Improved regularity assumptions for partial outer convexification of mixed-integer PDE-constrained optimization problems

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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“…For the sake of completeness we mention that an improved approximation result for this context has recently been proven by Manns and Kirches[19].…”

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confidence: 86%

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

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2020

Comput Optim Appl

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“…The consistency property gives rise to the main approximation relationship between (BC) and (RC) in the proposition below [17,18]. This relationship constitutes the theoretical justification for the described approximation methodology.…”

Section: Definition 1 Letmentioning

confidence: 99%

“…While solving a discretization of (MSCP) to optimality is often considered intractable in practice, roundings can be computed efficiently. What is more, the infimum in (MSCP) can be approximated arbitrarily well using this methodology [17,18]. Consequently, rounding approaches have been used to solve a variety of problems [11,13,20,21,23].…”

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confidence: 99%

“…First, the infimum is approximated in the limit of the mesh refinements, necessitating very fine grids in practice. Related to this, the rounding algorithms inevitably introduce an (often undesired) chattering behavior into the resulting discrete control because the fractional control is approximated in the weak * topology of L ∞ ((0, T ), R n V ) (see [18]). Second, the approximation properties usually do not hold anymore once the control costs C are present in (MSCP).…”

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confidence: 99%

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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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“…We use the results on convexification reformulations and the combinatorial integral approximation (CIA) decomposition from [12,15,21,23,24,26,27,36]. The CIA decomposition splits the optimization into 1. deriving and solving a continuous relaxation of the problem (P), and 2. computing a discrete-valued approximation of the control of the relaxation.…”

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confidence: 99%

Compactness and convergence rates in the combinatorial integral approximation decomposition

2020

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The combinatorial integral approximation decomposition splits the optimization of a discrete-valued control into two steps: solving a continuous relaxation of the discrete control problem, and computing a discrete-valued approximation of the relaxed control. Different algorithms exist for the second step to construct piecewise constant discrete-valued approximants that are defined on given decompositions of the domain. It is known that the resulting discrete controls can be constructed such that they converge to a relaxed control in the $$\hbox {weak}^*$$ weak ∗ topology of $$L^\infty $$ L ∞ if the grid constant of this decomposition is driven to zero. We exploit this insight to formulate a general approximation result for optimization problems, which feature discrete and distributed optimization variables, and which are governed by a compact control-to-state operator. We analyze the topology induced by the grid refinements and prove convergence rates of the control vectors for two problem classes. We use a reconstruction problem from signal processing to demonstrate both the applicability of the method outside the scope of differential equations, the predominant case in the literature, and the effectiveness of the approach.

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Improved regularity assumptions for partial outer convexification of mixed-integer PDE-constrained optimization problems

Cited by 25 publications

References 16 publications

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

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

Compactness and convergence rates in the combinatorial integral approximation decomposition

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