  The problem of finding the minimizer of a sum of convex functions is central
to the field of optimization. In cases where the functions themselves are not
fully known (other than their individual minimizers and convexity parameters),
it is of interest to understand the region containing the potential minimizers
of the sum based only on those known quantities. Characterizing this region in
the case of multivariate strongly convex functions is far more complicated than
the univariate case. In this paper, we provide both outer and inner
approximations for the region containing the minimizer of the sum of two
strongly convex functions, subject to a constraint on the norm of the gradient
at the minimizer of the sum. In particular, we explicitly characterize the
boundary and interior of both outer and inner approximations. Interestingly,
the boundaries as well as the interiors turn out to be identical and we show
that the boundary of the region containing the potential minimizers is also
identical to that of the outer and inner approximations.
