Quasi-polynomial tractability

Tractability of multivariate problems has become a popular research subject. Polynomial tractability means that the solution of a dd-variate problem can be solved to within εε with polynomial cost in ε−1ε−1 and dd. Unfortunately, many multivariate problems are not polynomially tractable. This holds for all non-trivial unweighted linear tensor product problems. By an unweighted problem we mean the case when all variables and groups of variables play the same role.It seems natural to ask what is the “smallest” non-exponential function T:[1,∞)×[1,∞)→[1,∞)T:[1,∞)×[1,∞)→[1,∞) for which we have TT-tractability of unweighted linear tensor product problems; that is, when the cost of a multivariate problem can be bounded by a multiple of a power of T(ε−1,d)T(ε−1,d). Under natural assumptions, it turns out that this function is Tqpol(x,y)=exp((1+lnx)(1+lny))for all x,y∈[1,∞). The function Tqpol goes to infinity faster than any polynomial although not “much” faster, and that is why we refer to Tqpol-tractability as quasi-polynomial tractability.The main purpose of this paper is to promote quasi-polynomial tractability, especially for the study of unweighted multivariate problems. We do this for the worst case and randomized settings and for algorithms using arbitrary linear functionals or only function values. We prove relations between quasi-polynomial tractability in these two settings and for the two classes of algorithms.