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Type: Text × Subject: Variational inequality ×

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    Optimal Control Problems with Sparsity for Tumor Growth Models Involving Variational Inequalities
    (Dordrecht [u.a.] : Springer Science + Business Media, 2022) Colli, Pierluigi; Signori, Andrea; Sprekels, Jürgen
    This paper treats a distributed optimal control problem for a tumor growth model of Cahn–Hilliard type. The evolution of the tumor fraction is governed by a variational inequality corresponding to a double obstacle nonlinearity occurring in the associated potential. In addition, the control and state variables are nonlinearly coupled and, furthermore, the cost functional contains a nondifferentiable term like the L1-norm in order to include sparsity effects which is of utmost relevance, especially time sparsity, in the context of cancer therapies as applying a control to the system reflects in exposing the patient to an intensive medical treatment. To cope with the difficulties originating from the variational inequality in the state system, we employ the so-called deep quench approximation in which the convex part of the double obstacle potential is approximated by logarithmic functions. For such functions, first-order necessary conditions of optimality can be established by invoking recent results. We use these results to derive corresponding optimality conditions also for the double obstacle case, by deducing a variational inequality in terms of the associated adjoint state variables. The resulting variational inequality can be exploited to also obtain sparsity results for the optimal controls.
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    Newton and Bouligand derivatives of the scalar play and stop operator
    (Les Ulis : EDP Sciences, 2020) Brokate, Martin
    We prove that the play and the stop operator possess Newton and Bouligand derivatives, and exhibit formulas for those derivatives. The remainder estimate is given in a strengthened form, and a corresponding chain rule is developed. The construction of the Newton derivative ensures that the mappings involved are measurable. © The authors. Published by EDP Sciences, 2020.
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    Tensor methods for strongly convex strongly concave saddle point problems and strongly monotone variational inequalities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Ostroukhov, Petr; Kamalov, Rinat; Dvurechensky, Pavel; Gasnikov, Alexander
    In this paper we propose three tensor methods for strongly-convex-strongly-concave saddle point problems (SPP). The first method is based on the assumption of higher-order smoothness (the derivative of the order higher than 2 is Lipschitz-continuous) and achieves linear convergence rate. Under additional assumptions of first and second order smoothness of the objective we connect the first method with a locally superlinear converging algorithm in the literature and develop a second method with global convergence and local superlinear convergence. The third method is a modified version of the second method, but with the focus on making the gradient of the objective small. Since we treat SPP as a particular case of variational inequalities, we also propose two methods for strongly monotone variational inequalities with the same complexity as the described above.