Active-Set Identification in Noisy and Stochastic Optimization

Abstract

Identifying active constraints from a point near a local minimizer is important both theoretically and practically in constrained continuous optimization as it can help identify optimal Lagrange multipliers and essentially reduce an inequality-constrained problem to an equality-constrained one. Traditional active-set identification guarantees have been proved under various types of constraint qualifications. Importantly, they also assume exact function and derivative values. This work extends such results to settings when both objective and constraint function and derivative values have deterministic or stochastic noise. Two strategies are analyzed. For each strategy, it is shown that, under constraint qualifications and other standard conditions, the active set of a local minimizer can be identified correctly when a point is close enough to the minimizer and the noise is sufficiently small. It is also shown that regularity of a minimizer is critical. Under strict complementarity, a fixed noise level suffices, whereas in the presence of a weakly active constraint the noise must vanish at least linearly in the distance to the minimizer in general, a rate that is also shown to be sufficient for correct identification. Guarantees are also stated for the use of active-set identification strategies within a stochastic algorithm, with explicit sample-size conditions. The theoretical findings are demonstrated with two simple illustrative examples and a more realistic constrained neural-network training task.