It is known that the gradient descent algorithm converges linearly when applied to a strongly convex function with Lipschitz gradient. In this case the algorithm's rate of convergence is determined by the condition number of the function. In a similar vein, it has been shown that a variant of the Frank-Wolfe algorithm with away steps converges linearly when applied to a strongly convex function with Lipschitz gradient over a polytope. In a nice extension of the unconstrained case, the algorithm's rate of convergence is determined by the product of the condition number of the function and a certain condition number of the polytope.We shed new light into the latter type of polytope conditioning. In particular, we show that previous and seemingly different approaches to define a suitable condition measure for the polytope are essentially equivalent to each other. Perhaps more interesting, they can all be unified via a parameter of the polytope that formalizes a key premise linked to the algorithm's linear convergence. We also give new insight into the linear convergence property. For a convex quadratic objective, we show that the rate of convergence is determined by a condition number of a suitably scaled polytope. *

No abstract

The von Neumann algorithm is a simple coordinate-descent algorithm to determine whether the origin belongs to a polytope generated by a finite set of points. When the origin is in the interior of the polytope, the algorithm generates a sequence of points in the polytope that converges linearly to zero. The algorithm's rate of convergence depends on the radius of the largest ball around the origin contained in the polytope.We show that under the weaker condition that the origin is in the polytope, possibly on its boundary, a variant of the von Neumann algorithm that includes away steps generates a sequence of points in the polytope that converges linearly to zero. The new algorithm's rate of convergence depends on a certain geometric parameter of the polytope that extends the above radius but is always positive. Our linear convergence result and geometric insights also extend to a variant of the Frank-Wolfe algorithm with away steps for minimizing a convex quadratic function over a polytope.

No abstract

Na área de Complexidade Algébrica tem-se estudado com muita profundidade como calcular o produto de dois elementos de uma álgebra qualquer, os assim chamados problemas bilineares. Neste trabalho, estendemos o estudo ao produto de n elementos de uma álgebra, o que chamamos de problemas multilineares. Apresentamos uma nova forma de definir e de caracterizar os algoritmos algébricos, mostramos cotas inferiores do problema de calcular um conjunto de funções em .Klai, . . . , a.l, deânimos algoritmos multilineares, damos um gerador de algoritmos multilineares quase-ótimos e estudamos com detalhe o produto de n polinõmios numa variável. Esses métodos se mostraram adequados para o produto de polinõmios em várias variáveis densos-Os polinâmios em várias variáveis esparsos são muito importantes na prática, sendo nosso modelo ineficiente para trata-los. Analisamos um algoritmo para calcular polinâmios esparsos por interpolação, comparando sua complexidade pelo modelo RAM e pelo modelo de computação multiplicativo.