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

Säger, Sebastian; Zeile, Clemens

doi:10.1007/s10589-020-00244-5

Cited by 24 publications

(24 citation statements)

References 23 publications

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“…The approach was again generalized to PDE-constrained problems [29]. To reduce the (undesired) chattering behavior of the rounded control the total variation is constrained [49] or switching cost aware rounding algorithms are considered [5,6].…”

Section: Introductionmentioning

confidence: 99%

Parabolic optimal control problems with combinatorial switching constraints -- Part I: Convex relaxations

Buchheim¹,

Grütering²,

Meyer³

2022

Preprint

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We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon, they can thus be seen as dynamic switches. The switching patterns may be subject to combinatorial constraints such as, e.g., an upper bound on the total number of switchings or a lower bound on the time between two switchings. While such combinatorial constraints are often seen as an additional complication that is treated in a heuristic postprocessing, the core of our approach is to investigate the convex hull of all feasible switching patterns in order to define a tight convex relaxation of the control problem. The convex relaxation is built by cutting planes derived from finite-dimensional projections, which can be studied by means of polyhedral combinatorics, and solved by an outer approximation algorithm. However, both the relaxation and the algorithm are independent of any fixed discretization and can thus be formulated in function space. In a numerical evaluation for the case of a bounded number of switchings, we show that our approach does not only improve the dual bounds given by the straightforward continuous relaxation, but that our relaxation can even be solved more efficiently than the latter. In other words, our approach profits from the binarity of the controls, rather than suffering from it.

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Section: Introductionmentioning

confidence: 99%

Parabolic optimal control problems with combinatorial switching constraints -- Part I: Convex relaxations

Buchheim¹,

Grütering²,

Meyer³

2022

Preprint

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“…Therefore, minimization and constraining of total variation terms have been included in the approximation step of the combinatorial integral approximation decomposition in recent articles [2,3,44]. Furthermore, the convex relaxation of the multibang regularizer can be integrated into the combinatorial integral decomposition approach as well; see [34].…”

Section: Introductionmentioning

confidence: 99%

Sequential linear integer programming for integer optimal control with total variation regularization

Leyffer¹,

Manns²

2022

ESAIM: COCV

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We propose a trust-region method that solves a sequence of linear integer programs to tackle integer optimal control problems regularized with a total variation penalty. The total variation penalty implies that the considered integer control problems admit minmizers. We introduce a local optimality concept for the problem, which arises from the infinite-dimensional perspective. In the case of a one-dimensional domain of the control function, we prove convergence of the iterates produced by our algorithm to points that satisfy first-order stationarity conditions for local optimality. We demonstrate the theoretical findings on a computational example.

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“…Combinatorical constraints coupling over time can be handled by combinatorial integer approximation problems; see, for example, Reference 13. Extensions to this approach include reductions of unrealistic frequent switching, 14 by constrained total variation of the control, 15 and by minimum dwell time constraints. Implementations include the open‐source software package pycombina 16 …”

Section: Introductionmentioning

confidence: 99%

State elimination for mixed‐integer optimal control of partial differential equations by semigroup theory

Thünen

Leyffer

Säger

2022

Optim Control Appl Methods

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Mixed-integer optimal control problems governed by partial differential equations (MIPDECOs) are powerful modeling tools but also challenging in terms of theory and computation. We propose a highly efficient state elimination approach for MIPDECOs that are governed by partial differential equations that have the structure of an abstract ordinary differential equation in function space. This allows us to avoid repeated calculations of the states for all time steps, and our approach is applied only once before starting the optimization. The presentation of theoretical results is complemented by numerical experiments.

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

Cited by 24 publications

References 23 publications

Parabolic optimal control problems with combinatorial switching constraints -- Part I: Convex relaxations

Parabolic optimal control problems with combinatorial switching constraints -- Part I: Convex relaxations

Sequential linear integer programming for integer optimal control with total variation regularization

State elimination for mixed‐integer optimal control of partial differential equations by semigroup theory

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