On the implementation of a quasi-Newton interior-point method for PDE-constrained optimization using finite element discretizations

ABSTRACT We present a quasi-Newton interior-point method appropriate for optimization problems with pointwise inequality constraints in Hilbert function spaces. Among others, our methodology applies to optimization problems constrained by partial differential equations (PDEs) that are posed in a reduced-space formulation and have bounds or inequality constraints on the optimized parameter function. We first introduce the formalization of an infinite-dimensional quasi-Newton interior-point algorithm using secant BFGS updates and then proceed to derive a discretized interior-point method capable of working with a wide range of finite element discretization schemes. We also discuss and address mathematical and software interface issues that are pervasive when existing off-the-shelf PDE solvers are to be used with off-the-shelf nonlinear programming solvers. Finally, we elaborate on the numerical and parallel computing strengths and limitations of the proposed methodology on several classes of PDE-constrained problems.

**摘要** 我们提出了一种拟牛顿内点法，适用于希尔伯特函数空间中具有逐点不等式约束的优化问题。尤其，我们的方法适用于由偏微分方程（PDE）约束的优化问题，这些问题以降维形式提出，并且在优化的参数函数上有界或不等式约束。我们首先介绍了一种使用割线BFGS更新的无穷维拟牛顿内点算法的形式化，然后着手推导一种能够与多种有限元离散化方案配合使用的离散化内点方法。我们还讨论并解决了在将现有的现成偏微分方程求解器与现成的非线性规划求解器一起使用时普遍存在的数学和软件接口问题。最后，我们详细阐述了所提出的方法在几类偏微分方程约束问题上的数值和并行计算优势及局限性。