A difference-of-convex functions approach for sparse PDE optimal control problems with nonconvex costs

We propose a local regularization of elliptic optimal control problems which involves the nonconvex L q quasi-norm penalization in the cost function. The proposed Huber type regularization allows us to formulate the PDE constrained optimization instance as a DC programming problem (difference of con...

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Bibliographic Details
Published in:Computational optimization and applications Vol. 74; no. 1; pp. 225 - 258
Main Author: Merino, Pedro
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2019
Springer Nature B.V
Subjects:
Algorithms
Convex analysis
Convex and Discrete Geometry
Economic models
Management Science
Mathematics
Mathematics and Statistics
Nonlinear programming
Numerical methods
Operations Research
Operations Research/Decision Theory
Optimal control
Optimization
Regularization
Statistics
DCA
Elliptic PDE
49K20
90C46
49J20
Optimal control
DC programming
90C26
Nonconvex
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:We propose a local regularization of elliptic optimal control problems which involves the nonconvex L q quasi-norm penalization in the cost function. The proposed Huber type regularization allows us to formulate the PDE constrained optimization instance as a DC programming problem (difference of convex functions) that is useful to obtain necessary optimality conditions and tackle its numerical solution by applying the well known DC algorithm used in nonconvex optimization problems. By this procedure we approximate the original problem in terms of a consistent family of parameterized nonsmooth problems for which there are efficient numerical methods available. Finally, we present numerical experiments to illustrate our theory with different configurations associated to the parameters of the problem.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-019-00101-0