Lp Relaxations Better than Convexification for Multicommodity Network Optimization Problems with Step Increasing Cost Functions

We address here a class of particularly hard-to-solve combinatorial optimization problems namely multicommodity network optimization when the link cost functions are discontinuous step increasing. The main focus is on the development of relaxations for such problems in order to derive lower bounds. A straightforward way of getting lower bounds is to solve the “convexification” of the problem, i.e. a minimum cost multicommodity flow problem in which each link cost function is replaced by its best lower convex approximation. We propose here an alternative relaxation of the problem in terms of a large scale LP model, which can be solved by a generalized linear programming approach. Computational results on a series of test problems, typical of telecommunication networks, are presented showing that this relaxation leads to lower bounds strictly improving upon convexification in all the examples treated (the values of the improvement typically ranging from 10 to 20%). As far as we know, this is the first time a systematic way of generating bounds better than convexification to multicommodity network optimization problems with discontinuous step increasing cost functions is proposed.