Routing problems appear frequently when dealing with the operation of communication or transportation networks. Among them, the message routing problem plays a determinant role in the optimization of network performance. Much of the motivation for this work comes from this problem which is shown to belong to the class of nonlinear convex multicommodity flow problems. This paper emphasizes the message routing problem in data networks, but it includes a broader literature overview of convex multicommodity flow problems. We present and discuss the main solution techniques proposed for solving this class of large-scale convex optimization problems. We conduct some numerical experiments on the message routing problem with four different techniques.Network Optimization, Multicommodity Flows, Message Routing, Convex Programming

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We present an algorithm to solve: Find (x,y) E A X A.L such that y E Tx, where A is a subspace and T is a maximal monotone operator. The algorithm is based on the proximal decomposition on the graph of a monotone operator and we show how to recover Spingarn's decom position method. We give a proof of convergence that does not use the concept of partial inverse and show how to choose a scaling factor to accelerate the convergence in the strongly monotone case. Numerical results performed on quadratic problems confirm the robust behaviour of the algorithm.