Abstract

We establish asymptotic expansions for nonautonomous gradient flows of the form u?(t) = - ?f(u(t), r(t)), where f(x, r) is a penalty approximation of a linear program and the penalty parameter r(t) tends to 0 as t? 8. Under appropriate conditions we show that every integral curve satisfies u(t) = u8 + r(t) d* 0 + ? (t)r(t) w* 0 + o(?(t)r(t)) for suitable vectors u8, d* 0, and w* 0. We deduce an asymptotic expansion for a related dual trajectory, and we show that the primal-dual limit point is a pair of strictly complementary optimal solutions for the linear program. © 2009 Society for Industrial and Applied Mathematics.