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On the Multi-Commodity Flow with convex objective function: Column-Generation approaches

Guillaume Beraud-Sudreau, Lucas Létocart, Youcef Magnouche, Sébastien Martin

TL;DR

This paper describes the convex multi‐commodity flow problem and presents methodologies to solve both its Splittable and Unsplittable variants, offering a robust framework for managing network flows in complex telecommunication environments.

Abstract

The purpose of this work is to develop an algorithmic optimization approach for a capacitated Multi-Commodity flow problem, where the objective is to minimize the total link costs, where the cost of each arc increases convexly with its utilization. This objective is particularly relevant in telecommunication networks, where device performance can deteriorate significantly as the available bandwidth on a link becomes limited. By optimizing this convex function, traffic is efficiently distributed across the network, ensuring optimal use of available resources and preserving capacity for future demands. This paper describes the Convex Multi-Commodity Flow Problem and presents methodologies to solve both its Splittable and Unsplittable variants. In the Splittable version, flows can be fractionally distributed across multiple paths, while in the Unsplittable version, each commodity must be routed through a single path. Our approach employs Column-Generation techniques to address the convexly increasing cost functions associated with arc utilization, effectively accommodating various forms of convex increasing cost functions, including nondifferentiable or black-box convex increasing functions. The proposed methods demonstrate strong computational efficiency, offering a robust framework for managing network flows in complex telecommunication environments.

On the Multi-Commodity Flow with convex objective function: Column-Generation approaches

TL;DR

This paper describes the convex multi‐commodity flow problem and presents methodologies to solve both its Splittable and Unsplittable variants, offering a robust framework for managing network flows in complex telecommunication environments.

Abstract

The purpose of this work is to develop an algorithmic optimization approach for a capacitated Multi-Commodity flow problem, where the objective is to minimize the total link costs, where the cost of each arc increases convexly with its utilization. This objective is particularly relevant in telecommunication networks, where device performance can deteriorate significantly as the available bandwidth on a link becomes limited. By optimizing this convex function, traffic is efficiently distributed across the network, ensuring optimal use of available resources and preserving capacity for future demands. This paper describes the Convex Multi-Commodity Flow Problem and presents methodologies to solve both its Splittable and Unsplittable variants. In the Splittable version, flows can be fractionally distributed across multiple paths, while in the Unsplittable version, each commodity must be routed through a single path. Our approach employs Column-Generation techniques to address the convexly increasing cost functions associated with arc utilization, effectively accommodating various forms of convex increasing cost functions, including nondifferentiable or black-box convex increasing functions. The proposed methods demonstrate strong computational efficiency, offering a robust framework for managing network flows in complex telecommunication environments.
Paper Structure (26 sections, 4 theorems, 16 equations, 12 figures, 3 tables, 1 algorithm)

This paper contains 26 sections, 4 theorems, 16 equations, 12 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Splittable and Unsplittable Convex Multi-Commoditiy Flow Problem
  3. State of the Art
  4. Contribution
  5. Splittable-CMCF Problem Modeling: Compact and Extended Formulations
  6. Compact Formulation
  7. Extended Formulation and Column Generation
  8. Problems Complexity
  9. Numerical Tests description
  10. Performances
  11. Splittable-CMCF Problem: Inner Approximation Formulation
  12. The Inner Approximation Formulation
  13. Extension to non-convex cost functions
  14. Performances
  15. Unsplittable-CMCF Problem
  16. ...and 11 more sections

Key Result

Proposition 3.2

The Column Generation algorithm associated to the $\mathcal{INNER}$ problem applied to a given graph $G=(V,A)$ and a set of commodities $K$ with continuously increasing non-convex functions $\{\pmb{r}_a\}_{a \in A}$, or for their convex $\{env(\pmb{r}_a)\}_{a \in A}$ converges to the same solution.

Figures (12)

  • Figure 1: Cost function used in numerical experiments: Linear, Quadratic and Kleinrock costs.
  • Figure 2: Example of cumulated distribution of available capacity of arcs on the SNDLib instance ZIB54 (|K=1501|, |A|=160)(|K=1501|, |A|=160), with optimal routing minimizing linear, quadratic or Kleinrock cost functions.
  • Figure 3: Performance Chart of the Splittable-CMCF problem on instances of the SNDLibon instances of the SNDLibwith 2 cost functions ; the problem tested for the Flow-Deviation algorithm is a relaxation of the Splittable-CMCF problem with uncapacitated arcs.; the problem tested for the Flow-Deviation algorithm is a relaxation of the Splittable-CMCF problem with uncapacitated arcs.
  • Figure 4: Example of cost function $r_a$ and an inner-approximation of its epigraph.
  • Figure 5: Example of convex, increasing, non-differentiable cost function, in blue, and vertices of a possible inner-approximation in orange.Example of convex, increasing, non-differentiable cost function, in blue, and vertices of a possible inner-approximation in orange.
  • ...and 7 more figures

Theorems & Definitions (9)

  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 4.1
  • Proposition 4.2
  • proof
  • Corollary 4.1
  • Proposition 4.3
  • proof