On the convergence of the affine-scaling algorithm

Cover title.Includes bibliographical references (p. 20-22).Research partially supported by the National Science Foundation. NSF-ECS-8519058 Research partially supported by the U.S. Army Research Office. DAAL03-86-K-0171 Research partially supported by the Science and Engineering Research Board of McMaster University.by Paul Tseng and Zhi-Quan Luo

Fast L1L_1-CkC^k polynomial spline interpolation algorithm with shape-preserving properties

International audienceIn this article, we address the interpolation problem of data points per regular $L_1$-spline polynomial curve that is invariant under a rotation of the data. We iteratively apply a minimization method on ¯ve data, belonging to a sliding window, in order to obtain this interpolating curve. We even show in the $C^k$-continuous interpolation case that this local minimization method preserves well the linear parts of the data, while a global $L_p$ (p >=1) minimization method does not in general satisfy this property. In addition, the complexity of the calculations of the unknown derivatives is a linear function of the length of the data whatever the order of smoothness of the curve

Barrier functions and interior-point algorithms for linear programming with zero-, one-, or two-sided bounds on the variables

Includes bibliographical references (p. 37-39).Supported by NSF, AFOSR, and ONR through NSF grant. DMS-8920550 Supported by the Center for Applied Mathematics.Robert M. Freund and Michael J. Todd

A software system for large-scale structural optimization

This work is driven by recent developments in mathematical programming, the state-of-the-art of structural optimization, the spectacular performance of linear programming algorithms, and computer hardware developments which imply that applications of structural optimization might be used commonly in engineering design. Currently, there are few general purpose optimization routines available to the structural engineer and much of the work has addressed specific classes of problems. Further, there is little widespread use of the available routines, partly due to the large amount of familiarity one must have with the specific details of both the problem and the optimization method. In response, it is the intention here to prototype a software system that implements a general approach for structural optimization using the latest in mathematical programming techniques. This work develops a general system that can be used for a variety of structural optimization problems in a manner analogous to the finite element method for structural analysis. The most commonly used structural elements, truss and beam, are included as well as techniques for plate optimization. Consideration is given to the software requirements of a general purpose structural optimization system and the demands of large structural systems typically encountered in design practice. This general approach is aimed at using classical methods taken directly from the area of mathematical programming, specifically linear programming, which has seen considerable change in the last ten years. Here, sequential linear programming (SLP) techniques are shown to handle a wide variety of structural constraints including stress constraints, displacement constraints, buckling, and frequency constraints. It is the purpose of this thesis to bring the latest developments in linear programming to the field of structural optimization in the form of a general purpose, state-of-the-art structural optimization system. The model was tested for sample structures and it was shown to effect a reduction in total structure volume of up to 80%

Approximation de fonctions et de données discrètes au sens de la norme L1 par splines polynomiales

Data and function approximation is fundamental in application domains like path planning or signal processing (sensor data). In such domains, it is important to obtain curves that preserve the shape of the data. Considering the results obtained for the problem of data interpolation, L1 splines appear to be a good solution. Contrary to classical L2 splines, these splines enable to preserve linearities in the data and to not introduce extraneous oscillations when applied on data sets with abrupt changes. We propose in this dissertation a study of the problem of best L1 approximation. This study includes developments on best L1 approximation of functions with a jump discontinuity in general spaces called Chebyshev and weak-Chebyshev spaces. Polynomial splines fit in this framework. Approximation algorithms by smoothing splines and spline fits based on a sliding window process are introduced. The methods previously proposed in the littérature can be relatively time consuming when applied on large datasets. Sliding window algorithm enables to obtain algorithms with linear complexity. Moreover, these algorithms can be parallelized. Finally, a new approximation approach with prescribed error is introduced. A pure algebraic algorithm with linear complexity is introduced. This algorithm is then applicable to real-time application.L'approximation de fonctions et de données discrètes est fondamentale dans des domaines tels que la planification de trajectoire ou le traitement du signal (données issues de capteurs). Dans ces domaines, il est important d'obtenir des courbes conservant la forme initiale des données. L'utilisation des splines L1 semble être une bonne solution au regard des résultats obtenus pour le problème d'interpolation de données discrètes par de telles splines. Ces splines permettent notamment de conserver les alignements dans les données et de ne pas introduire d'oscillations résiduelles comme c'est le cas pour les splines d'interpolation L2. Nous proposons dans cette thèse une étude du problème de meilleure approximation au sens de la norme L1. Cette étude comprend des développements théoriques sur la meilleure approximation L1 de fonctions présentant une discontinuité de type saut dans des espaces fonctionnels généraux appelés espace de Chebyshev et faiblement Chebyshev. Les splines polynomiales entrent dans ce cadre. Des algorithmes d'approximation de données discrètes au sens de la norme L1 par procédé de fenêtre glissante sont développés en se basant sur les travaux existants sur les splines de lissage et d'ajustement. Les méthodes présentées dans la littérature pour ces types de splines peuvent être relativement couteuse en temps de calcul. Les algorithmes par fenêtre glissante permettent d'obtenir une complexité linéaire en le nombre de données. De plus, une parallélisation est possible. Enfin, une approche originale d'approximation, appelée interpolation à delta près, est développée. Nous proposons un algorithme algébrique avec une complexité linéaire et qui peut être utilisé pour des applications temps réel