  The interval Distance Geometry Problem (iDGP) consists in finding a
realization in $\mathbb{R}^K$ of a simple undirected graph $G=(V,E)$ with
nonnegative intervals assigned to the edges in such a way that, for each edge,
the Euclidean distance between the realization of the adjacent vertices is
within the edge interval bounds. In this paper, we focus on the application to
the conformation of proteins in space, which is a basic step in determining
protein function: given interval estimations of some of the inter-atomic
distances, find their shape. Among different families of methods for
accomplishing this task, we look at mathematical programming based methods,
which are well suited for dealing with intervals. The basic question we want to
answer is: what is the best such method for the problem? The most meaningful
error measure for evaluating solution quality is the coordinate root mean
square deviation. We first introduce a new error measure which addresses a
particular feature of protein backbones, i.e. many partial reflections also
yield acceptable backbones. We then present a set of new and existing quadratic
and semidefinite programming formulations of this problem, and a set of new and
existing methods for solving these formulations. Finally, we perform a
computational evaluation of all the feasible solver$+$formulation combinations
according to new and existing error measures, finding that the best methodology
is a new heuristic method based on multiplicative weights updates.
