Abstract We consider the problem of setting revenue‐maximizing tolls on a subset of arcs of a transportation network, assuming that the users of the network are assigned to shortest paths with respect to the sum of tolls and initial costs. Our main results are concerned with a polyhedral study of the problem, i.e., the design of valid inequalities and facets for this pricing problem and some of its variants. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

Convexity has recently received a lot of attention in the machine learning community, and the lack of convexity has been seen as a major disadvantage of many learning algorithms, such as multi-layer artificial neural networks. We show that training multi-layer neural networks in which the number of hidden units is learned can be viewed as a convex optimization problem. This problem involves an infinite number of variables, but can be solved by incrementally inserting a hidden unit at a time, each time finding a linear classifier that minimizes a weighted sum of errors.

Abstract We consider the problem of maximizing the revenue raised from tolls set on the arcs of a transportation network, under the constraint that users are assigned to toll‐compatible shortest paths. We first prove that this problem is strongly NP‐hard. We then provide a polynomial time algorithm with a worst‐case precision guarantee of ${{1 \over 2}\log_2 m_T+1}$ , where mT denotes the number of toll arcs. Finally, we show that the approximation is tight with respect to a natural relaxation by constructing a family of instances for which the relaxation gap is reached. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(1), 57–67 2005

Abstract A noncooperative game is formulated on a transportation network with congestion. The players are associated with origin‐destination pairs, and are facing demand functions at their respective destination nodes. A Nash‐Cournot equilibrium is defined and conditions for existence and uniqueness of this solution are provided. The asymptotic behavior of the Nash‐Cournot equilibrium is then shown to yield (under appropriate assumptions) a total flow vector corresponding to a Wardrop equilibrium.