Column basis reduction and decomposable knapsack problems

We propose a very simple preconditioning method for integer programming feasibility problems: replacing the problem b’   ≤   Ax   ≤   b,   x ∈ Zn with b’   ≤   (AU)y   ≤   b,   y ∈ Zn, where U is a unimodular matrix computed via basis reduction, to make the … Read more

New Korkin-Zolotarev Inequalities

Korkin and Zolotarev showed that if $$\sum_i A_i(x_i-\sum_{j>i} \alpha_{ij}x_j)^2$$ is the Lagrange expansion of a Korkin–Zolotarev reduced positive definite quadratic form, then $A_{i+1}\geq \frac{3}{4} A_i$ and $A_{i+2}\geq \frac{2}{3}A_i$. They showed that the implied bound $A_{5}\geq \frac{4}{9}A_1$ is not attained by any KZ-reduced form. We propose a method to optimize numerically over the set of Lagrange … Read more

An improved algorithm for computing Steiner minimal trees in Euclidean d-space

We describe improvements to Smith’s branch-and-bound (B&B) algorithm for the Euclidean Steiner problem in R^d. Nodes in the B&B tree correspond to full Steiner topologies associated with a subset of the terminal nodes, and branching is accomplished by “merging” a new terminal node with each edge in the current Steiner tree. For a given topology … Read more

On a resource-constrained scheduling problem with application to distributed systems reconfiguration

This paper is devoted to the study of a resource-constrained scheduling problem which arises in relation to the operability of certain high availability real-time distributed systems. After a brief survey of the literature, we prove the NP-hardness of the problem and exhibit a few polynomial special cases. We then present a branch-and-bound algorithm for the … Read more

New solution approaches to the general single machine earliness-tardiness problem

This paper addresses the general single-machine earliness-tardiness problem with distinct release dates, due dates, and unit costs. The aim of this research is to obtain an exact nonpreemptive solution in which machine idle time is allowed. In a hybrid approach, we formulate and then solve the problem using dynamic programming (DP) while incorporating techniques from … Read more

MW: A Software Framework for Combinatorial Optimization on Computational Grids

Our goal in this paper is to demonstrate that branch-and-bound algorithms for combinatorial optimization can be effectively implemented on a relatively new type of multiprocessor platform known as a computational grid. We will argue that to easily and effectively harness the power of computational grids for branch-and-bound algorithms, the master-worker paradigm should be used to … Read more

Solving a Quantum Chemistry problem with Deterministic Global Optimization

The Hartree-Fock method is well known in quantum chemistry, and widely used to obtain atomic and molecular eletronic wave functions, based on the minimization of a functional of the energy. This gives rise to a multi-extremal, nonconvex, polynomial optimization problem. We give a novel mathematical programming formulation of the problem, which we solve by using … Read more

Semidefinite-Based Branch-and-Bound for Nonconvex Quadratic Programming

This paper presents a branch-and-bound algorithm for nonconvex quadratic programming, which is based on solving semidefinite relaxations at each node of the enumeration tree. The method is motivated by a recent branch-and-cut approach for the box-constrained case that employs linear relaxations of the first-order KKT conditions. We discuss certain limitations of linear relaxations when handling … Read more

Dual constrained single machine sequencing to minimize total weighted completion time

We study a single-machine sequencing problem with both release dates and deadlines to minimize the total weighted completion time. We propose a branch-and-bound algorithm for this problem. The algorithm exploits an effective lower bound and a dynamic programming dominance technique. As a byproduct of the lower bound, we have developed a new algorithm for the … Read more

New hybrid optimization algorithms for machine scheduling problems

Dynamic programming, branch-and-bound, and constraint programming are the standard solution principles for finding optimal solutions to machine scheduling problems. We propose a new hybrid optimization framework that integrates all three methodologies. The hybrid framework leads to powerful solution procedures. We demonstrate our approach through the optimal solution of the single-machine total weighted completion time scheduling … Read more