A conic quadratic formulation for a class of convex congestion functions in network flow problems


Gurel S.

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH, vol.211, no.2, pp.252-262, 2011 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 211 Issue: 2
  • Publication Date: 2011
  • Doi Number: 10.1016/j.ejor.2010.12.018
  • Journal Name: EUROPEAN JOURNAL OF OPERATIONAL RESEARCH
  • Journal Indexes: Science Citation Index Expanded, Scopus
  • Page Numbers: pp.252-262
  • Keywords: Integer programming, Network flows, Second-order cone programming, Capacity expansion, Congestion costs, Convex increasing power functions, ALGORITHMS, ASSIGNMENT, CAPACITY

Abstract

In this paper we consider a multicommodity network flow problem with flow routing and discrete capacity expansion decisions. The problem involves trading off congestion and capacity assignment (or expansion) costs. In particular, we consider congestion costs involving convex, increasing power functions of flows on the arcs. We first observe that under certain conditions the congestion cost can be formulated as a convex function of the capacity level and the flow. Then, we show that the problem can be efficiently formulated by using conic quadratic inequalities. As most of the research on this problem is devoted to heuristic approaches, this study differs in showing that the problem can be solved to optimum by branch-and-bound solvers implementing the second-order cone programming (SOCP) algorithms. Computational experiments on the test problems from the literature show that the continuous relaxation of the formulation gives a tight lower bound and leads to optimal or near optimal integer solutions within reasonable CPU times. (C) 2011 Elsevier B.V. All rights reserved.