Juan Carlos De Los Reyes

Sufficient Second-Order Optimality Conditions for Semilinear Control Problems with Pointwise State Constraints

Second-order sufficient optimality conditions are established for the optimal control of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state. In contrast to former publications on this subject, the cone of critical directions is the smallest possible in the sense that the second-order sufficient conditions are the closest to the associated necessary ones. The theory is developed for elliptic distributed controls in domains up to dimension three. Moreover, problems of elliptic boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one, respectively.

Optimal Control of a Class of Variational Inequalities of the Second Kind

Optimal control problems governed by a class of elliptic variational inequalities of the second kind are investigated. Applications include the optimal control of viscoplastic fluid flow and of simplified friction. Based on a Tikhonov regularization of the dual problem, a family of primal-dual regularized control problems is introduced, and convergence of the regularized solutions towards a solution of the original control problem is verified. For each regularized problem an optimality condition is derived, and an optimality system for the original control problem is obtained as a limit of the regularized ones. Thanks to the structure of the proposed regularization, complementarity relations between the variables involved are derived. Since the regularized optimality systems involve Newton differentiable functions, a semismooth Newton algorithm is proposed and its numerical performance investigated.

Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost based on a Huber-regularised TV seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a combined quasi-Newton/semismooth Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between

A duality based semismooth Newton framework for solving variational inequalities of the second kind

\text {TGV}^2

Numerical Study of the Optimization of Separation Control

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Optimal Control of Static Elastoplasticity in Primal Formulation

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