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

We study the asymptotic behavior of the integral curves of the differential equation u̇(t) = −∇ₓf(u(t), r(t)), $u(t_0) = u_0$ 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 $(t_0, u_0)$ the solution u(t) is defined on the whole interval $[t_0, \infty)$ and, under suitable hypothesis on the rate of decrease of r(t), we establish the convergence of u(t) towards an optimal solution $u_\infty$ of the linear program. In particular we find sufficient conditions for $u_\infty$ to coincide with the limit of the unique minimizer x(r) of f(·, r).