Criss-cross methods: A fresh view on pivot algorithms

Criss-cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution. The first finite criss-cross method was invented by Chang, Terlaky and Wang independently. Unlike the simplex method that follows a monotonic edge path on the feasible region, the trace of a criss-cross method is neither monotonic (with respect to the objective function) nor feasibility preserving. The main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss-cross pivot methods. A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms. The origins and the history of criss-cross methods are also touched upo

Criss-cross methods: a fresh view on pivot algorithms

Criss-cross methods are pivot algorithms that solve linear programming problems in one phase starting with any basic solution. The first finite criss-cross method was invented by Chang, Terlaky and Wang independently. Unlike the simplex method that follows a monotonic edge path on the feasible region, the trace of a criss-cross method is neither monotonic (with respect to the objective function) nor feasibility preserving. The main purpose of this paper is to present mathematical ideas and proof techniques behind finite criss-cross pivot methods. A recent result on the existence of a short admissible pivot path to an optimal basis is given, indicating shortest pivot paths from any basis might be indeed short for criss-cross type algorithms. The origins and the history of criss-cross methods are also touched upon

Optimization and Applications

Proceedings of a workshop devoted to optimization problems, their theory and resolution, and above all applications of them. The topics covered existence and stability of solutions; design, analysis, development and implementation of algorithms; applications in mechanics, telecommunications, medicine, operations research

An output-sensitive algorithm for multi-parametric LCPs with sufficient matrices

This paper considers the multi-parametric linear complementarity problem (pLCP) with sufficient matrices. The main result is an algorithm to find a polyhedral decomposition of the set of feasible parameters and to construct a piecewise affine function that maps each feasible parameter to a solution of the associated LCP in such a way that the function is affine over each cell of the decomposition. The algorithm is output-sensive in the sense that its time complexity is polynomial in the size of the input and linear in the size of the output, when the problem is non-degenerate. We give a lexicographic perturbation technique to resolve degeneracy as well. Unlike for the non-parametric case, the resolution turns out to be nontrivial, and in particular, it involves linear programming (LP) duality and multi-objective LP

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

CMB-S4 Science Book, First Edition

This book lays out the scientific goals to be addressed by the next-generation ground-based cosmic microwave background experiment, CMB-S4, envisioned to consist of dedicated telescopes at the South Pole, the high Chilean Atacama plateau and possibly a northern hemisphere site, all equipped with new superconducting cameras. CMB-S4 will dramatically advance cosmological studies by crossing critical thresholds in the search for the B-mode polarization signature of primordial gravitational waves, in the determination of the number and masses of the neutrinos, in the search for evidence of new light relics, in constraining the nature of dark energy, and in testing general relativity on large scales