In this paper, an inexact proximal-point penalty method is studied for constrained optimization problems, where the objective function is non-convex, and the constraint functions can also be non-convex. The proposed method approximately solves a sequence of subproblems, each of which is formed by adding to the original objective function a proximal term and quadratic penalty terms associated to the constraint functions. Under a weak-convexity assumption, each subproblem is made strongly convex and can be solved effectively to a required accuracy by an optimal gradient-based method. The computational complexity of the proposed method is analyzed separately for the cases of convex constraint and non-convex constraint. For both cases, the complexity results are established in terms of the number of proximal gradient steps needed to find an $\varepsilon$-stationary point. When the constraint functions are convex, we show a complexity result of $\tilde O(\varepsilon^{-5/2})$ to produce an $\varepsilon$-stationary point under the Slater's condition. When the constraint functions are non-convex, the complexity becomes $\tilde O(\varepsilon^{-3})$ if a non-singularity condition holds on constraints and otherwise $\tilde O(\varepsilon^{-4})$ if a feasible initial solution is available.

中文翻译：

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约束非凸优化的不精确近端惩罚方法

本文针对目标函数为非凸且约束函数也可以为非凸的约束优化问题，研究了一种不精确近端惩罚方法。所提出的方法近似地解决一系列子问题，每个子问题是通过向原始目标函数添加与约束函数相关联的近端项和二次惩罚项而形成的。在弱凸假设下，每个子问题都是强凸的，并且可以通过基于最佳梯度的方法有效地解决所需的精度。针对凸约束和非凸约束的情况分别分析了所提出方法的计算复杂度。对于这两种情况，复杂性结果是根据找到 $\varepsilon$-平稳点所需的近端梯度步数来确定的。当约束函数是凸的时，我们展示了 $\tilde O(\varepsilon^{-5/2})$ 的复杂性结果，以在 Slater 条件下产生 $\varepsilon$-静止点。当约束函数为非凸函数时，如果非奇异性条件对约束成立，则复杂度变为 $\tilde O(\varepsilon^{-3})$，否则为 $\tilde O(\varepsilon^{-4}) $ 如果可行的初始解决方案可用。