Asymptotic expansion of penalty-gradient flows in linear programming

We establish asymptotic expansions for nonautonomous gradient flows of the form u̇(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→ ∞. Under appropriate conditions we show that every integral curve satisfies u(t) = u^∞ + r(t) d^*₀ + ṙ (t)r(t) w^*₀ + o(ṙ(t)r(t)) for suitable vectors u∞, 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.