In this paper we derive new properties of extreme inequalities for infinite group problems. We develop tools to prove that given valid inequalities for the infinite group problem are extreme. These results show that integer infinite group problems have discontinuous extreme inequalities. These inequalities are strong when compared to related classes of continuous extreme inequalities. This give...

The sequential ordering problem (SOP) is the generalisation of the asymmetric TSP in which the tour must obey given precedence relations between pairs of nodes. Many solution approaches have been devised for the SOP, but the version with symmetric costs, which we call the SSOP, has received very little attention. To fill this gap, we present a new 0-1 linear programming formulation for the SSOP...

We study the convex hull of the bounded, nonconvex setMn = {(x1, . . . , xn, xn+1) ∈ R n+1 : xn+1 = ∏n i=1 xi; i ≤ xi ≤ ui, i = 1, . . . , n + 1} for any n ≥ 2. We seek to derive strong valid linear inequalities forMn; this is motivated by the fact that many exact solvers for nonconvex problems use polyhedral relaxations so as to compute a lower bound via linear programming solvers. We present ...

In cutting plane methods, the question of how to generate the “best possible” set of cuts is both central and crucial. We propose a lexicographic multi-objective cutting plane generation scheme that generates, among all the maximally violated valid inequalities of a given family, an inequality that is undominated and maximally diverse w.r.t. the cuts that were previously found. By optimizing a ...

This paper considers the precedence constrained knapsack problem. More speci cally, we are interested in classes of valid inequalities which are facet-de ning for the precedence constrained knapsack polytope. We study the complexity of obtaining these facets using the standard sequential lifting procedure. Applying this procedure requires solving a combinatorial problem. For valid inequalities ...

We discuss two families of valid inequalities for linear mixed integer programming problems with cone constraints of arbitrary order, which arise in the context of stochastic optimization with downside risk measures. In particular, we extend the results of Atamtürk and Narayanan (Math. Program., 2010, 2011), who developed mixed integer rounding cuts and lifted cuts for mixed integer programming...

We study mixed integer conic sets involving a general regular (closed, convex, full dimensional, and pointed) cone K such as the nonnegative orthant, the Lorentz cone or the positive semidefinite cone. In a unified framework, we introduce K-minimal inequalities and show that under mild assumptions, these inequalities together with the trivial cone-implied inequalities are sufficient to describe...

This paper contributes to the theory of cutting planes for mixed integer linear programs (MILPs). Minimal valid inequalities are well understood for a relaxation of an MILP in tableau form where all the nonbasic variables are continuous. In this paper we study lifting functions for the nonbasic integer variables starting from such minimal valid inequalities. We characterize precisely when the l...

The successful application of Branch and Cut methods to the TSP has drawn attention also to the polyhedral properties of the symmetric capacitated vehicle routing problem, which is the capacitated counterpart of the TSP. We investigate three classes of valid inequalities for the CVRP, multistars, pathbin inequalities and hypotours and give computational results we obtained with a Branch and Cut...