On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control

Jung, Michael N.; Kirches, Christian; Säger, Sebastian

doi:10.1007/978-3-642-38189-8_16

Cited by 28 publications

(19 citation statements)

References 61 publications

(103 reference statements)

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“…We can then repeat the above estimates for each of the components of the vectors in the right hand side of Equation 25. Since t * i < τ and since the estimate becomes trivial for t * i = 0, the existence of a constantC independent ofβ such that Equation 19 holds follows from induction. Finally, we note that X = C([0, T]; L 1 ((0, L); R n )) is a Banach space for which Equation 14 holds with c = e −KT andc = 1.…”

Section: A Priori Estimates On the Relaxation Gapmentioning

confidence: 96%

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Relaxation methods for hyperbolic PDE mixed‐integer optimal control problems

Hante

2017

Optim Control Appl Methods

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Abstract. We extend the convergence analysis for methods solving PDEconstrained optimal control problems containing both discrete and continuous control decisions based on relaxation and rounding strategies to the class of first order semilinear hyperbolic systems in one space dimension. The results are obtained by novel a-priori estimates for the size of the relaxation gap based on the characteristic flow, fixed-point arguments and particular regularity theory for such mixed-integer control problems. As an application we consider a relaxation model for optimal flux switching control in conservation laws motivated by traffic flow problems.

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Section: A Priori Estimates On the Relaxation Gapmentioning

confidence: 96%

“…Finally, we note that X = C([0, T]; L 1 ((0, L); R n )) is a Banach space for which Equation 14 holds with c = e −KT andc = 1. So the desired estimate Equation 18 follows from Equation 19 and Lemma 6 with X = Y andC = 2c −1c nC = 2e KT nC.…”

Section: A Priori Estimates On the Relaxation Gapmentioning

confidence: 99%

Relaxation methods for hyperbolic PDE mixed‐integer optimal control problems

Hante

2017

Optim Control Appl Methods

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“…If engine speed constraints need to be incorporated, a more elaborated approach is necessary, see [37] for a survey of possible problem formulations and methods.…”

Section: Treating Gear Shiftsmentioning

confidence: 99%

Model Predictive Control of Autonomous Vehicles

Zanon¹,

Frasch

Vukov³

et al. 2014

Lecture Notes in Control and Information Sciences

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The control of autonomous vehicles is a challenging task that requires advanced control schemes. Nonlinear Model Predictive Control (NMPC) and Moving Horizon Estimation (MHE) are optimization-based control and estimation techniques that are able to deal with highly nonlinear, constrained, unstable and fast dynamic systems. In this chapter, these techniques are detailed, a descriptive nonlinear model is derived and the performance of the proposed control scheme is demonstrated in simulations of an obstacle avoidance scenario on a low-fricion icy road. IntroductionDue to the well known vehicle-road dynamics, Model Predictive Control (MPC) is an excellent tool for precise trajectory planning in autonomous vehicle guidance, which can be of great importance in dangerous driving situations. High sampling rates (i.e., in the range of tenths of Hertz) and long prediction horizons however, which are required for a safe operation, pose a computational challenge, particularly in combination with the involved nonlinear vehicle dynamics. Many recent chapter chose a two-level MPC approach to overcome this computational challenge, being composed of a coarse path planning algorithm with a long prediction horizon, and a higher-fidelity path following algorithm on a shorter horizon, cf.

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“…The aim of this paper is to devise a fast heuristic for approximate solutions of (9) via the generation of feasible points of (22) with near-optimal objective value. We follow the approach in [27,18,19] and split the solution of the mixed-integer nonlinear program (22) Our computational approach can be outlined in the following steps: 1. Discretize by choosing suitable spatial mesh sizes ∆x (i) > 0, i ∈ E. Relax problem by omitting (14).…”

Section: 2mentioning

confidence: 99%

Partial Outer Convexification for Traffic Light Optimization in Road Networks

Göttlich¹,

Potschka²,

Ziegler³

2017

SIAM J. Sci. Comput.

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Abstract. We consider the problem of computing optimal traffic light programs for urban road intersections using traffic flow conservation laws on networks. Based on a partial outer convexification approach, which has been successfully applied in the area of mixed-integer optimal control for systems of ordinary or differential algebraic equations, we develop a computationally tractable two-stage solution heuristic. The two-stage approach consists of the solution of a (smoothed) nonlinear programming problem with dynamic constraints and a reconstruction mixed-integer linear program without dynamic constraints. The two-stage approach is founded on a discrete approximation lemma for partial outer convexification, whose grid-independence properties for (smoothed) conservation laws are investigated. We use the two-stage approach to compute traffic light programs for two scenarios on different discretizations and demonstrate that the solution candidates cannot be improved in a reasonable amount of time by global state-of-the-art mixed-integer nonlinear programming solvers. The two-stage solution candidates are not only better than results obtained by global optimization of piecewise linearized traffic flow models but also can be computed at a faster rate.Key words. partial outer convexification, traffic networks, discretized conservation laws, optimization, mixed-integer programming AMS subject classifications. 35L65, 49J20, 90C11, 90C35 DOI. 10.1137/15M10481971. Introduction. With their seminal papers [22,25] in the 1950s on timedependent models for traffic flow with continuous densities instead of individual vehicles, Lighthill and Whitham and Richards established a new area of mathematical research in traffic modeling. The area is still highly active, e.g., for the case of traffic intersections (see, e.g., [5,8]), which constitute a building block of larger road networks. Our goal in this paper is to devise efficient heuristics to approximately solve the problem of optimal traffic light settings for road networks. We base our work on the model and scenarios presented in [12] and refer the reader to the discussion therein for further references regarding the wide variety of different approaches for traffic light control or related problems such as spillover dissipation [11,23].The resulting mathematical optimization problems, which we elaborate in full detail in section 2, can be described as nonlinear mixed-integer optimal control problems constrained by scalar hyperbolic conservation laws on networks using boundary control. These problems are extremely challenging for several reasons: First, the problems are infinite dimensional in nature and must be discretized appropriately. Special care has to be exercised for the discretization of the scalar conservation laws in order to guarantee well-posedness and, on the discrete level, consistency and sta-

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On Perspective Functions and Vanishing Constraints in Mixed-Integer Nonlinear Optimal Control

Cited by 28 publications

References 61 publications

Relaxation methods for hyperbolic PDE mixed‐integer optimal control problems

Relaxation methods for hyperbolic PDE mixed‐integer optimal control problems

Model Predictive Control of Autonomous Vehicles

Partial Outer Convexification for Traffic Light Optimization in Road Networks

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