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This is a Quantitative Study study.

September 23, 2025Open Access

Active-set Newton-MR methods for nonconvex optimization problems with bound constraints

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Active-set methods minimize nonconvex functions under bound constraints, suggesting effective descent techniques.

The proposed methods achieve O(ε^{3/2}) descent within feasible faces without cubic regularization, demonstrating efficiency.
The SPG-based method requires at most O(nε^{-2}) oracle calls to find ε-approximate stationary points under Lipschitz conditions.
Using Minimum Residual instead of Conjugate Gradient for approximating Newton directions shows significant numerical advantages.
Theoretical guarantees indicate the effectiveness of descent methods with Armijo line search in nonconvex settings.

Birgin et al. (2025) studied this question.

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