@misc{WeiserDeuflhard, author = {Weiser, Martin and Deuflhard, Peter}, title = {The Central Path towards the Numerical Solution of Optimal Control Problems}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6380}, number = {01-12}, abstract = {A new approach to the numerical solution of optimal control problems including control and state constraints is presented. Like hybrid methods, the approach aims at combining the advantages of direct and indirect methods. Unlike hybrid methods, however, our method is directly based on interior-point concepts in function space --- realized via an adaptive multilevel scheme applied to the complementarity formulation and numerical continuation along the central path. Existence of the central path and its continuation towards the solution point is analyzed in some theoretical detail. An adaptive stepsize control with respect to the duality gap parameter is worked out in the framework of affine invariant inexact Newton methods. Finally, the performance of a first version of our new type of algorithm is documented by the successful treatment of the well-known intricate windshear problem.}, language = {en} } @misc{Potra, author = {Potra, Florian}, title = {A path-following method for linear complementarity problems based on the affine invariant Kantorovich Theorem}, doi = {10.1007/s10107-003-0501-8}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-5981}, number = {00-30}, abstract = {A path following algorithm for linear complementarity problems is presented. Given a point \$z\$ that approximates a point \$z(\tau)\$ on the central path with complementarity gap \$\tau\$, one determines a parameter \$\theta\in (0,1)\$ so that this point satisfies the hypothesis of the affine invariant Kantorovich Theorem for the equation defining \$z((1-\theta)\tau)\$. It is shown that \$\theta\$ is bounded below by a multiple of \$n^{-1/2}\$, where \$n\$ is the dimension of the problem. Since the hypothesis of of the Kantorovich Theorem is satisfied the sequence generated by Newton's method, or by the simplified Newton method, will converge to \$z((1-\theta)\tau)\$. We show that the number of steps required to obtain an acceptable approximation of \$z((1-\theta)\tau)\$ is bounded above by a number independent of \$n\$. Therefore the algorithm has \$O(\sqrt{n}L)\$-iteration complexity. The parameters of the algorithm can be determined in such a way that only one Newton step is needed each time the complementarity gap is decreased.}, language = {en} } @misc{SchielaWeiser, author = {Schiela, Anton and Weiser, Martin}, title = {Superlinear Convergence of the Control Reduced Interior Point Method for PDE Constrained Optimization}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8490}, number = {05-15}, abstract = {A thorough convergence analysis of the Control Reduced Interior Point Method in function space is performed. This recently proposed method is a primal interior point pathfollowing scheme with the special feature, that the control variable is eliminated from the optimality system. Apart from global linear convergence we show, that this method converges locally almost quadratically, if the optimal solution satisfies a function space analogue to a non-degeneracy condition. In numerical experiments we observe, that a prototype implementation of our method behaves in compliance with our theoretical results.}, language = {en} } @misc{WeiserSchiela, author = {Weiser, Martin and Schiela, Anton}, title = {Function space interior point methods for PDE constrained optimization}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8027}, number = {04-27}, abstract = {A primal-dual interior point method for optimal control problems with PDE constraints is considered. The algorithm is directly applied to the infinite dimensional problem. Existence and convergence of the central path are analyzed. Numerical results from an inexact continuation method applied to a model problem are shown.}, language = {en} } @misc{WeiserGaenzlerSchiela, author = {Weiser, Martin and G{\"a}nzler, Tobias and Schiela, Anton}, title = {A Control Reduced Primal Interior Point Method for PDE Constrained Optimization}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-8138}, number = {04-38}, abstract = {A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparable coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples.}, language = {en} } @misc{Weiser, author = {Weiser, Martin}, title = {Interior Point Methods in Function Space}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7578}, number = {03-35}, abstract = {A primal-dual interior point method for optimal control problems is considered. The algorithm is directly applied to the infinite dimensional problem. Existence and convergence of the central path are analyzed, and linear convergence of a short step pathfollowing method is established.}, language = {en} } @misc{WeiserDeuflhardErdmann, author = {Weiser, Martin and Deuflhard, Peter and Erdmann, Bodo}, title = {Affine conjugate adaptive Newton methods for nonlinear elastomechanics}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-7768}, number = {04-01}, abstract = {The paper extends affine conjugate Newton methods from convex to nonconvex minimization, with particular emphasis on PDE problems originating from compressible hyperelasticity. Based on well-known schemes from finite dimensional nonlinear optimization, three different algorithmic variants are worked out in a function space setting, which permits an adaptive multilevel finite element implementation. These algorithms are tested on two well-known 3D test problems and a real-life example from surgical operation planning.}, language = {en} } @misc{Schiela, author = {Schiela, Anton}, title = {A continuity result for Nemyckii Operators and some applications in PDE constrained optimal control}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9354}, number = {06-41}, abstract = {This work explores two applications of a classical result on the continuity of Nemyckii operators to optimal control with PDEs. First, we present an alternative approach to the analysis of Newton's method for function space problems involving semi-smooth Nemyckii operators. A concise proof for superlinear convergence is presented, and sharpened bounds on the rate of convergence are derived. Second, we derive second order sufficient conditions for problems, where the underlying PDE has poor regularity properties. We point out that the analytical structure in both topics is essentially the same.}, language = {en} } @misc{Schiela, author = {Schiela, Anton}, title = {Convergence of the Control Reduced Interior Point Method for PDE Constrained Optimal Control with State Constraints}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-9099}, number = {06-16}, abstract = {We propose a variant of the control reduced interior point method for the solution of state constrained problems. We show convergence of the corresponding interior point pathfollowing algorithm in function space. Morever, we provide error bounds for the iterates.}, language = {en} } @misc{Kornhuber, author = {Kornhuber, Ralf}, title = {Monotone Iterations for Elliptic Variational Inequalities}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-3539}, number = {SC-98-10}, abstract = {A wide range of free boundary problems occurring in engineering andindustry can be rewritten as a minimization problem for astrictly convex, piecewise smooth but non--differentiable energy functional.The fast solution of related discretized problemsis a very delicate question, because usual Newton techniquescannot be applied. We propose a new approach based on convex minimization and constrained Newton type linearization. While convex minimization provides global convergence of the overall iteration, the subsequent constrained Newton type linearization is intended to accelerate the convergence speed. We present a general convergence theory and discuss several applications.}, language = {en} }