Using mixed-integer programming to solve power grid blackout problems

We consider optimization problems related to the prevention of large-scale cascading blackouts in power transmission networks subject to multiple scenarios of externally caused damage. We present computation with networks with up to 600 nodes and 827 edges, and many thousands of damage scenarios. Citation CORC Report TR-2005-07, Columbia University Article Download View Using mixed-integer programming … Read more

MIP-based heuristics for multi-item capacitated lot-sizing problem with setup times and shortage costs

We address a multi-item capacitated lot-sizing problem with setup times that arises in real-world production planning contexts. Demand cannot be backlogged, but can be totally or partially lost. Safety stock is an objective to reach rather than an industrial constraint to respect. The problem is NP-hard. A mixed integer mathematical formulation is presented. We propose … Read more

A special ordered set approach to discontinuous piecewise linear optimization

Piecewise linear functions (PLFs) are commonly used to approximate nonlinear functions. They are also of interest in their own, arising for example in problems with economies of scale. Early approaches to piecewise linear optimization (PLO) assumed continuous PLFs. They include the incremental cost MIP model of Markowitz and Manne and the convex combination MIP model … Read more

The multi-item capacitated lot-sizing problem with setup times and shortage costs

We address a multi-item capacitated lot-sizing problem with setup times and shortage costs that arises in real-world production planning problems. Demand cannot be backlogged, but can be totally or partially lost. The problem is NP-hard. A mixed integer mathematical formulation is presented. Our approach in this paper is to propose some classes of valid inequalities … Read more

The polar of a simple mixed-integer set

We study the convex hull $P$ of the set $S = \{(x, y) \in \Re_{+} \times Z^{n}: x + B_{i} y_{ij} \geq b_{ij}, j \in N_{i}, i \in M\}$, where $M = \{1, \ldots, m\}$, $N_{i} = \{1, \ldots, n_{i}\}$ $\forall i \in M$, $\sum_{i = 1}^{m}n_{i} = n$, and $B_{1} | \cdots | B_{m}$. … Read more

On generalized branching methods for mixed integer programming

In this paper we present a restructuring of the computations in Lenstra’s methods for solving mixed integer linear programs. We show that the problem of finding a good branching hyperplane can be formulated on an adjoint lattice of the Kernel lattice of the equality constraints without requiring any dimension reduction. As a consequence the short … Read more

Sequential pairing of mixed integer inequalities

We present a scheme for generating new valid inequalities for mixed integer programs by taking pair-wise combinations of existing valid inequalities. Our scheme is related to mixed integer rounding and mixing. The scheme is in general sequence-dependent and therefore leads to an exponential number of inequalities. For some important cases, we identify combination sequences that … Read more

A Mixed-Integer Programming Approach to Multi-Class Data Classification Problem

This paper presents a new data classification method based on mixed-integer programming. Traditional approaches that are based on partitioning the data sets into two groups perform poorly for multi-class data classification problems. The proposed approach is based on the use of hyper-boxes for defining boundaries of the classes that include all or some of the … Read more

Semi-Continuous Cuts for Mixed-Integer Programming

We study the convex hull of the feasible set of the semi-continuous knapsack problem, in which the variables belong to the union of two intervals. Besides being important in its own right, the semi-continuous knapsack problem is a relaxation of general mixed-integer programming. We show how strong inequalities valid for the semi-continuous knapsack polyhedron can … Read more

The Network Packing Problem in Terrestrial Broadcasting

The introduction of digital technology all over Europe requires a complete and challenging re-planning of the actual terrestrial broadcasting system. In fact, in order to implement digital networks, transmitters and frequencies must be removed from the current analog networks. On the other hand, the service level (territory coverage) of analog networks must be preserved until … Read more