A Local-Linearly Convergent Algorithm for Nonconvex Equality-Constrained Optimization

• URL: http://arxiv.org/abs/2608.12665v1
• Date: Wed, 12 Aug 2026 23:55:59 GMT
• Title: A Local-Linearly Convergent Algorithm for Nonconvex Equality-Constrained Optimization
• Abstract summary: A Gradient-Eigenstep Algorithm is used to find an approximate second-order stationary point from an arbitrary starting point.<n>The algorithm can be used as an efficient subproblem solver in the context of a progressive sampling strategy.<n>It offers an algorithm with an improved worst-case sample complexity when compared to an approach that solves a full-sample problem directly.
• Score: 4.8718671103376865
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: For solving nonconvex equality-constrained optimization problems, a recent Gradient-Eigenstep Algorithm by Goyens et al.~is an iteration-efficient approach, based on minimizing Fletcher's augmented Lagrangian function, for finding an approximate second-order stationary point from an arbitrary starting point. In this paper, the analysis of this algorithm is extended, offering a two-fold contribution. First, it is shown that a local-linear rate of convergence can be obtained by this method if it is initiated sufficiently close to a strong second-order stationary point and employs a sufficiently small step-size parameter and sufficiently large penalty parameter. In this case, the algorithm reduces to a gradient descent algorithm applied to minimize Fletcher's augmented Lagrangian. Second, as a particularly useful application of the first result, it is shown that the Gradient-Eigenstep algorithm can be used as an iteration-efficient subproblem solver in the context of a progressive sampling strategy for solving equality-constrained optimization problems when the objective and constraint functions are defined by large sample averages, ultimately offering an algorithm with an improved worst-case sample complexity when compared to an approach that solves a full-sample problem directly.

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