This paper introduces a method for computing points satisfying the second-order necessary optimality conditions for nonconvex minimization problems subject to a closed and convex constraint set. The method comprises two independent steps corresponding to the first- and second-order conditions. The first-order step is a generic closed map algorithm, which can be chosen from a variety of first-order algorithms, making it adjustable to the given problem. The second-order step can be viewed as a second-order feasible direction step for nonconvex minimization subject to a convex set. We prove that any limit point of the resulting scheme satisfies the second-order necessary optimality condition, and establish the scheme’s convergence rate and complexity, under standard and mild assumptions. Numerical tests illustrate the proposed scheme.

中文翻译：

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在约束最小化中寻找二阶平稳点：一种可行的方向方法

本文介绍了一种计算满足二阶必要最优性条件的点的方法，该方法适用于受闭和凸约束集约束的非凸最小化问题。该方法包括对应于一阶和二阶条件的两个独立步骤。一阶步骤是一种通用的封闭映射算法，可以从多种一阶算法中选择，使其可针对给定问题进行调整。二阶步骤可以看作是受凸集约束的非凸最小化的二阶可行方向步骤。我们证明了所得方案的任何极限点都满足二阶必要最优性条件，并在标准和温和的假设下建立了该方案的收敛速度和复杂度。数值测试说明了所提出的方案。