An inexact [ell] sub(1) penalty SQP algorithm for PDE-constrained optimization with an application to shape optimization in linear elasticity

We develop an optimization algorithm which is able to deal with inexact evaluations of the objective function. The proposed algorithm employs sequential quadratic programming with a line search that uses the [ell] sub(1) penalty function for an Armijo-like condition. Both the objective gradient comp...

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Bibliographic Details
Published in:Optimization methods & software Vol. 28; no. 5; pp. 943 - 968
Main Authors: Hess, Wolfgang, Ulbrich, Stefan
Format: Journal Article
Language:English
Published: 01.10.2013
Subjects:
Algorithms
Iterative methods
Mathematical analysis
Mathematical models
Optimization
Partial differential equations
Searching
Shape optimization
Solvers
ISSN:1055-6788, 1029-4937
Online Access:Get full text
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Summary:We develop an optimization algorithm which is able to deal with inexact evaluations of the objective function. The proposed algorithm employs sequential quadratic programming with a line search that uses the [ell] sub(1) penalty function for an Armijo-like condition. Both the objective gradient computations for the quadratic subproblems and the objective function computations for the line search admit some inexactness which is controlled by the algorithm. We prove convergence results for the presented algorithm under suitable assumptions. The kind of optimization problems handled by the algorithm arises, for example, when we apply iterative solvers in the evaluation of the reduced objective function of a shape optimization problem subject to the linear elasticity equations. In the second part of this paper, we present an application of the algorithm to the shape optimization of steel profiles using nested iterations with adaptively refined meshes. It is shown how the inexactness in the algorithm translates to an acceptable residual of the state and adjoint state solutions in the iterative solver. We look at three different stopping criteria for the outer iterations. Numerical results are presented.
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ISSN:1055-6788
1029-4937
DOI:10.1080/10556788.2011.651082