A convergence result for nonautonomous subgradient evolution equations and its application to the steepest descent exponential penalty trajectory in linear programming

We present a new result on the asymptotic behavior of nonautonomous subgradient evolution equations of the form u(t) ∈ - ∂o(u(t)), where {o_t: t ≥ 0} is a family of closed proper convex functions. The result is used to study the flow generated by the family o_t(x) = f(x, r(t)), where f(x, r) := c^Tx + r ∑ exp[(A_ix-b_i)/r] is the exponential penalty approximation of the linear program min {c^Tx: Ax ≤ b}, and r(t) is a positive function tending to 0 when t → ∞. We prove that the trajectory u(t) converges to an optimal solution u^∞ of the linear program, and we give conditions for the convergence of an associated dual trajectory μ(t) toward an optimal solution of the dual program.