Reduction of Linear Programming to Linear Approximation

It is well known that every l∞ linear approximation problem can be reduced to a linear program. In this paper we show that conversely every linear program can be reduced to an l∞ linear approximation problem. Now we recall relevent definitions. An affine function of variables x1, . . . , xn is b0 + c1x1 + · · · + cnxn where b0, ci are given numbers. A linear constraint is any of the following constraints: f ≤ g, f ≥ g, f = g, where f, g are affine functions. A linear program is an optimization (maximization or minimization) of an affine function subject to a finite system of linear constraints. An l∞ linear approximation problem, also known as (discrete) Chebyshev approximation problem or finding the least-absolute-deviation fit, is the problem of minimization of the following function: