A penalty continuation method for the ℓ∞ solution of overdetermined linear systems

A new algorithm for the ℓ∞ solution of overdetermined linear systems is given. The algorithm is based on the application of quadratic penalty functions to a primal linear programming formulation of the ℓ∞ problem. The minimizers of the quadratic penalty function generate piecewise-linear non-interior paths to the set of ℓ∞ solutions. It is shown that the entire set of ℓ∞ solutions is obtained from the paths for sufficiently small values of a scalar parameter. This leads to a finite penalty/continuation algorithm for ℓ∞ problems. The algorithm is implemented and extensively tested using random and function approximation problems. Comparisons with the Barrodale-Phillips simplex based algorithm and the more recent predictor-corrector primal-dual interior point algorithm are given. The results indicate that the new algorithm shows a promising performance on random (non-function approximation) problems

Packing While Traveling: Mixed Integer Programming for a Class of Nonlinear Knapsack Problems

Packing and vehicle routing problems play an important role in the area of supply chain management. In this paper, we introduce a non-linear knapsack problem that occurs when packing items along a fixed route and taking into account travel time. We investigate constrained and unconstrained versions of the problem and show that both are NP-hard. In order to solve the problems, we provide a pre-processing scheme as well as exact and approximate mixed integer programming (MIP) solutions. Our experimental results show the effectiveness of the MIP solutions and in particular point out that the approximate MIP approach often leads to near optimal results within far less computation time than the exact approach

Managing facility disruption in hub-and-spoke networks: formulations and efficient solution methods

Hub disruption result in substantially higher transportation cost and customer dissatisfaction. In this study, first a mathematical model to design hub-and-spoke networks under hub failure is presented. For a fast and inexpensive recovery, the proposed model constructs networks in which every single demand point will have a backup hub to be served from in case of disruption. The problem is formulated as a mixed integer quadratic program in a way that could be linearized without significantly increasing the number of variables. To further ease the model’ computational burden, indicator constraints are employed in the linearized model. The resulting formulation produced optimal solutions for small and some medium size instances. To tackle large problems, three efficient particle swarm optimisation-based metaheuristics which incorporate efficient solution representation, short-term memory and special crossover operator are proposed. We present the results for two scenarios relating to high and low probabilities of hub failures and provide managerial insight. The computational results, using problem instances with various sizes taken from CAB and TR datasets, confirm the effectiveness and efficiency of the proposed problem formulation and our new solution techniques

Efficient Production-Distribution System Design

The production-distribution system design is an integral part of the general supply chain design. This paper proposes a novel solution methodology for this problem that is based on Lagrangean relaxation, interior-point methods, and branch and bound. Unlike classical approaches, Lagrangean relaxation is applied in a two-level hierarchy, branch and bound is based on a Lagrangean lower bound and column generation (branch and price), while interior-point methods are used within a cutting-plane context (analytic centre cutting-plane method---ACCPM). Numerical results demonstrate that the two-level approach outperforms the classical approach and provides a very sharp lower bound that is the (proven) optimal in most cases.production-distribution systems, Lagrangean relaxation, nested decomposition, interior-point cutting-plane methods, branch and price