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.

翻译：本文针对限制优化问题研究了一种不精确的准点惩罚方法, 其目标函数为非电解质, 约束功能也可以为非电解质。 拟议的方法可以解决子问题序列, 每个子问题都是通过在原始目标函数中添加一个与约束功能相关的近似值和四度惩罚条件来形成的。 在弱调度假设下, 每种子问题都具有很强的共性, 并且可以通过一种基于最佳梯度的方法有效地解决到所要求的准确性 。 拟议的方法的计算复杂性将针对 convex 约束和非电解质制约的情况分别分析 。 对于这两种情况, 复杂性的结果都是根据找到 $\ varepsil$- 固定点所需的准度梯度步骤的数量来确定的。 当制约功能是 contilde O(\ vareplon_ 5⁄ /2} 美元, 我们展示了一个复杂的结果 $\\ varepilexlon$- $- ylor- explain requil=nal deal destrivil a constal destal destrations) a contistrital destritations. 如果 Slats- revl=n=n ex