Polyhedra study of mixed integer programs with variable upper bounds

We investigate the convex hull of the set defined by a single constraint with continues and binary variables and in addition, variable upper bound constraints are present. We introduce the flow cover inequality, which is valid for a projection. We also give conditions under which this inequality is facet defining. We use sequence independent lifting to obtain valid inequalities for the entire set. In general, computing the lifting function is NP-hard, but under an additional restriction on the cover we obtain a closed form. In the second part of the thesis we first elaborate on general sequence dependent lifting for this set and present a dynamic program for calculating lifting coefficients. Then we consider projections of this set to knapsack cover and to the single binary variable polytope. We present the result of sequence independent lifting of the knapsack cover inequality and obtain lifting some coefficients for sequence dependent lifting with for specific sequences. We generate two new families of facet defining inequalities for the single binary variable polytope and prove that combined with the trivial inequalities they give a full description of this polytope. Finally, we give an optimization problem for the lifting coefficients.