applies an adaptive step size policy and only relies on objective gradient estimates, constraint function estimates, and constraint derivative estimates to update iterates. Both asymptotic and non-asymptotic convergence properties of the algorithm are analyzed. Under reasonable assumptions, the algorithm generates a sequence of iterates whose first-order stationary measure diminishes in expectation. In addition, we identify the iteration and sample complexity for obtaining a first-order $\varepsilon$-stationary iterate in expectation. The results of numerical experiments demonstrate the efficiency and efficacy of our proposed algorithm compared to a penalty method and an augmented Lagrangian method.

Sequential Quadratic Optimization for Solving Expectation Equality Constrained Stochastic Optimization Problems

Abstract

A sequential quadratic programming method is designed for solving general smooth nonlinear stochastic optimization problems subject to expectation equality constraints. We consider the setting where the objective and constraint function values, as well as their derivatives, are not directly available. The algorithm applies an adaptive step size policy and only relies on objective gradient estimates, constraint function estimates, and constraint derivative estimates to update iterates. Both asymptotic and non-asymptotic convergence properties of the algorithm are analyzed. Under reasonable assumptions, the algorithm generates a sequence of iterates whose first-order stationary measure diminishes in expectation. In addition, we identify the iteration and sample complexity for obtaining a first-order -stationary iterate in expectation. The results of numerical experiments demonstrate the efficiency and efficacy of our proposed algorithm compared to a penalty method and an augmented Lagrangian method.

Paper Structure

This paper contains 27 sections, 35 theorems, 220 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Notation
  4. Organization
  5. Problem Setting
  6. Algorithm Framework
  7. Convergence Properties and Complexity Analysis
  8. Fundamental Technical Results
  9. Convergence Results
  10. Complexity Results
  11. An overview of our complexity result
  12. Comparision with the complexity result in CurtRobiOnei24
  13. Analysis of the relation between $\bar{\mathcal{T}}_k$ and $\mathcal{T}_k^{{\rm trial}}$
  14. Analysis of the noise term
  15. Complexity of Algorithm \ref{['alg.main']}
  16. ...and 12 more sections

Key Result

Lemma 1

Suppose Assumptions ass.prob--ass.estimate hold. There exist constants $q_{\mathop{\textrm{max}}} \geq q_{\mathop{\textrm{min}}} > 0$ such that for any iteration $k\in\mathbb{N}$, the singular values of $$ and $$ always stay in $[q_{\mathop{\textrm{min}}}, q_{\mathop{\textrm{max}}}]$.

Figures (4)

  • Figure 1: Box plots of infeasibility errors (the left column) and stationarity errors (the right column) on a total of 44 CUTEst problems with $\epsilon_g \in \{10^{-8},10^{-4},10^{-2}\}$ (from top to bottom).
  • Figure 2: Box plots of infeasibility errors (the left column) and stationarity errors (the right column) on a total of 44 CUTEst problems with diminishing $\{\beta_k\}$ sequences under large-noise settings.
  • Figure 3: Histogram of the final values of $\log_{10}(\bar{\mathcal{T}}')$.
  • Figure 4: Data profiles, including mean values and 95% confidence intervals, of true KKT errors vs. iterations on 8 problems from the CUTEst collection gould2015cutest. Red and cyan curves represent noise levels $(\epsilon_g,\epsilon_c,\epsilon_J) = (0.25,0.25,0.5)$ and $(\epsilon_g,\epsilon_c,\epsilon_J) = (10^{-2},10^{-2},10^{-1})$, respectively.

Theorems & Definitions (77)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 67 more