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.

We present a new and much more efficient implementation of the proximal decomposition algorithm for routing in congested telecommunication networks. The routing model that we analyze is a static one intended for use as a subproblem in a network design context. After describing our new implementation of the proximal decomposition algorithm and reviewing the flow deviation algorithm, we compare the solution times for (1) the original proximal decomposition algorithm, (2) our new implementation of the proximal decomposition algorithm, and (3) the flow deviation algorithm. We report extensive computational comparisons of solution times using actual and randomly generated networks. These results show that our new proximal decomposition algorithm is substantially faster than the earlier proximal decomposition algorithm in every case. Our new proximal decomposition is also faster than the flow deviation algorithm if the network is not too congested and a highly accurate solution is desired, such as one within 0.1% of optimality. For moderate accuracy requirements, such as 1.0% optimality, and for congested networks, the flow deviation algorithm is faster. More importantly, solutions that we obtained from the proximal decomposition algorithm always involve flow on only a small number of routes between source‐destination pairs. The flow deviation algorithm, however, can produce solutions with flows on a very large number of different routes between individual source‐destination pairs. © 1998 John Wiley & Sons, Inc. Networks 31: 227–238, 1998

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We present briefly some results we obtained with known methods to solve minimum cost tension problems, comparing their performance on non-specific graphs and on series-parallel graphs. These graphs are shown to be of interest to approximate many tension problems, like synchronization in hypermedia documents. We propose a new aggregation method to solve the minimum convex piecewise linear cost tension problem on series-parallel graphs in O(m 3 ) operations.

Planning issues in a continuous domain in the presence of noise lead to important modeling and computational difficulties. The game of billiards has offered many interesting challenges to both communities of AI and Optimization. We present a two-layered approach consisting in a high level planner and a low level controller. We propose here a refined controller for billiards based on robust optimization combined with specific adjustments to take advantage of the domain knowledge. A multi-objective formulation of a robust controller will be presented to provide the tools needed to execute any desired shot on the table, as part of a two-layered approach for the game of billiards. Some results will be then shown, followed by a short discussion on future work.

International audienceThe Stackelberg Minimum Spanning Tree Game (StackMST) is defined in terms of a graph G = (V, B ∪ R), with two disjoint sets of edges, blue B and red R, and costs {c e ≥ 0 : e ∈ R} defined for the red edges. Once the leader of the game defines prices {p e : e ∈ B} to the blue edges, the follower chooses a minimum weight spanning tree (V, E T), at cost e∈B∩E T p e + e∈R∩E T c e. The goal is to find prices to maximize the revenue e∈B∩E T p e collected by the leader. We introduce a reformulation and a Branch-and-cut-and-price algorithm for StackMST. The reformulation is obtained after applying KKT optimality conditions to a StackMST non-compact Bilevel Linear Programming formulation and is strengthened with a partial rank-1 RLT and with valid inequalities from the literature. We also implemented a Branch-and-cut algorithm for an extended formulation derived from another in the literature. A preliminary computational study comparing both methods is also presented