In this paper, we study the perturbation analysis of a class of composite optimization problems, which is a very convenient and unified framework for developing both theoretical and algorithmic issues of constrained optimization problems. The underlying theme of this paper is very important in both theoretical and computational study of optimization problems. Under some mild assumptions on the objective function, we provide a definition of a strong second order sufficient condition (SSOSC) for the composite optimization problem and also prove that the following conditions are equivalent to each other: the SSOSC and the nondegeneracy condition, the nonsingularity of Clarke's generalized Jacobian of the nonsmooth system at a Karush-Kuhn-Tucker (KKT) point, and the strong regularity of the KKT point. These results provide an important way to characterize the stability of the KKT point. As for the convex composite optimization problem, which is a special case of the general problem, we establish the equivalence between the primal/dual second order sufficient condition and the dual/primal strict Robinson constraint qualification, the equivalence between the primal/dual SSOSC and the dual/primal nondegeneracy condition. Moreover, we prove that the dual nondegeneracy condition and the nonsingularity of Clarke's generalized Jacobian of the subproblem corresponding to the augmented Lagrangian method are also equivalent to each other. These theoretical results lay solid foundation for designing an efficient algorithm.

翻译：本文研究了一类复合优化问题的扰动分析，该类问题为约束优化问题的理论与算法研究提供了极为便捷且统一的框架。本文的核心主题在优化问题的理论与计算研究中具有重要地位。在目标函数的某些温和假设下，我们给出了复合优化问题的强二阶充分条件（SSOSC）的定义，并证明了以下条件相互等价：SSOSC与非退化条件、Karush-Kuhn-Tucker（KKT）点处Clarke广义Jacobian矩阵的非奇异性，以及KKT点的强正则性。这些结果为刻画KKT点的稳定性提供了重要途径。针对作为一般问题特例的凸复合优化问题，我们建立了原/对偶二阶充分条件与对偶/原严格Robinson约束规范之间的等价性，以及原/对偶SSOSC与对偶/原非退化条件之间的等价性。此外，我们还证明了对偶非退化条件与增广拉格朗日方法对应子问题的Clarke广义Jacobian矩阵的非奇异性也是等价的。这些理论结果为设计高效算法奠定了坚实基础。