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Moreau–Yosida regularization in shape optimization with geometric constraints

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  • Moritz Keuthen
  • Michael Ulbrich

Abstract

In the context of shape optimization with geometric constraints we employ the method of mappings (perturbation of identity) to obtain an optimal control problem with a nonlinear state equation on a fixed reference domain. The Lagrange multiplier associated with the geometric shape constraint has a low regularity (similar to state constrained problems), which we circumvent by penalization and a continuation scheme. We employ a Moreau–Yosida-type regularization and assume a second-order condition to hold. The regularized problems can then be solved with a semismooth Newton method and we study the properties of the regularized solutions and the rate of convergence towards a solution of the original problem. A model for the value function in the spirit of Hintermüller and Kunisch (SIAM J Control Optim 45(4): 1198–1221, 2006 ) is introduced and used in an update strategy for the regularization parameter. The theoretical findings are supported by numerical tests. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Moritz Keuthen & Michael Ulbrich, 2015. "Moreau–Yosida regularization in shape optimization with geometric constraints," Computational Optimization and Applications, Springer, vol. 62(1), pages 181-216, September.
    • Handle: RePEc:spr:coopap:v:62:y:2015:i:1:p:181-216
    DOI: 10.1007/s10589-014-9661-0
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    References listed on IDEAS

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    1. C. Meyer & I. Yousept, 2009. "Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions," Computational Optimization and Applications, Springer, vol. 44(2), pages 183-212, November.
    2. Eduardo Casas & Fredi Tröltzsch, 2012. "A general theorem on error estimates with application to a quasilinear elliptic optimal control problem," Computational Optimization and Applications, Springer, vol. 53(1), pages 173-206, September.
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