On Absorbing Cycles in Min–Max Digraphs

We consider smooth finite dimensional optimization problems with a compact, connected feasible set M and objective function f. The basic problem, on which we focus, is: how to get from one local minimum to all the other ones. To this aim we introduce a bipartite digraph Γ as follows. Its nodes are formed by the set of local minima and maxima of f| M , respectively. Given a smooth Riemannian (i.e. variable) metric, there is an arc from a local minimum x to a local maximum y if the ascent (semi-)flow induced by the projected gradients of f connects points from a neighborhood of x with points from a neighborhood of y. The existence of an arc from y to x is defined with the aid of the descent (semi-)flow. Strong connectedness of Γ ensures that, starting from one local minimum, we may reach any other one using ascent and descent trajectories in an alternating way. In case that no inequality constraints are present or active, it is well known that for a generic Riemannian metric the resulting min-max digraph Γ is indeed strongly connected. However, if inequality constraints are active, then there might appear obstructions. In fact, we show that Γ may contain absorbing two-cycles. If one enters such a cycle, one cannot leave it anymore via ascent and descent trajectories. Moreover, the cycles being constructed are stable with respect to small perturbations (in the C 1 -topology) of the Riemannian metric. Copyright Springer Science+Business Media New York 2005