On the time transformation of mixed integer optimal control problems using a consistent fixed integer control function

AbstractWe present a trust-region steepest descent method for dynamic optimal control problems with binary-valued integrable control functions. Our method interprets the control function as an indicator function of a measurable set and makes set-valued adjustments derived from the sublevel sets of a topological gradient function. By combining this type of update with a trust-region framework, we are able to show by theoretical argument that our method achieves asymptotic stationarity despite possible discretization errors and truncation errors during step determination. To demonstrate the practical applicability of our method, we solve two optimal control problems constrained by ordinary and partial differential equations, respectively, and one topological optimization problem.

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Relaxation methods for hyperbolic PDE mixed-integer optimal control problems

Optimal Control Applications and Methods ◽

10.1002/oca.2315 ◽

2017 ◽

Vol 38 (6) ◽

pp. 1103-1110 ◽

Cited By ~ 12

Author(s):

Falk M. Hante

Keyword(s):

Optimal Control ◽

Optimal Control Problems ◽

Mixed Integer ◽

Control Problems ◽

Relaxation Methods ◽

Hyperbolic Pde

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On the relaxation gap for PDE mixed-integer optimal control problems

PAMM ◽

10.1002/pamm.201610380 ◽

2016 ◽

Vol 16 (1) ◽

pp. 783-784

Author(s):

Falk M. Hante

Keyword(s):

Optimal Control ◽

Optimal Control Problems ◽

Mixed Integer ◽

Control Problems

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Mixed-integer optimal control problems with switching costs: a shortest path approach

Mathematical Programming ◽

10.1007/s10107-020-01581-3 ◽

2020 ◽

Author(s):

Felix Bestehorn ◽

Christoph Hansknecht ◽

Christian Kirches ◽

Paul Manns

Keyword(s):

Optimal Control ◽

Shortest Path ◽

Switching Costs ◽

Optimal Control Problems ◽

Directed Acyclic Graphs ◽

Shortest Path Problem ◽

Mixed Integer ◽

Programming Approach ◽

Control Problems ◽

Decomposition Approach

Abstract We investigate an extension of Mixed-Integer Optimal Control Problems by adding switching costs, which enables the penalization of chattering and extends current modeling capabilities. The decomposition approach, consisting of solving a partial outer convexification to obtain a relaxed solution and using rounding schemes to obtain a discrete-valued control can still be applied, but the rounding turns out to be difficult in the presence of switching costs or switching constraints as the underlying problem is an Integer Program. We therefore reformulate the rounding problem into a shortest path problem on a parameterized family of directed acyclic graphs (DAGs). Solving the shortest path problem then allows to minimize switching costs and still maintain approximability with respect to the tunable DAG parameter $$\theta $$ θ . We provide a proof of a runtime bound on equidistant rounding grids, where the bound is linear in time discretization granularity and polynomial in $$\theta $$ θ . The efficacy of our approach is demonstrated by a comparison with an integer programming approach on a benchmark problem.

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