SIAM Journal on Optimization, Volume 31, Issue 1, Page 1-29, January 2021.
The prevalence of uncertainty in models of engineering and the natural sciences necessitates the inclusion of random parameters in the underlying partial differential equations (PDEs). The resulting decision problems governed by the solution of such random PDEs are infinite dimensional stochastic optimization problems. In order to obtain risk-averse optimal decisions in the face of such uncertainty, it is common to employ risk measures in the objective function. This leads to risk-averse PDE-constrained optimization problems. We propose a method for solving such problems in which the risk measures are convex combinations of the mean and conditional value-at-risk (CVaR). Since these risk measures can be evaluated by solving a related inequality-constrained optimization problem, we suggest a log-barrier technique to approximate the risk measure. This leads to a new continuously differentiable convex risk measure: the log-barrier risk measure. We show that the log-barrier risk measure fits into the setting of optimized certainty equivalents of Ben-Tal and Teboulle and the expectation quadrangle of Rockafellar and Uryasev. Using the differentiability of the log-barrier risk measure, we derive first-order optimality conditions reminiscent of classical primal and primal-dual interior-point approaches in nonlinear programming. We derive the associated Newton system, propose a reduced symmetric system to calculate the steps, and provide a sufficient condition for local superlinear convergence in the continuous setting. Furthermore, we provide a $\Gamma$-convergence result for the log-barrier risk measures to prove convergence of the minimizers to the original nonsmooth problem. The results are illustrated by a numerical study.

中文翻译：

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用相干风险测度解决规避风险的偏微分方程约束优化问题的内点方法

SIAM优化杂志，第31卷，第1期，第1-29页，2021年1月。
工程模型和自然科学模型中普遍存在不确定性，因此必须将随机参数包含在基础偏微分方程（PDE）中。由此类随机PDE的求解决定的最终决策问题是无限维随机优化问题。为了在面对此类不确定性时获得规避风险的最佳决策，通常在目标函数中采用风险度量。这导致了规避风险的PDE约束的优化问题。我们提出了一种解决此类问题的方法，其中风险度量是均值和条件风险值（CVaR）的凸组合。由于可以通过解决相关的不等式约束优化问题来评估这些风险措施，因此我们建议采用对数屏障技术对风险措施进行近似。这导致了一种新的，连续可区分的凸风险度量：对数壁垒风险度量。我们表明，对数壁垒风险度量适合本-塔尔和特布勒的最佳确定性当量以及洛克菲拉尔和乌里亚塞夫的期望四边形的设置。利用对数壁垒风险测度的可微性，我们推导了一阶最优条件，该条件使人联想到非线性规划中的经典原始和原始对偶内点方法。我们推导了相关的牛顿系统，提出了一个简化的对称系统来计算步长，并为连续设置中的局部超线性收敛提供了充分的条件。此外，我们提供了对数障碍风险度量的$ \ Gamma $收敛结果，以证明最小化器收敛到原始的非光滑问题。