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In this paper, the optimal boundary control of a time-discrete Cahn-Hilliard-Navier-Stokes system is studied. A general class of free energy potentials is considered which, in particular, includes the double-obstacle potential. The latter homogeneous free energy density yields an optimal control problem for a family of coupled systems, which result from a time discretization of a variational inequality of fourth order and the Navier--Stokes equation. The existence of an optimal solution to the time-discrete control problem as well as an approximate version is established. The latter approximation is obtained by mollifying the Moreau--Yosida approximation of the double-obstacle potential. First order optimality conditions for the mollified problems are given, and in addition to the convergence of optimal controls of the mollified problems to an optimal control of the original problem, first order optimality conditions for the original problem are derived through a limit process. The newly derived stationarity system is related to a function space version of C-stationarity.

A priori estimates of the length of the primal-dual path resulting from a Moreau--Yosida approximation of the feasible set for state constrained optimal control problems are derived. These bounds depend on the regularity of the state and the dimension of the problem. Numerical results indicate that the bounds are indeed sharp and are typically attained in cases where the active set consists of isolated active points. Further conditions on the multiplier approximation are identified which guarantee higher convergence rates for the feasibility violation due to the Moreau--Yosida approximation process. Numerical experiments show again that the results are sharp and accurately predict the convergence behavior.

A mathematical framework for optimal bilinear control of nonlinear Schrödingerequations of Gross–Pitaevskii type arising in the description of Bose–Einstein condensates is presented.The obtained results generalize earlier efforts found in the literature in several aspects. Inparticular, the cost induced by the physical workload over the control process is taken into accountrather than the often used L2- or H1-norms for the cost of the control action. Well-posedness ofthe problem and existence of an optimal control are proved. In addition, the first order optimalitysystem is rigorously derived. Also a numerical solution method is proposed, which is based on aNewton-type iteration, and used to solve several coherent quantum control problems.

A nonconvex variational model is introduced which contains the q-"norm," q ∈ (0, 1), of the gradientof the underlying image in the regularization part together with a least squares–type datafidelity term which may depend on a possibly spatially dependent weighting parameter. Hence,the regularization term in this functional is a nonconvex compromise between the minimization ofthe support of the reconstruction and the classical convex total variation model. In the discretesetting, existence of a minimizer is proved, and a Newton-type solution algorithm is introduced andits global as well as local superlinear convergence toward a stationary point of a locally regularizedversion of the problem is established. The potential nonpositive definiteness of the Hessian of theobjective during the iteration is handled by a trust-region–based regularization scheme. The performanceof the new algorithm is studied by means of a series of numerical tests. For the associatedinfinite dimensional model an existence result based on the weakly lower semicontinuous envelopeis established, and its relation to the original problem is discussed.

This paper considers a class of nonlinear evolution quasi-variational inequality (QVI)problems with pointwise gradient constraints in vector-valued function spaces. The existence andapproximation of solutions is addressed based on a combination of C0-semigroup methods, Moscoconvergence, and monotone operator techniques developed by Br´ezis. An algorithm based on semidiscretizationin time is proposed and its numerical implementation based on a penalty approach andsemismooth Newton methods is studied. This paper ends with a report on numerical tests whichinvolve the p-Laplacian and several types of nonlinear constraints.

The minimization of a functional composed of a nonsmooth and nonadditive regularization term and a combined L1 and L2 data-fidelity term is proposed. It is shown analytically and numerically that the new model has noticeable advantages over popular models in image processing tasks. For the numerical minimization of the new objective, subspace correction methods are introduced which guarantee the convergence and monotone decay of the associated energy along the iterates. Moreover, an estimate of the distance between the outcome of the subspace correction method and the global minimizer of the nonsmooth objective is derived. This estimate and numerical experiments for Image denoising, inpainting, and deblurring indicate that in practice the proposed subspace correction methods indeed approach the global solution of the underlying minimization problem.

In this paper we study the distributed optimal control for the Cahn–Hilliard system. A general class of free energy potentials is allowed which, in particular, includes the double-obstacle potential. The latter potential yields an optimal control problem of a parabolic variational inequality which is of fourth order in space. We show the existence of optimal controls to approximating problems where the potential is replaced by a mollified version of its Moreau–Yosida approximation. Corresponding first-order optimality conditions for the mollified problems are given. For this purpose a new result on the continuous Fréchet differentiability of superposition operators with values in Sobolev spaces is established. Besides the convergence of optimal controls of the mollified problems to an optimal control of the original problem, we also derive first-order optimality conditions for the original problem by a limit process. The newly derived stationarity system corresponds to a function space version of C-stationarity.

A class of nonlinear elliptic quasi-variational inequality (QVI) problems with gradientconstraints in function space is considered. Problems of this type arise, for instance, in themathematical description of the magnetization of superconductors, in problems in elastoplasticity, orin electrostatics as well as in game theory. The paper addresses the iterative solution of the QVIs bya sequential minimization technique relying on the repeated solution of variational inequality–typeproblems. A monotone operator theoretic approach is developed which does not resort to Moscoconvergence as is often done in connection with existence analysis for QVIs. For the numerical solutionof the QVIs a penalty approach combined with a semismooth Newton iteration is proposed.The paper ends with a report on numerical tests involving the p-Laplace operator and various typesof nonlinear constraints.

Mathematical programs in which the constraint set is partially defined by the solutions of an elliptic variational inequality, so-called "elliptic MPECs", are formulated in reflexive Banach spaces. With the goal of deriving explicit first-order optimality conditions amenable to the development of numerical procedures, variational analytic concepts are both applied and further developed. The paper is split into two main parts. The first part concerns the derivation of conditions in which the (lower-level) state constraints are assumed to be polyhedric sets. This part is then completed by two examples, the latter of which involves pointwise bilateral bounds on the gradient of the state. The second part focuses on an important nonpolyhedric example, namely, when the lower-level state constraints are presented by pointwise bounds on the Euclidean norm of the gradient of the state. A formula for the second-order (Mosco) epiderivative of the indicator function for this convex set is derived. This result is then used to demonstrate the (Hadamard) directional differentiability of the solution mapping of the variational inequality, which then leads to the derivation of explicit strong stationarity conditions for this problem.

A mathematical framework for optimal bilinear control of nonlinear Schrödingerequations of Gross–Pitaevskii type arising in the description of Bose–Einstein condensates is presented.The obtained results generalize earlier efforts found in the literature in several aspects. Inparticular, the cost induced by the physical workload over the control process is taken into accountrather than the often used L2- or H1-norms for the cost of the control action. Well-posedness ofthe problem and existence of an optimal control are proved. In addition, the first order optimalitysystem is rigorously derived. Also a numerical solution method is proposed, which is based on aNewton-type iteration, and used to solve several coherent quantum control problems.

Motivated by an obstacle problem for a membrane subject to cohesion forces, constrained minimization problems involving a nonconvex and nondifferentiable objective functional representing the total potential energy are considered. The associated first-order optimality system leads to a hemivariational inequality, which can also be interpreted as a special complementarity problem in function space. Besides an analytical investigation of first-order optimality, a primal-dual active set solver is introduced. It is associated to a limit case of a semismooth Newton method for a regularized version of the underlying problem class. For the numerical algorithms studied in this paper, global as well as local convergence properties are derived and verified numerically.

We derive primal-dual weighted goal-oriented a posteriori error estimates for pointwise state constrained optimal control problems for second order elliptic partial differential equations. The constraints give rise to a primal-dual weighted error term representing the mismatch in the complementarity system due to discretization. In the case of sufficiently regular active (or coincidence) sets and problem data, a further decomposition of the multiplier into a regular L2-part on the active set and a singular part concentrated on the boundary between the active and inactive set allows us to further characterize the mismatch error. The paper ends with a report on the behavior of the error estimates for test cases including the case of singular active sets consisting of only smooth curves or points.

An inverse problem in the pricing of American options is considered. The problem is formulated as an output least-squares problem governed by a parabolic variational inequality in non-divergence form. The existence of an optimal solution is proved, and first-order optimality conditions of C-stationarity-type are derived by using a relaxation-penalization technique. Numerically, the discrete optimality system is solved by an active-set-Newton solver with feasibility restoration.

An optimal control problem governed by an elliptic variational inequality is studied. The feasible set of the problem is relaxed, and a path–following-type method is used to regularize the constraint on the state variable. First order optimality conditions for the relaxed-regularized subproblems are derived, and convergence of stationary points with respect to the relaxation and regularization parameters is shown. In particular, C- and strong stationarity as well as variants thereof are studied. The subproblems are solved by using semismooth Newton methods. The overall algorithmic concept is provided, and its performance is discussed by means of examples, including problems with bilateral constraints and a nonsymmetric operator.

A general Moreau–Yosida-based framework for minimization problems subject to partial differential equations and pointwise constraints on the control, the state, and its derivative is considered. A range space constraint qualification is used to argue existence of Lagrange multipliers and to derive a KKT-type system for characterizing first-order optimality of the unregularized problem. The theoretical framework is then used to develop a semismooth Newton algorithm in function space and to prove its locally superlinear convergence when solving the regularized problems. Further, for maintaining the local superlinear convergence in function space it is demonstrated that in some cases it might be necessary to add a lifting step to the Newton framework in order to bridge an L2-Lr-norm gap, with r > 2. The paper ends by a report on numerical tests.

An adjustment scheme for the regularization parameter of a Moreau–Yosida-based regularization, or relaxation, approach to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method utilizes error estimates of an associated finite element discretization of the regularized problems for the optimal selection of the regularization parameter in dependence on the mesh size of discretization and error estimates for the approximation error due to regularization. The theoretical results are verified numerically.

Dual-weighted goal-oriented error estimates for a class of pointwise control constrained optimal control problems for second order elliptic partial differential equations are derived. It is demonstrated that the constraints give rise to a primal-dual weighted error term representing the mismatch in the complementarity system due to discretization. The paper also contains a posteriori error estimators for the $L 2$-norm of the error in the state and in the adjoint state.

Path-following methods for primal-dual active set strategies requiring a regularization parameter are introduced. Existence of a primal-dual path and its differentiability properties are analyzed. Monotonicity and convexity of the primal-dual path value function are investigated. Both feasible and infeasible approximations are considered. Numerical path-following strategies are developed and their efficiency is demonstrated by means of examples.

In this paper, a primal-dual algorithm for total bounded variation (TV)–type image restoration is analyzed and tested. Analytically it turns out that employing a global Ls-regularization, with 1 < s ≤ 2, in the dual problem results in a local smoothing of the TVregularization term in the primal problem. The local smoothing can alternatively be obtained as the infimal convolution of the lr-norm, with r−1 + s−1 = 1, and a smooth function. In the case r = s = 2, this results in Gauss-TV–type image restoration. The globalized primal-dual algorithm introduced in this paper works with generalized derivatives, converges locally at a superlinear rate, and is stable with respect to noise in the data. In addition, it utilizes a projection technique which reduces the size of the linear system that has to be solved per iteration. A comprehensive numerical study ends the paper.

Primal‐dual path‐following methods for constrained minimization problems in function space with low multiplier regularity are introduced and analyzed. Regularity properties of the path are proved. The path structure allows us to define approximating models, which are used for controlling the path parameter in an iterative process for computing a solution of the original problem. The Moreau–Yosida regularized subproblems of the new path‐following technique are solved efficiently by semismooth Newton methods. The overall algorithmic concept is provided, and numerical tests (including a comparison with primal‐dual path‐following interior point methods) for state constrained optimal control problems show the efficiency of the new concept.