Abstract

We present a new result on the asymptotic behavior of nonautonomous subgradient evolution equations of the form u(t) ? - ?o(u(t)), where {ot: t ? 0} is a family of closed proper convex functions. The result is used to study the flow generated by the family ot(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 ?(t) toward an optimal solution of the dual program. © 2001 Elsevier Science.