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

• Published in: Optimization methods & software Vol. 38; no. 1; pp. 59 - 90
• Main Authors: Petra, Cosmin G., Salazar De Troya, Miguel, Petra, Noemi, Choi, Youngsoo, Oxberry, Geoffrey M., Tortorelli, Daniel
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 02.01.2023; Taylor & Francis Ltd
• Subjects: Algorithms; constrained optimization in function spaces; Constraints; Discretization; Function space; mesh independent optimization; Nonlinear programming; Optimization; Partial differential equations; Quasi-Newton interior-point method; Solvers
• ISSN: 1055-6788; 1029-4937
• DOI: 10.1080/10556788.2022.2117354

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.