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54 OP08 Abstracts CP1 Dept. of Mathematics Improving Ultimate Convergence of An Aug- [email protected] mented Lagrangian Method Optimization methods that employ the classical Powell- CP1 Hestenes-Rockafellar Augmented Lagrangian are useful A Second-Derivative SQP Method for Noncon- tools for solving Nonlinear Programming problems. Their vex Optimization Problems with Inequality Con- reputation decreased in the last ten years due to the com- straints parative success of Interior-Point Newtonian algorithms, which are asymptotically faster. In the present research We consider a second-derivative 1 sequential quadratic a combination of both approaches is evaluated. The idea programming trust-region method for large-scale nonlin- is to produce a competitive method, being more robust ear non-convex optimization problems with inequality con- and efficient than its ”pure” counterparts for critical prob- straints. Trial steps are composed of two components; a lems. Moreover, an additional hybrid algorithm is defined, Cauchy globalization step and an SQP correction step. A in which the Interior Point method is replaced by the New- single linear artificial constraint is incorporated that en- tonian resolution of a KKT system identified by the Aug- sures non-accent in the SQP correction step, thus ”guiding” mented Lagrangian algorithm. the algorithm through areas of indefiniteness. A salient feature of our approach is feasibility of all subproblems. Ernesto G. Birgin IME-USP Daniel Robinson Department of Computer Science Oxford University [email protected] [email protected] Jos´e Mario Martinez Nick Gould University of Campinas Oxford University [email protected] Rutherford Appleton Laboratory [email protected] CP1 Augmented Lagrangian Algorithm with Recursive CP2 Trust-Region Method: Application to Optimal A Multilevel Algorithm for Inverse Problems with Control Problems with Equality Constraints Bound Constraints In this talk, preliminary numerical results of augmented We present a provably scalable algorithm for solving a class Lagrangian algorithm ALDISCR by Sartenaer and Sachs of inverse problems in the presence of bound constraints. are presented. This algorithm is tailored for infinite- Previous analysis proved that the associated unconstrained dimensional optimization problems with equality con- problems can be solved efficiently using specially designed straints, like optimal control problems. ALDISCR consid- multilevel preconditioners. An extension of that technol- ers a sequence of discretized augmented Lagrangian sub- ogy is combined with a semismooth Newton method to problems with dynamically determined discretization levels show that the bound-constrained inverse problem can be to ensure global convergence. An application of the recur- solved at a cost that is decreasing with increasing resolu- sive multiscale trust-region method from Gratton, Sarte- tion relative to that of the forward problem. naer and Toint within this framework to solve the sub- problems is also proposed and discussed. Andrei Draganescu Department of Mathematics and Statistics Romain Dujol University of Maryland, Baltimore County Facult´es Universitaires Notre-Dame de la Paix [email protected] D´epartement de Math´ematiques [email protected] CP2 Annick Sartenaer An Interior Method for Computing a Trust-Region University of Namur Step Department of Mathematics We consider methods for large-scale unconstrained opti- [email protected] mization based on finding an approximate solution of a quadratically constrained trust-region subproblem. The CP1 solver is based on sequential subspace minimization with a modified barrier ”accelerator” direction in the subspace On the Use of Iterative Methods for Solving Linear basis. The method is able to find solutions of the subprob- Equations in Nonlinear Optimization lem to any prescribed accuracy. Numerical results will be Solving linear equations is an important step in many presented. This is joint work with Philip Gill and Joshua methods for nonlinear optimization, e.g., interior meth- Griffin. ods. In some cases, the matrices are symmetric and ill- Jennifer Erway conditioned. This may be due to e.g. symmetrization of Department of Mathematics the linear equation in an interior method or ill-posedness of Wake Forest University an inverse problem. We discuss the behavior of conjugate- [email protected] gradient type methods on such ill-conditioned linear equa- tions. Philip E. Gill Anders Forsgren University of California, San Diego Royal Institute of Technology (KTH) Department of Mathematics OP08 Abstracts 55 [email protected] [email protected] Joshua D. Griffin Roberto Andreani Informatics and Decision Sciences Department DMA-IMECC-UNICAMP Sandia National Laboratories [email protected] jgriffi@sandia.gov Giovane Cesar University of Campinas CP2 [email protected] Modified Line Search for High Dimensional Func- tions Optimization Roberto Cesar JR. This paper proposes a modified line search method which University of Sao Paulo makes use of partial derivatives and re-starts the search [email protected] process after a given number of iterations by modifying the boundaries based on the best solution obtained at the pre- Mario MARTINEZ vious iterations. Performance of the proposed algorithm is University of Campinas compared with genetic algorithm and particle swarm opti- [email protected] mization for functions having up to 10,000 dimensions for which line search clearly outperforms the other methods. CP3 Crina Grosan Goal Oriented Adaptivity and Multilevel Tech- Babes-Bolyai University niques for Shape Optimization with Inviscid Flows [email protected] We present a continuous adjoint approach to shape opti- Ajith Abraham mization for turbulent flows modelled by the incompress- Norwegian University of Science and Technology ible instationary Navier-Stokes equations. [email protected] After introducing the analytical adjoint calculus in the presence of stabilization terms, we will focus on adaptivity based on a goal oriented error estimator and multilevel op- CP2 timization with inexact trust-region SQP techniques. Nu- A Safeguarded Line Search Algorithm Using Trust merical results will be presented. Region Approach Christian Brandenburg In this talk, we present an hybrid algorithm for uncon- TU Darmstadt strained nonlinear optimization which combines line search Germany and trust region strategies. More precisely, the proposed [email protected] method implements a line search technique but allows the exploitation of directions of negative curvature by switch- Florian Lindemann ing to a trust region approach when such a direction is en- Technical University of Munich countered. We detail an adaptation of the Steihaug-Toint Germany conjugate gradient algorithm and report some preliminary [email protected] numerical results. Michael Ulbrich Laetitia Legrain Technical University of Munich University of Namur Chair of Mathematical Optimization Belgium [email protected] [email protected] Stefan Ulbrich Annick Sartenaer Technische Universitaet Darmstadt University of Namur Fachbereich Mathematik Department of Mathematics [email protected] [email protected] CP3 CP2 A Newton-Multigrid Method for One-Shot Pde- Efficient Curve Detection Using the Gauss-Newton Constrained Optimization Method We present a Newton-Multigrid method for the fast solu- The detection of geometric primitives, like lines, circles or tion of PDE-constrained optimization problems. Applying ellipses, in an image plays a crucial role in many com- an inexact Newton method to the nonlinear optimality sys- puter vision applications. In this talk, we present a new tem yields a sequence of quadratic optimization problems. approach to such problem using a Low-Order-Value Opti- An efficient preconditioner is mandatory for the coupled it- mization model that can be tackled by variations of the erative solution of these ill-conditioned and indefinite sys- Gauss-Newton method. The proposed model is robust to tems. To this end, a multigrid algorithm together with noise and very efficient to find patterns described by many appropriate smoothing iterations is developed. We discuss parameters. We present extensive performance compar- different relaxation methods and their properties for a va- isons to many classical algorithms. riety of optimal control problems. Paulo J. S. Silva Martin Engel University of Sao Paulo Institute for Numerical Simulation 56 OP08 Abstracts University of Bonn, Germany temperatures. [email protected] Widodo Samyono Michael Griebel Dept. of Appl. Phy, & Appl. Math., Columbia University University of Bonn, Germany New York, NY [email protected] [email protected] David Keyes CP3 Columbia University The Minimization of an L∞-Functional Subject to Brookhaven National Laboratory an Elliptic PDE and Pointwise State Constraints [email protected] We study the optimal control of a maximum-norm ob- Dana A. Knoll jective functional subjected to an elliptic-type PDE and Idaho National Laboratory point-wise state constraints. The problem is transformed [email protected] to a problem where the on-differentiable L∞-norm in the functional will be replaced by a scalar variable and addi- tional state constraints. This problem is solved by barrier CP4 methods. We will show the existence and convergence of Solution of Integer Bilevel Linear Programming the central path for a class of barrier functions. Numerical Problems experiments
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