  


Combinatorial Optimal Control of Semilinear Elliptic PDEs
Christoph Buchheim (christoph.buchheimtudortmund.de) Abstract: Optimal control problems (OCP) containing both integrality and partial differential equation (PDE) constraints are very challenging in practice. The most widespread solution approach is to first discretize the problem, it results in huge and typically nonconvex mixedinteger optimization problems that can be solved to proven optimality only in very small dimensions. In this paper, we propose a novel outer approximation approach to efficiently solve such OCPs in the case of certain semilinear elliptic PDEs with static integer controls over arbitrary combinatorial structures, where we assume the nonlinear part of the PDE to be nondecreasing and convex. The basic idea is to decompose the OCP into an integer linear programming (ILP) master problem and a subproblem for calculating linear cutting planes. These cutting planes rely on the pointwise concavity or submodularity of the PDE solution operator in terms of the control variables. The decomposition allows us to use standard solution techniques for ILPs as well as for PDEs. We further benefit from reoptimization strategies due to the iterative structure of the algorithm. Experimental results show that the new approach is capable of solving the combinatorial OCP of a semilinear Poisson equation with up to 180 binary controls to global optimality within a five hour time limit. In the case of the screened Poisson equation, which yields semiinfinite integer linear programs, problems with even 1400 binary controls are solved. Keywords: Optimal Control, Partial Differential Equations, Submodularity, Outer Approximation, Integer Nonlinear Programming Category 1: Nonlinear Optimization (Systems governed by Differential Equations Optimization ) Category 2: Integer Programming (Cutting Plane Approaches ) Category 3: Infinite Dimensional Optimization (Semiinfinite Programming ) Citation: Download: [PDF] Entry Submitted: 10/15/2015 Modify/Update this entry  
Visitors  Authors  More about us  Links  
Subscribe, Unsubscribe Digest Archive Search, Browse the Repository

Submit Update Policies 
Coordinator's Board Classification Scheme Credits Give us feedback 
Optimization Journals, Sites, Societies  