Active-Set Identification in Noisy and Stochastic Optimization

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Abstract
Identifying active constraints from a point near an optimal solution is important both theoretically and practically in constrained continuous optimization, as it can help identify optimal Lagrange multipliers and essentially reduces an inequality-constrained problem to an equality-constrained one. Traditional active-set identification guarantees have been proved under assumptions of smoothness and constraint qualifications, and assume exact function and derivative values. This work extends these results to settings when both objective and constraint function and derivative values have deterministic or stochastic noise. Two strategies are proposed that, under mild conditions, are proved to identify the active set of a local minimizer correctly when a point is close enough to the local minimizer and the noise is sufficiently small. Guarantees are also stated for the use of active-set identification strategies within a stochastic algorithm. We demonstrate our findings with two simple illustrative examples and a more realistic constrained neural-network training task.
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