Relaxation methods for hyperbolic PDE mixed‐integer optimal control problems

Hante, Falk M.

doi:10.1002/oca.2315

Cited by 20 publications

(34 citation statements)

References 33 publications

(80 reference statements)

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“…As a last step we use the optimal traffic light setting obtained in Stage II and compute the forward solution (including the intersection flux maximization (14) left out in (21)) yielding the corresponding values for traffic density and flow of the original nonlinear traffic light optimization problem (9).…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Then, a forward simulation using the nonlinear flow function and also the intersection flow maximization condition (14) computes the corresponding OFV for the original flow problem.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…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). 2.…”

Section: 2mentioning

confidence: 99%

“…Due to certain smoothness assumptions on the solution, these results cover partial differential equation (PDE) constraints of parabolic type. Recently, Hante [14] developed a POC theory for the case of mixed-integer optimal control problems constrained by first order semilinear hyperbolic systems in one space dimension with distributed control. His problem setting is close to our setting, but so far the generalization of his proof techniques to the case of boundary control seems to be out of reach, if not impossible.…”

mentioning

confidence: 99%

“…For realistic solutions in traffic light control, however, we need to satisfy additional combinatorial constraints that tightly couple control decisions over certain time intervals, e.g., maximum red light and minimum green light phases. These constraints are only marginally addressed in [14], with a direct reference to the combinatorial integer approximation problem (CIAP) introduced by Sager, Jung, and Kirches [27]. In the presence of such coupling combinatorial constraints, the integer optimality gap in the POC approach can no longer be reduced to arbitrarily small values.…”

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confidence: 99%

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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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Section: Numerical Resultsmentioning

confidence: 99%

“…Then, a forward simulation using the nonlinear flow function and also the intersection flow maximization condition (14) computes the corresponding OFV for the original flow problem.…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Partial Outer Convexification for Traffic Light Optimization in Road Networks

Göttlich¹,

Potschka²,

Ziegler³

2017

SIAM J. Sci. Comput.

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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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POD-Based Mixed-Integer Optimal Control of Evolution Systems

Jäkle

Volkwein

2020

Advances in Dynamics, Optimization and Computation

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In this chapter the authors consider the numerical treatment of a mixedinteger optimal control problem governed by linear convection-diffusion equations and binary control variables. Using relaxation techniques (introduced by [31] for ordinary differential equations) the original mixed-integer optimal control problem is transferred into a relaxed optimal control problem with no integrality constraints. After an optimal solution to the relaxed problem has been computed, binary admissible controls are constructed by a sum-up rounding technique. This allows us to construct-in an iterative process-binary admissible controls such that the corresponding optimal state and the optimal cost value approximate the original ones with arbitrary accuracy. However, using finite element (FE) methods to discretize the state and adjoint equations yield often to extensive systems which make the frequently calculations time-consuming. Therefore, a model-order reduction based on the proper orthogonal decomposition (POD) method is applied. Compared to the FE case, the POD approach yields to a significant acceleration of the CPU times while the error stays sufficiently small.

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Relaxation methods for hyperbolic PDE mixed‐integer optimal control problems

Cited by 20 publications

References 33 publications

Partial Outer Convexification for Traffic Light Optimization in Road Networks

Partial Outer Convexification for Traffic Light Optimization in Road Networks

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

POD-Based Mixed-Integer Optimal Control of Evolution Systems

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