09:50
Two approaches to solve 'high-dimensional' Hamilton Jacobi Bellman equations arising in optimal control
Prof. Dr. Karl Kunisch | University of Graz | Austria
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Authors:
Prof. Dr. Karl Kunisch | University of Graz | Austria
Dr. Tobias Breiten | University of Graz | Austria
Dr. Dante Kalise | Austria
Dr. Laurent Pfeiffer | Austria
The HJB equation provides optimal feedback solutions to control problems. Consequently these optimal solutions are stable with respect to system perturbation.
The price to pay for this advantage is the necesessity to cope with the curse of dimensionality. Different approaches are currently be developped to overcome this difficulty. In this talk I shall address a series expansion technique for a class of bilinear optimal control problems and on the generalized HJB equation for semi-linear parabolic equations.
10:10
Accelerated conditional gradient methods for sparse measure-valued optimization problems
Dr. Konstantin Pieper | Florida State University | United States
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Authors:
Dr. Konstantin Pieper | Florida State University | United States
Daniel Walter | TUM | Germany
We consider a minimization problem over the space of Radon measures, containing a smooth objective function and the total-variation norm as the cost term. Such problems are known to promote sparse solutions, which consist of a finite sum of Dirac-delta functions in many cases. For the algorithmic solution we consider a conditional gradient method, which iteratively inserts Dirac-delta functions and optimizes the corresponding coefficients. Under general assumptions, a sub-linear $C/k$ rate in the objective functional is obtained, which is sharp in most cases. To improve efficiency, one can fully resolve the
finite-dimensional subproblems occurring in each step of the method. We provide an analysis for the resulting procedure: under a structural assumption on the optimal solution, a linear $C\lambda^k$ convergence rate is obtained locally. Numerical experiments on the example of a sparse deconvolution problem confirm the theoretical findings and the practical efficiency of the method.
10:30
A semismooth implicit function theorem and its application to optimal control
Dr. Florian Mannel | University of Graz | Austria
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Author:
Dr. Florian Mannel | University of Graz | Austria
We present results about semismooth implicit functions defined on Banach spaces. In particular, we will discuss how these results can be applied to optimal control.