Semidefinite optimization relaxations are among the widely used approaches to find global optimal or approximate solutions for many nonconvex problems. Here, we consider a specific quadratically constrained quadratic problem with an additional linear constraint. We prove that under certain conditions the semidefinite relaxation approach enables us to find a global optimal solution of the unde...

We study a well known linear programming relaxation of the p-median problem. We characterize the oriented graphs for which this linear relaxation defines an integral polytope.

We consider the large time behavior of solutions for a hyperbolic relaxation system. For a certain class of initial data the solution is shown to converge to relaxation rarefaction prooles at a determined asymptotic rate. The result is established without the smallness conditions of the wave strength and the initial disturbances.

This paper considers the capacitated multi-level lot-sizing problem with setup times, a class of difficult problems often faced in practical production planning settings. In the literature, relax-and-fix is a technique commonly applied to solve this problem due to the fact that setup decisions in later periods of the planning horizon are sensitive to setup decisions in the early periods but not...

Gomory mixed-integer cuts (GMICs) are widely used in modern branchand-cut codes for the solution of mixed-integer programs. Typically, GMICs are iteratively generated from the optimal basis of the current linear programming (LP) relaxation, and immediately added to the LP before the next round of cuts is generated. Unfortunately, this approach is prone to instability. In this paper we analyze a...

It is well-known that the integrality gap of the usual linear programming relaxation for Maxcut is 2 − ǫ. For general graphs, we prove that for any ǫ and any fixed boundk, adding linear constraints of support bounded by k does not reduce the gap below 2−ǫ. We generalize this to prove that for any ǫ and any fixed bound k, strengthening the usual linear programming relaxation by doing k rounds of...

Given a complete graph Kn = (V,E) with non-negative edge costs c ∈ RE , the problem 2EC is that of finding a 2-edge connected spanning multi-subgraph of Kn of minimum cost. The integrality gap α2EC of the linear programming relaxation 2EC for 2EC has been conjectured to be 65 , although currently we only know that 65 ≤ α2EC ≤ 3 2 . In this paper, we explore the idea of using the structure of so...

The lot–sizing polytope is a fundamental structure contained in many practical production planning problems. Here we study this polytope and identify facet–defining inequalities that cut off all fractional extreme points of its linear programming relaxation, as well as liftings from those facets. We give a polynomial– time combinatorial separation algorithm for the inequalities when capacities ...

We analyze split cuts from the perspective of cut generating functions via geometric lifting. We show that α-cuts, a natural higher-dimensional generalization of the k-cuts of Cornuéjols et al., gives all the split cuts for the mixed-integer corner relaxation. As an immediate consequence we obtain that the k-cuts are equivalent to split cuts for the 1-row mixed-integer relaxation. Further, we s...

In the coordinated lot-size problem, a major setup cost is incurred when at least one member of a product family is produced and a minor setup cost for each different item produced. This research consolidates the various modeling and algorithmic approaches reported in the literature for the coordinated replenishment problem with deterministic dynamic demand. For the two most effective approache...