In this article we study convex integer maximization problems with composite
objective functions of the form f(Wx), where f is a convex function on
Rd and W is a d×n matrix with small or binary entries, over
finite sets S⊂Zn of integer points presented by an oracle or by
linear inequalities.
Continuing the line of research advanced by Uri Rothblum and his colleagues
on edge-directions, we introduce here the notion of {\em edge complexity} of
S, and use it to establish polynomial and constant upper bounds on the number
of vertices of the projection \conv(WS) and on the number of linear
optimization counterparts needed to solve the above convex problem.
Two typical consequences are the following. First, for any d, there is a
constant m(d) such that the maximum number of vertices of the projection of
any matroid S⊂{0,1}n by any binary d×n matrix W is m(d)
regardless of n and S; and the convex matroid problem reduces to m(d)
greedily solvable linear counterparts. In particular, m(2)=8. Second, for any
d,l,m, there is a constant t(d;l,m) such that the maximum number of
vertices of the projection of any three-index l×m×n
transportation polytope for any n by any binary d×(l×m×n)
matrix W is t(d;l,m); and the convex three-index transportation problem
reduces to t(d;l,m) linear counterparts solvable in polynomial time