Combinatorial integral approximation

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

doi:10.1007/s00186-011-0355-4

Cited by 79 publications

(61 citation statements)

References 22 publications

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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%

“…Optionally, a refinement step of the temporal mesh size ∆t can be performed after step 4 to improve the accuracy of the switching if necessary, but this step is only useful in the case without additional combinatorial constraints that couple over time [26,27]. We do not consider this option in the numerical computations in this paper.…”

Section: 2mentioning

confidence: 99%

“…We use Abel summation (the discrete version of integration by parts) and assumptions (23), (25), and (26) to bound the penultimate term in (27) according to…”

Section: Case Imentioning

confidence: 99%

“…We now determine an integer vector β t ∈ H ∩ {0, 1} nΩ as the solution of the Stage II mixed-integer linear program without dynamic constraints (the CIAP) [27],…”

Section: Case Imentioning

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.…”

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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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: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

“…We use Abel summation (the discrete version of integration by parts) and assumptions (23), (25), and (26) to bound the penultimate term in (27) according to…”

Section: Case Imentioning

confidence: 99%

“…We now determine an integer vector β t ∈ H ∩ {0, 1} nΩ as the solution of the Stage II mixed-integer linear program without dynamic constraints (the CIAP) [27],…”

Section: Case Imentioning

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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Joint subchannel assignment and power control in OFDMA in‐band full‐duplex heterogeneous networks

Rajabi

Rasti

Pedram

et al. 2020

Trans Emerging Tel Tech

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Among the emerging technologies for the fifth‐generation (5G) cellular networks, in‐band full‐duplex (FD) wireless communication and heterogeneous networks (HetNets) are two promising technologies, which can significantly improve the network capacity. Due to self‐interference (SI) and multiple sources of intercell interference, base station (BS)‐BS interference and user equipment (UE)‐UE interference in FD HetNets, interference management is a significant and challenging issue in such a network. To this end, in this article, we study the joint subchannel assignment and power control problem in the OFDMA FD HetNet. The joint subchannel assignment and power control problem in this network is formulated as an optimization problem to maximize the uplink and downlink sum throughput of the femtocell user equipments (FUEs) taking into account the maximum interference constraint imposed on the macrocell user equipments (MUEs) and the minimum throughput constraint of FUEs. Since the formulated problem is a nonconvex and intractable mixed integer nonlinear programming (MINLP) problem, we invoke variable change and linear approximation to reformulate and convert the problem into a tractable mixed‐integer linear programming (MILP) problem. Numerical results show that the optimal solution of the new MILP problem is very close to the optimal objective value of the original MINLP problem. The results also demonstrate that in an OFDMA FD HetNet, when all the UEs and the BSs are FD, approximately 50% improvement in the network throughputcan be achieved compared with a half‐duplex HetNet.

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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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Combinatorial integral approximation

Cited by 79 publications

References 22 publications

Partial Outer Convexification for Traffic Light Optimization in Road Networks

Partial Outer Convexification for Traffic Light Optimization in Road Networks

Joint subchannel assignment and power control in OFDMA in‐band full‐duplex heterogeneous networks

Relaxation methods for hyperbolic PDE mixed‐integer optimal control problems

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