We consider a particular problem which arises when apply-ing the method of gradient projection for solving constrained optimiza-tion and finite dimensional variational inequalities on the convex set formed by the convex hull of the standard basis unit vectors. The method is especially important for relaxation labeling techniques applied to problems in artificial intelligence. Zoutendijk's method for finding feasible directions, which is relatively complicated in general situations, yields a very simple finite algorithm for this problem. We present an extremely simple algorithm for performing the gradient projection and an independent verification of its correctness.

Relaxation labeling processes are a class of iterative algorithms for using contextual information to reduce local ambiguities. This paper introduces a new perspective toward relaxation-that of considering it as a process for reordering labels attached to nodes in a graph. This new perspective is used to establish the formal equivalence between relaxation and another widely used algorithm, local maxima selection. The equivalence specifies conditions under which a family of cooperative relaxation algorithms, which generalize the well-known ones, decompose into purely local ones. Since these conditions are also sufficient for guaranteeing the convergence of relaxation processes, they serve as stopping criteria. We feel that equivalences such as these are necessary for the proper application of relaxation and maxima selection in complex speech and vision understanding systems.