The feasibility problem for a system of linear inequalities can be con
verted into an unconstrained optimization problem by using ideas from
the ellipsoid method, which can be viewed as a very simple minimizatio
n technique for the resulting nonlinear function. This function is rel
ated to the volume of an ellipsoid containing all feasible solutions,
which is parametrized by certain weights which we choose to minimize t
he function. The center of the resulting ellipsoid turns out to be a f
easible solution to the inequalities. Using more sophisticated nonline
ar minimization algorithms, we develop and investigate more efficient
methods, which lead to two kinds of weighted centers for the feasible
set. Using these centers, we develop new algorithms for solving linear
programming problems.