Tractability of linear problems defined over Hilbert spaces

We study d$d$-variate approximation problems in the worst and average case settings. We consider algorithms that use finitely many evaluations of arbitrary linear functionals. In the worst case setting, we obtain necessary and sufficient conditions for quasi-polynomial tractability and uniform weak tractability. Furthermore, we give an estimate of the exponent of quasi-polynomial tractability which cannot be improved in general. In the average case setting, we obtain necessary and sufficient conditions for uniform weak tractability. As applications we discuss some examples.     

A survey of average case complexity for linear multivariate problems

We survey recent results on the average case complexity for linear multivariate problems. Our emphasis is on problems defined on spaces of functions of d variables with large d. We present the sharp order of the average case complexity for a number of linear multivariate problems as well as necessary and sufficient conditions for the average case complexity not to be exponential in d. Dedicated to the 50th anniversary of the journal. The text was submitted by the authors in English.     

On the tractability of linear tensor product problems in the worst case

It has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem to be weakly tractable in the worst case. The complexity of linear tensor product problems in the worst case depends on the eigenvalues _{λi}i∈N${\left\{{\lambda }_i\right\}}_{i\in \mathbb{N}}$ of a certain operator. It is known that if _λ1=1${\lambda }_1=1$ and _λ2∈(0,1)${\lambda }_2\in (0,1)$ then _λn=o((lnn)⁻²)${\lambda }_n=o\left({(lnn)}^{-2}\right)$, as n→∞$n\to \infty$, is a necessary condition for a problem to be weakly tractable. We show that this is a sufficient condition as well.     

Average case complexity results for a centering algorithm for linear programming problems under Gaussian distributions

To solve linear programming problems by interior point methods an approximately centered interior point has to be known. Such a point can be found by an algorithmic approach – a so-called phase 1 algorithm or centering algorithm. For random linear programming problems distributed according to the rotation symmetry model, especially with normal distribution, we present probabilistic results on the quality of the origin as starting point and the average number of steps of a centering algorithm.