Optimistic Noise-Aware Sequential Quadratic Programming for Equality Constrained Optimization with Rank-Deficient Jacobians
Albert S. Berahas, Jiahao Shi, Baoyu Zhou
TL;DR
The paper addresses noisy equality-constrained optimization where both the objective and constraints, as well as their derivatives, are contaminated by bounded noise and the constraint Jacobian may be rank-deficient. It develops an optimistic, noise-aware SQP framework with a Byrd-Omojokun-style step decomposition into normal and tangential components, two adaptable step-size schemes, and an early stopping rule based on optimistic feasibility. Theoretical results establish neighborhood convergence to a first-order stationary point in the full-rank LICQ setting and convergence to a neighborhood of an infeasible stationary point when rank-deficiency occurs, with nonasymptotic bounds and iteration- complexity guarantees. Numerical experiments on CUTEst show robustness to noise and competitive performance against existing noise-aware SQP methods, including effective handling of rank-deficient scenarios and early termination benefits. Overall, the approach provides practical, theoretically grounded guidance for solving noisy equality-constrained problems in the presence of rank deficiencies and measurement noise.
Abstract
We propose and analyze a sequential quadratic programming algorithm for minimizing a noisy nonlinear smooth function subject to noisy nonlinear smooth equality constraints. The algorithm uses a step decomposition strategy and, as a result, is robust to potential rank-deficiency in the constraints, allows for two different step size strategies, and has an early stopping mechanism. Under the linear independence constraint qualification, convergence is established to a neighborhood of a first-order stationary point, where the radius of the neighborhood is proportional to the noise levels in the objective function and constraints. Moreover, in the rank-deficient setting, the merit parameter may converge to zero, and convergence to a neighborhood of an infeasible stationary point is established. Numerical experiments demonstrate the efficiency and robustness of the proposed method.