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

Branch and Cut based on the volume algorithm: Steiner trees in graphs and Max-cut

We present a Branch-and-Cut algorithm where the volume algorithm is applied instead of the traditionally used dual simplex algorithm to the linear programming relaxations in the root node of the search tree. This means that we use fast approximate solutions to these linear programs instead of exact but slower solutions. We present computational results with the Steiner tree and Max-Cut problems. We show evidence that one can solve these problems much faster with the volume algorithm based Branch-and-Cut code than with a dual simplex based one. We discuss when the volume based approach might be more efficient than the simplex based approach

Implementation of warm-start strategies in interior-point methods for linear programming in fixed dimension

We implement several warm-start strategies in interior-point methods for linear programming (LP). We study the situation in which both the original LP instance and the perturbed one have exactly the same dimensions. We consider different types of perturbations of data components of the original instance and different sizes of each type of perturbation. We modify the state-of-the-art interior-point solver PCx in our implementation. We evaluate the effectiveness of each warm-start strategy based on the number of iterations and the computation time in comparison with "cold start" on the NETLIB test suite. Our experiments reveal that each of the warm-start strategies leads to a reduction in the number of interior-point iterations especially for smaller perturbations and for perturbations of fewer data components in comparison with cold start. On the other hand, only one of the warm-start strategies exhibits better performance than cold start in terms of computation time. Based on the insight gained from the computational results, we discuss several potential improvements to enhance the performances of such warm-start strategies. © 2007 Springer Science+Business Media, LLC

Robust Linear Optimization with Recourse: Solution Methods and Other Properties.

The unifying theme of this dissertation is robust optimization; the study of solving certain types of convex robust optimization problems and the study of bounds on the distance to ill-posedness for certain types of robust optimization problems. Robust optimization has recently emerged as a new modeling paradigm designed to address data uncertainty in mathematical programming problems by finding an optimal solution for the worst-case instances of unknown, but bounded, parameters. Parameters in practical problems are not known exactly for many reasons: measurement errors, round-off computational errors, even forecasting errors, which created a need for a robust approach. The advantages of robust optimization are two-fold: guaranteed feasible solutions against the considered data instances and not requiring the exact knowledge of the underlying probability distribution, which are limitations of chance-constraint and stochastic programming. Adjustable robust optimization, an extension of robust optimization, aims to solve mathematical programming problems where the data is uncertain and sets of decisions can be made at different points in time, thus producing solutions that are less conservative in nature than those produced by robust optimization. This dissertation has two main contributions: presenting a cutting-plane method for solving convex adjustable robust optimization problems and providing preliminary results for determining the relationship between the conditioning of a robust linear program under structured transformations and the conditioning of the equivalent second-order cone program under structured perturbations. The proposed algorithm is based on Kelley's method and is discussed in two contexts: a general convex optimization problem and a robust linear optimization problem with recourse under right-hand side uncertainty. The proposed algorithm is then tested on two different robust linear optimization problems with recourse: a newsvendor problem with simple recourse and a production planning problem with general recourse, both under right-hand side uncertainty. Computational results and analyses are provided. Lastly, we provide bounds on the distance to infeasibility for a second-order cone program that is equivalent to a robust counterpart under ellipsoidal uncertainty in terms of quantities involving the data defining the ellipsoid in the robust counterpart.Ph.D.Industrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/64714/1/tlterry_1.pd

Interior Point Cutting Plane Methods in Integer Programming

This thesis presents novel approaches that use interior point concepts in solving mixed integer programs. Particularly, we use the analytic center cutting plane method to improve three of the main components of the branch-and-bound algorithm: cutting planes, heuristics, and branching. First, we present an interior point branch-and-cut algorithm for structured integer programs based on Benders decomposition. We explore using Benders decomposition in a branch-and-cut framework where the Benders cuts are generated using the analytic center cutting plane method. The algorithm is tested on two classes of problems: the capacitated facility location problem and the multicommodity capacitated fixed charge network design problem. For the capacitated facility location problem, the proposed approach was on average 2.5 times faster than Benders-branch-and-cut and 11 times faster than classical Benders decomposition. For the multicommodity capacitated fixed charge network design problem, the proposed approach was 4 times faster than Benders-branch-and-cut while classical Benders decomposition failed to solve the majority of the tested instances. Second, we present a heuristic algorithm for mixed integer programs based on interior points. As integer solutions are typically in the interior, we use the analytic center cutting plane method to search for integer feasible points within the interior of the feasible set. The algorithm searches along two line segments that connect the weighted analytic center and two extreme points of the linear programming relaxation. Candidate points are rounded and tested for feasibility. Cuts aimed to improve the objective function and restore feasibility are then added to displace the weighted analytic center until a feasible integer solution is found. The algorithm is composed of three phases. In the first, points along the two line segments are rounded gradually to find integer feasible solutions. Then in an attempt to improve the quality of the solutions, the cut related to the bound constraint is updated and a new weighted analytic center is found. Upon failing to find a feasible integer solution, a second phase is started where cuts related to the violated feasibility constraints are added. As a last resort, the algorithm solves a minimum distance problem in a third phase. For all the tested instances, the algorithm finds good quality feasible solutions in the first two phases and the third phase is never called. Finally, we present a new approach to generate good general branching constraints based on the shape of the polyhedron. Our approach is based on approximating the polyhedron using an inscribed ellipsoid. We use Dikin's ellipsoid which we calculate using the analytic center. We propose to use the disjunction that has a minimum width on the ellipsoid. We use the fact that the width of the ellipsoid in a given direction has a closed form solution in order to formulate a quadratic problem whose optimal solution is a thin direction of the ellipsoid. While solving a quadratic problem at each node of the branch-and-bound tree is impractical, we use a local search heuristic for its solution. Computational testing conducted on hard integer problems from MIPLIB and CORAL showed that the proposed approach outperforms classical branching

Global Optimization of the Maximum K-Cut Problem

RÉSUMÉ: Le problème de la k-coupe maximale (max-k-cut) est un problème de partitionnement de graphes qui est un des représentatifs de la classe des problèmes combinatoires NP-difficiles. Le max-kcut peut être utilisé dans de nombreuses applications industrielles. L’objectif de ce problème est de partitionner l’ensemble des sommets en k parties de telle façon que le poids total des arrêtes coupées soit maximisé. Les méthodes proposées dans la littérature pour résoudre le max-k-cut emploient, généralement, la programmation semidéfinie positive (SDP) associée. En comparaison avec les relaxations de la programmation linéaire (LP), les relaxations SDP sont plus fortes mais les temps de calcul sont plus élevés. Par conséquent, les méthodes basées sur la SDP ne peuvent pas résoudre de gros problèmes. Cette thèse introduit une méthode efficace de branchement et de résolution du problème max-k-cut en utilisant des relaxations SDP et LP renforcées. Cette thèse présente trois approches pour améliorer les solutions du max-k-cut. La première approche se concentre sur l’identification des classes d’inégalités les plus pertinentes des relaxations de max-k-cut. Cette approche consiste en une étude expérimentale de quatre classes d’inégalités de la littérature : clique, general clique, wheel et bicycle wheel. Afin d’inclure ces inégalités dans les formulations, nous utilisons un algorithme de plan coupant (CPA) pour ajouter seulement les inégalités les plus importantes . Ainsi, nous avons conçu plusieurs procédures de séparation pour trouver les violations. Les résultats suggèrent que les inégalités de wheel sont les plus fortes. De plus, l’inclusion de ces inégalités dans le max-k-cut peut améliorer la borne de la SDP de plus de 2%. La deuxième approche introduit les contraintes basées sur formulation SDP pour renforcer la relaxation LP. De plus, le CPA est amélioré en exploitant la technique de terminaison précoce d’une méthode de points intérieurs. Les résultats montrent que la relaxation LP avec les inégalités basées sur la SDP surpasse la relaxation SDP pour de nombreux cas, en particulier pour les instances avec un grand nombre de partitions (k � 7). La troisième approche étudie la méthode d’énumération implicite en se basant sur les résultats des dernières approches. On étudie quatre composantes de la méthode. Tout d’abord, nous présentons quatre méthodes heuristiques pour trouver des solutions réalisables : l’heuristique itérative d’agrégation, l’heuristique d’opérateur multiple, la recherche à voisinages variables, et la procédure de recherche aléatoire adaptative gloutonne. La deuxième procédure analyse les stratégies dichotomiques et polytomiques pour diviser un sous-problème. La troisième composante étudie cinq règles de branchement. Enfin, pour la sélection des noeuds de l’arbre de branchement, nous considérons les stratégies suivantes : meilleur d’abord, profondeur d’abord, et largeur d’abord. Pour chaque stratégie, nous fournissons des tests pour différentes valeurs de k. Les résultats montrent que la méthode exacte proposée est capable de trouver de nombreuses solutions. Chacune de ces trois approches a contribué à la conception d’une méthode efficace pour résoudre le problème du max-k-cut. De plus, les approches proposées peuvent être étendues pour résoudre des problèmes génériques d’optimisation en variables mixtes.----------ABSTRACT: In graph theory, the maximum k-cut (max-k-cut) problem is a representative problem of the class of NP-hard combinatorial optimization problems. It arises in many industrial applications and the objective of this problem is to partition vertices of a given graph into at most k partitions such that the total weight of the cut is maximized. The methods proposed in the literature to optimally solve the max-k-cut employ, usually, the associated semidefinite programming (SDP) relaxation in a branch-and-bound framework. In comparison with the linear programming (LP) relaxation, the SDP relaxation is stronger but it suffers from high CPU times. Therefore, methods based on SDP cannot solve large problems. This thesis introduces an efficient branch-and-bound method to solve the max-k-cut problem by using tightened SDP and LP relaxations. This thesis presents three approaches to improve the solutions of the problem. The first approach focuses on identifying relevant classes of inequalities to tighten the relaxations of the max-k-cut. This approach carries out an experimental study of four classes of inequalities from the literature: clique, general clique, wheel and bicycle wheel. In order to include these inequalities, we employ a cutting plane algorithm (CPA) to add only the most important inequalities in practice and we design several separation routines to find violations in a relaxed solution. Computational results suggest that the wheel inequalities are the strongest by far. Moreover, the inclusion of these inequalities in the max-k-cut improves the bound of the SDP formulation by more than 2%. The second approach introduces the SDP-based constraints to strengthen the LP relaxation. Moreover, the CPA is improved by exploiting the early-termination technique of an interior-point method. Computational results show that the LP relaxation with the SDP-based inequalities outperforms the SDP relaxations for many instances, especially for a large number of partitions (k � 7). The third approach investigates the branch-and-bound method using both previous approaches. Four components of the branch-and-bound are considered. First, four heuristic methods are presented to find a feasible solution: the iterative clustering heuristic, the multiple operator heuristic, the variable neighborhood search, and the greedy randomized adaptive search procedure. The second procedure analyzes the dichotomic and polytomic strategies to split a subproblem. The third feature studies five branching rules. Finally, for the node selection, we consider the following strategies: best-first search, depth-first search, and breadth-first search. For each component, we provide computational tests for different values of k. Computational results show that the proposed exact method is able to uncover many solutions. Each one of these three approaches contributed to the design of an efficient method to solve the max-k-cut problem. Moreover, the proposed approaches can be extended to solve generic mixinteger SDP problems

Explanatory model of antecedents and outcomes of health and safety climate in the South African construction industry

Includes bibliographical references.Workplace injuries and fatalities are a major cause of concern for government and organisations in South Africa. The cost incurred by government as compensation for injuries that occur in the workplace has increased steadily over the past 10 years. This has raised the need for alternative approaches to dealing with causes of workplace injuries and fatalities. The loss of employees due to workplace fatalities and the cost of medical care have both direct and indirect cost implications for organisations. The cost of hiring replacement labour while the injured employee is on leave, the cost of training a new employee, paying for medical care for the original employee, and reduced productivity due to lack of experience of the incoming replacement are financially draining and detrimental to the functions of organisations. The primary objective of the study being reported here was to develop an explanatory model of the health and safety (H&S) climate in the local construction industry. A secondary objective was to provide a theoretical and practical framework for the study of the health and safety climate in the South African construction industry. A literature review, observations and structured interviews informed the development of a survey questionnaire. The survey was completed by construction workers who were members of the Master Builders Association South Africa (MBASA) in the Western Cape. On-site observations and structured interviews by the researcher informed the development of a pen-and-paper survey, which was completed by construction workers at selected building sites from organisations who were members of MBASA. A pilot study was conducted for refinement of the survey measurement tool. Hypotheses were tested using regression analysis techniques. Partial least squares path analysis was used to test the structure of the proposed model. In total, 1 200 surveys were administered, and a total of 851 participants completed the survey. This study provided empirical evidence of the link between antecedents of the health and safety climate and health and safety performance. Overall, the proposed health and safety model showed significant predictive ability for health and safety incident reporting (R² = .464, p = <.001), health and safety motivation (R² = .450, p = <.001) and health and safety performance (R² = .508, p = <.001). Path analysis found a predictive ability of health and safety performance to injuries (R² = .028, p = <.001). The findings provided evidence-based support for the variables of top management's commitment to health and safety and health and safety communications and the predictive ability of these on positive health and safety behaviour. Predicting injuries in the construction industry can help to reduce the high costs of compensation and make employees in the sector safer. Insights gained from this study will contribute to the field of occupational health psychology in particular at both academic and practical level