  The concept of jump system, introduced by Bouchet and Cunningham (1995), is a
set of integer points satisfying a certain exchange property. We consider the
minimization of a separable convex function on a jump system. It is known that
the problem can be solved by a greedy algorithm. In this paper, we are
interested in whether the greedy algorithm has the geodesic property, which
means that the trajectory of solution generated by the algorithm is a geodesic
from the initial solution to a nearest optimal solution. We show that a special
implementation of the greedy algorithm enjoys the geodesic property, while the
original algorithm does not. As a corollary to this, we present a new greedy
algorithm for linear optimization on a delta-matroid and show that the
algorithm has the geodesic property.
