Cônes de matrices et programmation mathématique : quelques applications. (Cones of matrices and mathematical programming : some applications)

All along this dissertation we present our works related to the scope of integer linear pro gramming This work come from those done by L Lov asz and A Schrijver in L Lov asz and A Schrijver Cones of matrices set functions and optimization SIAM First we present extensively their work in order to make it more accessible Thus we show clearly the relations between integer programming and positive semi de nite programming Then we derive from the Lov asz and Schrijver s construction a cutting plane algorithm solving linear integer programs Second we present the most famous applications of positive semi de nite devoted to com binatorial optimisation problems the work of L lov asz related to maximum independant set and those of M Goemans and D Williamson to the maximum cut Then we explain an application to the minimum cost ow subjected to end to end delay constraint Then we look for embeddings of lattices in convex cones and more precisely in the cones de ned by L Lov asz and A Schrijver We show that the two cones handled in the rst part of the dissertation share the same unique Hilbert basis At last we show that the latter cones can be viewed as subsets of a cone of metrics as an application we modelise the problem of the connected subgraph on this cone of metrics and de ne a positive semi de nite relaxation of this problem In the annexes we explain a work done together with A Jarry related to the minimum number of edges of a connected graph with a diameter constraint A work done together with S Bertrand and P Mahey is also explained it concerns the minimum cost ow in the special behaviour of non conservative ows