Asymptotic Convergence of the Steepest Descent Method for the Exponential Penalty in Linear Programming

u̇(t) = −∇xf(u(t), r(t)), u(t0) = u0 where f(x, r) is the exponential penalty function associated with the linear program min{c′x : Ax ≤ b}, and r(t) decreases to 0 as t goes to ∞. We show that for each initial condition (t0, u0) the solution u(t) is defined on the whole interval [t0,∞) and, under suitable hypothesis on the rate of decrease of r(t), we establish the convergence of u(t) towards an optimal solution u∞ of the linear program. In particular we find sufficient conditions for u∞ to coincide with the limit of the unique minimizer x(r) of f(·, r).