Quasi Newton methods for bound constrained problems

Diese Arbeit befasst sich mit Methoden zur Lösung von hochdimensionalen Optimierungsproblemen mit einfachen Schranken. Probleme dieser Art tauchen in einer grossen Anzahl von unterschiedlichen Anwendungen auf und spielen eine entscheidende Rolle in einigen Methoden zur Lösung von Optimierungsproblemen mit allgemeinen Nebenbedingungen, Variationsungleichungen und Komplementaritätsproblemen. Im ersten Teil dieser Arbeit beschreiben wir den allgemeinen mathematischen Hintergrund der Optimierungstheorie für Optimierungsprobleme mit Schrankenbedingungen. Danach werden die nützlichsten auf aktiven Mengen beruhenden Verfahren zur Lösung dieser Probleme diskutiert. Im zweiten Teil dieser Arbeit stellen wir ein neues Limited Memory Quasi Newton-Verfahren für hochdimensionale und einfach eingeschränkte Probleme vor. Der neue Algorithmus verwendet eine Kombination aus den steilsten Abstiegsrichtungen und Quasi Newton Richtungen, um die Menge der optimalen aktiven Variablen zu identifizieren. Die Quasi Newton-Richtungen werden mit Hilfe der Limited Memory SR1 Matrizen und, falls erforderlich, durch die Anwendung einer Regularisierung berechnet. Zum Schluss präsentieren wir die Ergebnisse von numerischen Experimenten, die die relative Performance unseres Algorithmuses bezüglich unterschiedlicher Parametereinstellungen und im Vergleich mit einem anderen Algorithmus darstellen.In this thesis, we are concerned with methods for solving large scale bound constrained optimization problems. This kind of problems appears in a wide range of applications and plays a crucial role in some methods for solving general constrained optimization problems, variational inequalities and complementarity problems. In the first part, we provide the general mathematical background of optimization theory for bound constrained problems. Then the most useful methods for solving these problems based on the active set strategy are discussed. In second part of this thesis, we introduce a new limited memory quasi Newton method for bound constrained problems. The new algorithm uses a combination of the steepest decent directions and quasi Newton directions to identify the optimal active bound constraints. The quasi Newton directions are computed using limited memory SR1 matrices and, if needed, by applying regularization. At the end, we present results of numerical experiments showing the relative performance of our algorithm in different parameter settings and in comparison with another algorithm

Adaptive Parameter Optimization For An Elliptic-Parabolic System Using The Reduced-Basis Method With Hierarchical A-Posteriori Error Analysis

In this paper the authors study a non-linear elliptic-parabolic system, which is motivated by mathematical models for lithium-ion batteries. One state satisfies a parabolic reaction diffusion equation and the other one an elliptic equation. The goal is to determine several scalar parameters in the coupled model in an optimal manner by utilizing a reliable reduced-order approach based on the reduced basis (RB) method. However, the states are coupled through a strongly non-linear function, and this makes the evaluation of online-efficient error estimates difficult. First the well-posedness of the system is proved. Then a Galerkin finite element and RB discretization is described for the coupled system. To certify the RB scheme hierarchical a-posteriori error estimators are utilized in an adaptive trust-region optimization method. Numerical experiments illustrate good approximation properties and efficiencies by using only a relatively small number of reduced bases functions.Comment: 24 pages, 3 figure

Thermal diffusivity measurement for p-Si and Ag/p-Si by photoacoustic technique

Thermal diffusivity (TD) of p-Si and Ag/p-Si samples were measured by photoacoustic technique using open photoacoustic cell (OPC). The samples were annealed by heating them at 960, 1050, 1200, and 1300 °C for 3 h in air. The thermal diffusivity of Ag-coated samples was obtained by fitting the photoacoustic experimental data to the thermally thick equation for Rosencwaig and Gersho (RG) theory. For the single layer samples, the thermal diffusivity can be obtained by fitting as well as by obtaining the critical frequency f c . In this study, the thermal diffusivity of the p-Si samples increased with increasing the annealing temperature. The thermal diffusivity of the Ag/p-Si samples, after reaching the maximum value of about 2.73 cm2/s at a temperature of 1200 °C, decreased due to the silver complete melt in the surface of the silicon

Stabilization of 3D Navier–Stokes Equations to Trajectories by Finite-Dimensional RHC

Local exponential stabilization of the three-dimensional Navier–Stokes system to a given reference trajectory via receding horizon control (RHC) is investigated. The RHC enters as the linear combinations of a finite number of actuators. The actuators are spatial functions and can be chosen in particular as indicator functions whose supports cover only a part of the spatial domain.publishe

Receding horizon control for the stabilization of the wave equation

Stabilization of the wave equation by the receding horizon framework is investigated. Distributed control, Dirichlet boundary control, and Neumann boundary control are considered. Moreover for each of these control actions, the well-posedness of the control system and the exponential stability of Receding Horizon Control (RHC) with respect to a proper functional analytic setting are investigated. Observability conditions are necessary to show the suboptimality and exponential stability of RHC. Numerical experiments are given to illustrate the theoretical results.ERC advanced Grant 668998 (OCLOC)(VLID)278339

LMBOPT : a limited memory method for bound-constrained optimization

Recently, Neumaier and Azmi gave a comprehensive convergence theory for a generic algorithm for bound constrained optimization problems with a continuously differentiable objective function. The algorithm combines an active set strategy with a gradient-free line search CLS along a piecewise linear search path defined by directions chosen to reduce zigzagging. This paper describes LMBOPT, an efficient implementation of this scheme. It employs new limited memory techniques for computing the search directions, improves CLS by adding various safeguards relevant when finite precision arithmetic is used, and adds many practical enhancements in other details. The paper compares LMBOPT and several other solvers on the unconstrained and bound constrained problems from the CUTEst collection and makes recommendations on which solver to use and when. Depending on the problem class, the problem dimension, and the precise goal, the best solvers are LMBOPT, ASACG, and LMBFG-EIG-MS.publishe

Optimal Feedback Law Recovery by Gradient-Augmented Sparse Polynomial Regression

A sparse regression approach for the computation of high-dimensional optimal feedback laws arising in deterministic nonlinear control is proposed. The approach exploits the control-theoretical link between Hamilton-Jacobi-Bellman PDEs characterizing the value function of the optimal control problems, and rst-order optimality conditions via Pontryagin's Maximum Principle. The latter is used as a representation formula to recover the value function and its gradient at arbitrary points in the space-time domain through the solution of a two-point boundary value problem. After generating a dataset consisting of di erent state-value pairs, a hyperbolic cross polynomial model for the value function is tted using a LASSO regression. An extended set of low and high-dimensional numerical tests in nonlinear optimal control reveal that enriching the dataset with gradient information reduces the number of training samples, and that the sparse polynomial regression consistently yields a feedback law of lower complexity.publishe