A Bilevel Programming Method for Pipe Network Optimization

Zhang, Jianzhong; Zhu, Daming

doi:10.1137/s1052623493260696

Cited by 45 publications

(31 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The reported correlation coefficient is r 2 = 0.998. We have computed the best approximation of (15) by a function of type (14) (in the sense of the L 2 norm) and obtained…”

Section: A Convex Version Of the Joint Optimization Problemmentioning

confidence: 99%

“…A trust-region successive linear programming method is proposed in [8]. The authors in [14] add binary variables per diameter on each arc and use the continuous relaxation in their MIP techniques. In those approaches the computation of the continuous relaxation problem is crucial because it has a direct impact on the overall efficiency algorithm.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Design and Operations of Gas Transmission Networks

Babonneau

Nesterov

Vial

2012

Operations Research

View full text Add to dashboard Cite

Problems dealing with the design and the operations of gas transmission networks are challenging. The difficulty mainly arises from the simultaneous modeling of gas transmission laws and of the investment costs. The combination of the two yields a nonlinear non-convex optimization problem. To obviate this shortcoming, we propose a new formulation as a multi-objective problem, with two objectives. The first one is the investment cost function or a suitable approximation of it; the second is the cost of energy that is required to transmit the gas. This energy cost is approximated by the total energy dissipated into the network. This bi-criterion problem turns out to be convex and easily solvable by convex optimization solvers. Our continuous optimization formulation can be used as an efficient continuous relaxation for problems with non-divisible restrictions such as a limited number of available commercial pipe dimensions.

show abstract

“…The reported correlation coefficient is r 2 = 0.998. We have computed the best approximation of (15) by a function of type (14) (in the sense of the L 2 norm) and obtained…”

Section: A Convex Version Of the Joint Optimization Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Design and Operations of Gas Transmission Networks

Babonneau

Nesterov

Vial

2012

Operations Research

View full text Add to dashboard Cite

show abstract

“…From a numerical point of view, a backtrack algorithm exploring relevant Steiner topologies is proposed in [25] to solve a problem of water treatment network. A different algorithmic approach can be found in [26].…”

Section: The Gilbert-steiner Problem Modeling a Discrete Branched Strmentioning

confidence: 99%

“…In that case, by Corollary 6.7, we have M α (µ + , µ − ) ≤ E α (µ + , µ − ) and (26) holds. Assume that the energies in (26) are finite. Now, observe that we may assume that G n are loop-free.…”

Section: Transport Paths (Qinglan Xia) and Traffic Plansmentioning

confidence: 99%

The structure of branched transportation networks

2007

View full text Add to dashboard Cite

The transportation problem can be formalized as the problem of finding the optimal paths to transport a measure µ + onto a measure µ − with the same mass. In contrast with the Monge-Kantorovich formalization, recent approaches model the branched structure of such supply networks by an energy functional whose essential feature is to favor wide roads. Given a flow s in a road or a tube or a wire, the transportation cost per unit length is supposed to be proportional to s α with 0 < α < 1. For the Monge-Kantorovich energy α = 1 so that it is equivalent to have two roads with flow 1/2 or a larger one with flow 1. If instead 0 < α < 1, a road with flow s 1 + s 2 is preferable to two individual roads s 1 and s 2 because (s 1 + s 2 ) α < s α 1 + s α 2 . Thus, this very simple model intuitively leads to branched transportation structures. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electric power supply systems and in natural objects like the blood vessels or the trees. When α > 1 − 1 N such structures can irrigate a whole bounded open set of R N . The aim of this paper is to give a mathematical proof of several structure and regularity properties empirically observed in transportation networks. It is first proven that optimal transportation networks have a tree structure and can be monotonically approximated by finite graphs. An interior regularity result is then proven according to which an optimal network is a finite graph away from the irrigated measure. It is also proven that the branching number of optimal networks has everywhere a universal explicit bound. These results answer questions raised in two recent papers by Xia. M. Bernot UMPA, ENS Lyon, 46, Allée d'Italie, 123 280 M. Bernot et al. Mathematics Subject Classification (2000)Primary: 49Q10; Secondary: 90B10 · 90B06 · 90B20 List of Symbolsor a fixed subset with finite measure of R, set of fiber indices µ + , µ − positive Borel measures in X with equal mass π a probability measure on X × X or "transference plan" t → χ(ω, t), constant for sufficiently large t, Lipschitz path, fiber indexed by ω (ω, t) → χ(ω, t), traffic plan or pattern, set of fibers |χ| := | | the total mass transported by χ P χ the law of ω → χ(ω) ∈ Lip 1 (X ) T (ω)) final point of a fiber π = (τ, σ ) λ transference plan of χ µ + (χ)(A) := |{ω : χ(ω, 0) ∈ A}|, irrigating measure of χ µ + χ = τ λ, irrigating measure of χ µ − (χ)(A) := |{ω : χ(ω, T χ (ω)) ∈ A}|, measure irrigated by χ µ − χ = σ λ, measure irrigated by χ TP(µ + , µ − ) set of traffic plans χ such that µ − χ = µ − and µ + χ = µ + . TP C set of traffic plans such that (G) f (e) α H 1 (e) Gilbert energy M α (µ + , µ − ) minimal transport cost between µ + and µ − for the Xia functional [ω] t ⊂ branch of the fiber ω at time t in a pattern |[ω] t | measure of the branch containing ω at time t E α (χ) =of a traffic plan χ D restriction of χ to a sub-domain of D(χ ) χ x := {ω : x ∈ χ(ω, R)} path class of x in χ x := χ x abbreviation |x| χ = | χx | multiplicity of χ ...

show abstract

“…Some numerical studies involving the optimization of the graph, when the topology is given (i.e. a topology for the graph is given and the position of the vertices is optimized) may be found in [2], [11] and [12]. Nevertheless, for practical applications, one needs numerical simulations of optimal transport paths using algorithms that allow changing topology.…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation of optimal transport paths

Xia

2010

2010 Second International Conference on Computer Modeling and Simulation

View full text Add to dashboard Cite

Abstract-Transport networks with branching structures are observable not only in nature as in trees, blood vessels, etc. but also in efficiently designed transport systems such as used in railway configurations and postage delivery networks. Mathematically, such a branching transport network is modeled by an optimal transport path between two probability measures (representing the source and the target). An essential feature of such a transport path is to favor group transportation in a large amount. This article provides numerical simulation of optimal transport paths. We first construct an initial transport path, and then modify the path as much as possible by using both local and global minimization algorithms.

show abstract

A Bilevel Programming Method for Pipe Network Optimization

Cited by 45 publications

References 9 publications

Design and Operations of Gas Transmission Networks

Design and Operations of Gas Transmission Networks

The structure of branched transportation networks

Numerical simulation of optimal transport paths

Contact Info

Product

Resources

About