Developments in linear and integer programming

In this review we describe recent developments in linear and integer (linear) programming. For over 50 years Operational Research practitioners have made use of linear optimisation models to aid decision making and over this period the size of problems that can be solved has increased dramatically, the time required to solve problems has decreased substantially and the flexibility of modelling and solving systems has increased steadily. Large models are no longer confined to large computers, and the flexibility of optimisation systems embedded in other decision support tools has made on-line decision making using linear programming a reality (and using integer programming a possibility). The review focuses on recent developments in algorithms, software and applications and investigates some connections between linear optimisation and other technologies

An Efficient Method for Pivoting Free Variables in Linear Programming: A Computational Approach

Commonly used simplex method to solve linear programming problem do not allow variables to be negative during solution process and suggest to break each free variable (variable allowed to be negative) into difference of two non-negative variables. This transformation significantly increases the number of variables as well as after this the problem leaves its original variable space. , thus making the geometry of problem (during solution process) difficult to handle and understand. In this paper, we developed a natural generalization of simplex pivots for free variables. Described approach is capable of handling any general linear programming in its original variable space. In our computational study, the primary results showed that the new method outperforms simplex method on general LPs

Treatment of Degeneracy in Linear and Quadratic Programming

RÉSUMÉ : Dans cette thèse, nous considérons la résolution de problèmes d’optimisation quadratique dégénérés sur base de techniques initialement développées pour l’optimisation linéaire, ca-pables de tirer avantage de la dégénérescence. Nous commençons par améliorer l’efficacité de la règle de pivotage positive edge en fournissant une implémentation basée sur le logiciel libre CLP. Nous proposons ensuite le logiciel de haut niveau CyLP permettant de définir et d’expérimenter facilement avec de nouvelles règles de pivotage pour la méthode du Simplexe. CyLP offre de plus des services de modélisation puissants réduisant l’effort nécessaire à la modélisation de problèmes linéaires, en variables entières et quadratiques. À l’aide de CyLP, nous appliquons la règle positive edge à la variante du Simplexe suggérée par Wolfe pour résoudre les problèmes quadratiques. Nous incorporons également positive edge dans une méthode de gradient réduit. Nos tests démontrent l’efficacité de positive edge sur les problèmes quadratiques pour lesquels le terme linéaire est dominant. Chaque méthode est capable de fournir des niveaux substantiels d’accélération sur un certain sous-ensemble de problèmes lorsqu’elle est équipée de positive edge. Nous suggérons des pistes de recherche pour la conception de nouvelles méthodes qui incorporent positive edge et accélèrent la résolution sur une plus large gamme de problèmes.----------ABSTRACT : We consider solving degenerate quadratic programs (QPs) by means of degeneracy-benefiting techniques designed for linear programs (LPs). Specifically, we use a Simplex pivot method, called positive edge, that is able to take advantage of degeneracy in LPs. First, we improve the efficiency of the positive edge method by providing an internal implementation of it using CLP—an open-source LP solver. In the next stage, we develop a software, called CyLP, which allows easy definition of, and experimentation with, Simplex pivot rules. In addition, CyLP has a powerful modeling facility that reduces the effort of modeling LPs, mixed-integer programs (MIPs), and QPs. Using CyLP, we apply the positive edge rule to Wolfe’s method—a Simplex-like method for QPs. We also incorporate positive edge into a reduced-gradient method. Our experiments demonstrate the effectiveness of positive edge on QPs with relatively large linear terms. Each method is able to yield substantial improvements on a subset of test problems. We provide research leads to devise novel methods that incorporate the positive edge rule and are more generally applicable

The interplay of network structure and dispatch solutions in power grid cascading failures

For a given minimum cost of the electricity dispatch, multiple equivalent dispatch solutions may exist. We explore the sensitivity of networks to these dispatch solutions and their impact on the vulnerability of the network to cascading failure blackouts. It is shown that, depending on the heterogeneity of the network structure, the blackout statistics can be sensitive to the dispatch solution chosen, with the clustering coefficient of the network being a key ingredient. We also investigate mechanisms or configurations that decrease discrepancies that can occur between the different dispatch solutions

SIMPLEX v. 2.17: an Implementation of the Simplex Algorithm for Large Scale Linear Problems. User's Guide

This document presents SIMPLEX -- a linear optimizer for large scale linear programs. Issues like command line syntax, data input and output formats and interpretation of results are addressed. In order to make the interpretation of results and program's reports easier, some of the features of tee implementation are also discussed. Finally, numerical test results give a measure of the code's true efficiency

Advances in design and implementation of optimization software

Developing optimization software that is capable of solving large and complex real-life problems is a huge effort. It is based on a deep knowledge of four areas: theory of optimization algorithms, relevant results of computer science, principles of software engineering, and computer technology. The paper highlights the diverse requirements of optimization software and introduces the ingredients needed to fulfill them. After a review of the hardware/software environment it gives a survey of computationally successful techniques for continuous optimization. It also outlines the perspective offered by parallel computing, and stresses the importance of optimization modeling systems. The inclusion of many references is intended to both give due credit to results in the field of optimization software and help readers obtain more detailed information on issues of interest