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

Energy and content aware multi-homing video transmission in heterogeneous networks,”

Abstract-This paper studies video transmission using a multihoming service in a heterogeneous wireless access medium. We propose an energy and content aware video transmission framework that incorporates the energy limitation of mobile terminals (MTs) and the quality-of-service (QoS) requirements of video streaming applications, and employs the available opportunities in a heterogeneous wireless access medium. In the proposed framework, the MT determines the transmission power for the utilized radio interfaces, selectively drops some packets under the battery energy limitation, and assigns the most valuable packets to different radio interfaces in order to minimize the video quality distortion. First, the problem is formulated as MINLP which is known to be NP-hard. Then we employ a piecewise linearization approach and solve the problem using a cutting plane method which reduces the associated complexity from MINLP to a series of MIPs. Finally, for practical implementation in MTs, we approximate the video transmission framework using a two-stage optimization problem. Numerical results demonstrate that the proposed framework exhibits very close performance to the exact problem solution. In addition, the proposed framework, unlike the existing solutions in literature, offers a choice for desirable trade-off between the achieved video quality and the MT operational period per battery charging. Index Terms-Multi-homing video transmission, video packet scheduling, heterogeneous wireless access medium, precedenceconstrained multiple knapsack problem (PC-MKP)

The congested multicommodity network design problem

This paper studies a version of the fixed-charge multicommodity network design problem where in addition to the traditional costs of flow and design, congestion at nodes is explicitly considered. The problem is initially modeled as a nonlinear integer programming formulation and two solution approaches are proposed: (i) a reformulation of the problem as a mixed integer second order cone program to optimally solve the problem for small to medium scale problem instances, and (ii) an evolutionary algorithm using elements of iterated local search and scatter search to provide upper bounds. Extensive computational results on new benchmark problem instances and on real case data are presented

Using the primal-dual interior point algorithm within the branch-price-and-cut method

AbstractBranch-price-and-cut has proven to be a powerful method for solving integer programming problems. It combines decomposition techniques with the generation of both columns and valid inequalities and relies on strong bounds to guide the search in the branch-and-bound tree. In this paper, we present how to improve the performance of a branch-price-and-cut method by using the primal-dual interior point algorithm. We discuss in detail how to deal with the challenges of using the interior point algorithm with the core components of the branch-price-and-cut method. The effort to overcome the difficulties pays off in a number of advantageous features offered by the new approach. We present the computational results of solving well-known instances of the vehicle routing problem with time windows, a challenging integer programming problem. The results indicate that the proposed approach delivers the best overall performance when compared with a similar branch-price-and-cut method which is based on the simplex algorithm

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

Non-interior piecewise-linear pathways to l-infinity solutions of overdetermined linear systems

Ankara : Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent University, 1996.Thesis (Master's) -- Bilkent University, 1996.Includes bibliographical references leaves 70-71In this thesis, a new characterization of solutions to overdetermined systems of linear equations is described based on a simple quadratic penalty function, which is used to change the problem into an unconstrained one. Piecewiselinear non-interior pathways to the set of optimal solutions are generated from the minimization of the unconstrained function. It is shown that the entire set of solutions is obtained from the paths for sufficiently small values of a scalar parameter. As a consequence, a new finite penalty algorithm is given for fx, problems. The algorithm is implemented and exhaustively tested using random and function approximation problems. .A comparison with the Barrodale-Phillips algorithm is also done. The results indicate that the new algorithm shows promising performance on random (non-function approximation) problems.Elhedhli, SamirM.S