In this paper, we develop an algorithm to efficiently solve risk-averse optimization problems posed in reflexive Banach space. Such problems often arise in many practical applications as, e.g., optimization problems constrained by partial differential equations with uncertain inputs. Unfortunately, for many popular risk models including the coherent risk measures, the resulting risk-averse objective function is nonsmooth. This lack of differentiability complicates the numerical approximation of the objective function as well as the numerical solution of the optimization problem. To address these challenges, we propose a primal–dual algorithm for solving large-scale nonsmooth risk-averse optimization problems. This algorithm is motivated by the classical method of multipliers and by epigraphical regularization of risk measures. As a result, the algorithm solves a sequence of smooth optimization problems using derivative-based methods. We prove convergence of the algorithm even when the subproblems are solved inexactly and conclude with numerical examples demonstrating the efficiency of our method.

在本文中，我们开发了一种算法来有效解决自反Banach空间中提出的规避风险的优化问题。这些问题通常在许多实际应用中出现，例如，优化问题受到输入不确定的偏微分方程的约束。不幸的是，对于许多流行的风险模型（包括连贯的风险度量）而言，所产生的规避风险的目标函数并不平滑。这种可微性的缺乏使目标函数的数值逼近以及优化问题的数值解变得复杂。为了解决这些挑战，我们提出了一种原始对偶算法来解决大规模的非平稳风险规避优化问题。该算法受经典乘数方法和风险度量的表位正则化的启发。作为结果，该算法使用基于导数的方法解决了一系列平滑优化问题。即使在子问题不精确求解的情况下，我们也证明了算法的收敛性，并通过数值示例得出结论，证明了我们方法的有效性。