Turnpike Properties in Discrete-Time Mixed-Integer Optimal Control

Faulwasser, Timm; Murray, Alexander

doi:10.1109/lcsys.2020.2988943

Cited by 9 publications

(5 citation statements)

References 31 publications

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“…More generally, whenever the set of control values taken is finite, the turnpike will be exact with respect to the inputs. This occurs frequently in case of mixed-integer OCPs, see Faulwasser & Murray (2020) for first steps in this direction. We remark that recently it has been shown L 1 tracking terms can also induce exact turnpikes phenomena in finite dimensional OCPs (Gugat et al 2020) and that certain objectives induce exactness of turnpikes also in Hilbert spaces.…”

Section: Generating Mechanismsmentioning

confidence: 99%

“…Both in discrete-time and continuous-time the existing turnpike results focus mostly on the turnpike being in the interior of the constraints, i.e., a comprehensive treatment with active constraints at the turnpike appears to be open. Moreover, problems with mixedinteger inputs and/or hybrid dynamics have only been discussed to a very limited extent, see (Faulwasser & Murray 2020) for discrete-time dissipativity-based results and (Gugat & Hante 2019) for results on infinite-dimensional hyperbolic systems with integrality constraints on the inputs under strict convexity assumptions for the objective.…”

Section: Topics Not Discussed and Open Problemsmentioning

confidence: 99%

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Turnpike Properties in Optimal Control: An Overview of Discrete-Time and Continuous-Time Results

Faulwasser,

Grüne

2020

Preprint

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The turnpike property refers to the phenomenon that in many optimal control problems, the solutions for different initial conditions and varying horizons approach a neighborhood of a specific steady state, then stay in this neighborhood for the major part of the time horizon, until they may finally depart. While early observations of the phenomenon can be traced back to works of Ramsey and von Neumann on problems in economics in 1928 and 1938, the turnpike property received continuous interest in economics since the 1960s and recent interest in systems and control. The present chapter provides an introductory overview of discrete-time and continuous-time results in finite and infinite-dimensions. We comment on dissipativitybased approaches and infinite-horizon results, which enable the exploitation of turnpike properties for the numerical solution of problems with long and infinite horizons. After drawing upon numerical examples, the chapter concludes with an outlook on timevarying, discounted, and open problems.

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Section: Generating Mechanismsmentioning

confidence: 99%

Section: Topics Not Discussed and Open Problemsmentioning

confidence: 99%

Turnpike Properties in Optimal Control: An Overview of Discrete-Time and Continuous-Time Results

Faulwasser,

Grüne

2020

Preprint

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“…Strict dissipativity of an OCP allows deriving sufficient stability conditions with terminal constraints [2], [8], [1] and without them [11], [20], [40], [19]. Dissipativity and turnpike concepts can be extended to time-varying cases [41], [45], [23] and to OCPs with discrete controls [17]. Finally, there exists a close relation between dissipativity notions of OCPs and infinite-horizon optimal control problems [15].…”

Section: The Dissipativity Route To Optimal Control and Mpcmentioning

confidence: 99%

Primal or Dual Terminal Constraints in Economic MPC? -- Comparison and Insights

Faulwasser¹,

Zanon²

2020

Preprint

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This chapter compares different formulations for Economic nonlinear Model Predictive Control (EMPC) which are all based on an established dissipativity assumption on the underlying Optimal Control Problem (OCP). This includes schemes with and without stabilizing terminal constraints, respectively, or with stabilizing terminal costs. We recall that a recently proposed approach based on gradient correcting terminal penalties implies a terminal constraint on the adjoints of the OCP. We analyze the feasibility implications of these dual/adjoint terminal constraints and we compare our findings to approaches with and without primal terminal constraints. Moreover, we suggest a conceptual framework for approximation of the minimal stabilizing horizon length. Finally, we illustrate our findings considering a chemical reactor as an example.

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“…It has been coined by Dorfman, Solow and Samuelson [11] and has received considerable interest in economics, see [12,13]. There has been interest in turnpike properties and their relation to dissipation inequalities [14,15], in turnpikes of PDE-constrained OCPs [16], mixed-integer problems [17] and economic MPC [18]. We refer to [19] for an overview.…”

Section: Introductionmentioning

confidence: 99%

On the Turnpike to Design of Deep Neural Nets: Explicit Depth Bounds

Faulwasser,

Hempel,

Streif

2021

Preprint

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It is well-known that the training of Deep Neural Networks (DNN) can be formalized in the language of optimal control. In this context, this paper leverages classical turnpike properties of optimal control problems to attempt a quantifiable answer to the question of how many layers should be considered in a DNN. The underlying assumption is that the number of neurons per layer-i.e., the width of the DNN-is kept constant. Pursuing a different route than the classical analysis of approximation properties of sigmoidal functions, we prove explicit bounds on the required depths of DNNs based on asymptotic reachability assumptions and a dissipativityinducing choice of the regularization terms in the training problem. Numerical results obtained for the two spiral task data set for classification indicate that the proposed estimates can provide non-conservative depth bounds.

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Turnpike Properties in Discrete-Time Mixed-Integer Optimal Control

Cited by 9 publications

References 31 publications

Turnpike Properties in Optimal Control: An Overview of Discrete-Time and Continuous-Time Results

Turnpike Properties in Optimal Control: An Overview of Discrete-Time and Continuous-Time Results

Primal or Dual Terminal Constraints in Economic MPC? -- Comparison and Insights

On the Turnpike to Design of Deep Neural Nets: Explicit Depth Bounds

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