On the implementation of a quasi-Newton interior-point method for PDE-constrained optimization using finite element discretizations. (2nd January 2023)

• Record Type: Journal Article
• Authors: Petra, Cosmin G.; Salazar De Troya, Miguel; Petra, Noemi; Choi, Youngsoo; Oxberry, Geoffrey M.; Tortorelli, Daniel
• Is Part Of: Optimization methods and software. Volume 38:Number 1(2023)
• Page Start: 59
• Page End: 90
• Publication Date: 2023-01-02
• Subjects: Quasi-Newton interior-point method -- constrained optimization in function spaces -- mesh independent optimization; 49M15 -- 49M37 -- 65K10 -- 90C26 -- 35Q93; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2022.2117354
• Languages: English
• ISSNs: 1055-6788

Abstract

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.