@TechReport{ it:2010-006,
author = {Carl Nettelblad and Sverker Holmgren},
title = {Stochastically Guaranteed Global Optimums Achievable with
a Divide-and-Conquer Approach to Multidimensional {QTL}
Searches},
institution = {Department of Information Technology, Uppsala University},
department = {Division of Scientific Computing},
year = {2010},
number = {2010-006},
month = mar,
abstract = {The problem of searching for multiple quantitative trait
loci (QTL) in an experimental cross population of
considerable size poses a significant challenge, if general
interactions are to be considered. Different global
optimization approaches have been suggested, but without an
analysis of the mathematical properties of the objective
function, it is hard to devise reasonable criteria for when
the optimum found in a search is truly global.
We reformulate the standard residual sum of squares
objective function for QTL analysis by a simple
transformation, and show that the transformed function will
be Lipschitz continuous in an infinite-size population,
with a well-defined Lipschitz constant. We discuss the
different deviations possible in an experimental
finite-size population, suggesting a simple bound for the
minimum value found in the vicinity of any point in the
model space.
Using this bound, we modify the DIRECT optimization
algorithm to exclude regions where the optimum cannot be
found according to the bound. This makes the algorithm more
attractive than previously realized, since optimality is
now in practice guaranteed. The consequences are realized
in permutation testing, used to determine the significance
of QTL results. DIRECT previously failed in attaining the
correct thresholds. In addition, the knowledge of a
candidate QTL for which significance is tested allows
spectacular increases in permutation test performance, as
most searches can be abandoned at an early stage.}
}