Convex programming based robust optimization is an active research topic in the past two decades, partially because of its computational tractability for many classes of optimization problems and uncertainty sets. However, many problems arising from modern operations research and statistical learning applications are nonconvex even in the nominal case, let alone their robust counterpart. In this paper, we introduce a systematic approach for tackling the nonconvexity of the robust optimization problems that is usually coupled with the nonsmoothness of the objective function brought by the worst-case value function. A majorization-minimization algorithm is presented to solve the penalized min-max formulation of the robustified problem that deterministically generates a “better” solution compared with the starting point (that is usually chosen as an unrobustfied optimal solution). A generalized saddle-point theorem regarding the directional stationarity is established and a game-theoretic interpretation of the computed solutions is provided. Numerical experiments show that the computed solutions of the nonconvex robust optimization problems are less sensitive to the data perturbation compared with the unrobustfied ones.

在过去的二十年中，基于凸规划的鲁棒优化是一个活跃的研究主题，部分原因是它对许多类优化问题和不确定性集具有计算易处理性。但是，即使是名义上的情况，现代运筹学和统计学习应用程序所产生的许多问题也不是凸面的，更不用说它们的可靠对应了。在本文中，我们介绍了一种系统的方法来解决鲁棒优化问题的不凸性，该方法通常与最坏情况下的值函数带来的目标函数的不光滑性相关。提出了一种最小化-最小化算法，用于解决鲁棒性问题的罚分最小-最大提法，该提法确定性地生成了一个与起点相比更好的解（通常选择为非鲁棒的最优解）。建立了关于方向平稳性的广义鞍点定理，并提供了计算解的博弈论解释。数值实验表明，与非鲁棒优化问题相比，非凸鲁棒优化问题的计算解对数据扰动较不敏感。