Sequential Linear Integer Programming for Mixed-Integer Optimal Control Problems

• Applicant: Professor Dr. Paul Manns
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 515118017

Mixed-integer optimal control problems are powerful mathematical modeling tools for many questions from natural and engineering sciences with versatile applications from material design over optimum experimental design to control of supply networks. The high versatility of the problem class is opposed to a lack of scalable solution algorithms that are widely applicable.The main goal of the proposed project is the analysis and advancement of a novel algorithmic method that takes on and connects different ideas in the field of applied mathematics, in particular of optimal control of partial differential equations, nonlinear and discrete optimization, and mathematical image processing.Regularizing mixed-integer optimal control problems with the total variation seminorm is at the core of the proposed method because it yields the opportunity to treat the resulting optimization problems with the algorithmic methods of nonlinear optimization, in particular so-called trust-region methods. Consequently, the existence of optimal solutions and the derivation of optimality conditions shall be complemented with the convergence analysis of a trust-region method in function space to solve the regularized problems. Then a discretized variant of the algorithm shall be designed, which couples two discretization grids for the control input function and the dual formulation of the total variation seminorm. Then a numerical analysis shall be carried out for the subproblems and the overall algorithm based on this discretization. The resulting discretized subproblems are combinatorial optimization problems and shall be treated with methods from combinatorial optimizationThis procedure is intended to extend the class of mixed-integer optimal control problems that are algorithmically treatable as well as to provide a rigorous analysis of the problems and the proposed algorithm. For all steps of the algorithm the proposed numerical analysis is intended to supply approximation guarantees. In combination with combinatorial algorithms to solve the discretized subproblems one obtains a completely implementable algorithm with approximation errors that can controlled by means of the discretization grids.

• DFG Programme: Research Grants