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)∈−∂ϕt(u(t)), where {ϕt: t⩾0} is a family of closed proper convex functions. The result is used to study the flow generated by the family ϕt(x)=f(x, r(t)), where f(x, r)≔cTx+r ∑ exp[(Aix−bi)/r] is the exponential penalty approximation of the linear program min{cTx: 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.