Mixed-integer optimal control under minimum dwell time constraints

Zeile, Clemens; Robuschi, Nicolò; Säger, Sebastian

doi:10.1007/s10107-020-01533-x

Cited by 22 publications

(20 citation statements)

References 26 publications

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“…Binary optimal control problems are solved both locally and globally by using various approaches, including indirect methods based on the global maximum principle [14,15], dynamic programming [8,18], moment relaxations [33], combinatorial integral approximation decompositions [16,34], or direct first-discretize-then-optimize methods that result in mixed-integer nonlinear programs [3]. We refer to [33,37] for broader surveys.…”

Section: Introductionmentioning

confidence: 99%

Binary optimal control by trust-region steepest descent

Leyffer

Säger

2022

Math. Program.

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We 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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Section: Introductionmentioning

confidence: 99%

Binary optimal control by trust-region steepest descent

Leyffer

Säger

2022

Math. Program.

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“…Proof For the (CIA) problem without TV constraints, but with minimum up time constraints the sharp bound is proven in Theorem 2 in [32], where the constant C U ≥ 0 represents the given minimum up time. If we require for (CIA) that an activated control remains active for a time period of at least t f −t 0 max +1 , at most max switches take place.…”

Section: Corollary 4 Consider An Equidistant Gridmentioning

confidence: 99%

On mixed-integer optimal control with constrained total variation of the integer control

Säger

Zeile

2020

Comput Optim Appl

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The combinatorial integral approximation (CIA) decomposition suggests solving mixed-integer optimal control problems by solving one continuous nonlinear control problem and one mixed-integer linear program (MILP). Unrealistic frequent switching can be avoided by adding a constraint on the total variation to the MILP. Within this work, we present a fast heuristic way to solve this CIA problem and investigate in which situations optimality of the constructed feasible solution is guaranteed. In the second part of this article, we show tight bounds on the integrality gap between a relaxed continuous control trajectory and an integer feasible one in the case of two controls. Finally, we present numerical experiments to highlight the proposed algorithm’s advantages in terms of run time and solution quality.

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“…We show that the algorithm can be adapted to obey additional constraints, such as minimum dwell times (see [26]) and vanishing constraints without any increase in complexity. What is more, the objective function to be optimized can be adapted to mirror problems such as the Combinatorial Integral Approximation (CIA) problem (see [13]).…”

Section: Contributionmentioning

confidence: 99%

“…2. Minimum dwell time constraints [26] can be incorporated by choosing a path through the DAG which satisfies the dwell times for all controls. 3.…”

Section: Remark 15mentioning

confidence: 99%

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

et al. 2020

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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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Mixed-integer optimal control under minimum dwell time constraints

Cited by 22 publications

References 26 publications

Binary optimal control by trust-region steepest descent

Binary optimal control by trust-region steepest descent

On mixed-integer optimal control with constrained total variation of the integer control

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

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