{"title":"Directional differentiability for shape optimization with variational inequalities as constraints","authors":"V. A. Kovtunenko, K. Kunisch","doi":"10.1051/cocv/2023056","DOIUrl":null,"url":null,"abstract":"For equilibrium constrained optimization problems subject to nonlinear state equations, the property of directional differentiability with respect to a parameter is studied. An abstract class of parameter dependent shape optimization problems is investigated with penalty constraints linked to variational inequalities. Based on the Lagrange multiplier approach, on smooth penalties due to Lavrentiev regularization, and on adjoint operators, a shape derivative is obtained. The explicit formula provides a descent direction for the gradient algorithm identifying the shape of the breaking-line from a boundary measurement. A numerical example is presented for a nonlinear Poisson problem modeling Barenblatt’s surface energies and non-penetrating cracks.","PeriodicalId":50500,"journal":{"name":"Esaim-Control Optimisation and Calculus of Variations","volume":"58 1","pages":""},"PeriodicalIF":1.3000,"publicationDate":"2023-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Esaim-Control Optimisation and Calculus of Variations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1051/cocv/2023056","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
For equilibrium constrained optimization problems subject to nonlinear state equations, the property of directional differentiability with respect to a parameter is studied. An abstract class of parameter dependent shape optimization problems is investigated with penalty constraints linked to variational inequalities. Based on the Lagrange multiplier approach, on smooth penalties due to Lavrentiev regularization, and on adjoint operators, a shape derivative is obtained. The explicit formula provides a descent direction for the gradient algorithm identifying the shape of the breaking-line from a boundary measurement. A numerical example is presented for a nonlinear Poisson problem modeling Barenblatt’s surface energies and non-penetrating cracks.
期刊介绍：
ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations.
Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines.
Targeted topics include:
in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory;
in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis;
in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.
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