Tubularity and Asymptotic Convergence of Penalty Trajectories in Convex Programming

In this paper, we give a sufficient condition for the asymptotic convergence of penalty trajectories in convex programming with multiple solutions. We show that, for a wide class of penalty methods, the associated optimal trajectory converges to a particular solution of the original problem, characterized through a minimization selection principle. Our main assumption for this convergence result is that all the functions involved in the convex program are tubular. This new notion of regularity, weaker than that of quasianalyticity, is defined and studied in detail.