TY - GEN A1 - Weiser, Martin A1 - Deuflhard, Peter T1 - The Central Path towards the Numerical Solution of Optimal Control Problems N2 - 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. T3 - ZIB-Report - 01-12 KW - optimal control KW - interior point methods KW - affine invariance Y1 - 2001 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6380 ER - TY - GEN A1 - Potra, Florian T1 - A path-following method for linear complementarity problems based on the affine invariant Kantorovich Theorem N2 - 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. T3 - ZIB-Report - 00-30 KW - Linear complementarity problem KW - interior-point algorithm KW - path-following KW - Kantorovich Theorem Y1 - 2000 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5981 ER - TY - GEN A1 - Schiela, Anton A1 - Weiser, Martin T1 - Superlinear Convergence of the Control Reduced Interior Point Method for PDE Constrained Optimization N2 - 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. T3 - ZIB-Report - 05-15 KW - interior point methods in function space KW - optimal control KW - superlinear convergence Y1 - 2005 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8490 ER - TY - GEN A1 - Weiser, Martin A1 - Schiela, Anton T1 - Function space interior point methods for PDE constrained optimization N2 - 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. T3 - ZIB-Report - 04-27 KW - interior point methods in function space KW - optimal control KW - complementarity functions Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8027 ER - TY - GEN A1 - Weiser, Martin A1 - Gänzler, Tobias A1 - Schiela, Anton T1 - A Control Reduced Primal Interior Point Method for PDE Constrained Optimization N2 - 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. T3 - ZIB-Report - 04-38 KW - interior point methods in function space KW - optimal control KW - finite elements KW - discretization error Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8138 ER - TY - GEN A1 - Weiser, Martin T1 - Interior Point Methods in Function Space N2 - 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. T3 - ZIB-Report - 03-35 KW - interior point methods in function space KW - optimal control KW - complementarity functions Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7578 ER - TY - GEN A1 - Weiser, Martin A1 - Deuflhard, Peter A1 - Erdmann, Bodo T1 - Affine conjugate adaptive Newton methods for nonlinear elastomechanics N2 - 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. T3 - ZIB-Report - 04-01 KW - affine conjugate Newton methods KW - nonconvex minimization KW - nonlinear elastomechnics KW - cranio-maxillofacial surgery KW - soft tissue simulation KW - multilev Y1 - 2004 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7768 ER - TY - GEN A1 - Schiela, Anton T1 - A continuity result for Nemyckii Operators and some applications in PDE constrained optimal control N2 - 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. T3 - ZIB-Report - 06-41 KW - continuity of Nemyckii Operators KW - Newton methods in function space KW - optimal control KW - second order sufficient conditions Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9354 ER - TY - GEN A1 - Schiela, Anton T1 - Convergence of the Control Reduced Interior Point Method for PDE Constrained Optimal Control with State Constraints N2 - 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. T3 - ZIB-Report - 06-16 KW - interior point methods in function space KW - optimal control KW - state constraints Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9099 ER - TY - GEN A1 - Kornhuber, Ralf T1 - Monotone Iterations for Elliptic Variational Inequalities N2 - 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. T3 - ZIB-Report - SC-98-10 KW - finite elements KW - multigrid methods KW - variational inequalities Y1 - 1998 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3539 ER -