Simplexinspired algorithms for solving a class of convex programming problems

We consider a class of convex programming problems whose
objective function is given as a linear function plus a convex
function whose arguments are linear functions of the decision
variables and whose feasible region is a polytope. We show that
there exists an optimal solution to this class of problems on a face
of the constraint polytope of dimension no more than the number of
arguments of the convex function. Based on this result, we develop
a method to solve this problem that is inspired by the simplex
method for linear programming. It is shown that this method
terminates in a finite number of iterations in the special case that
the convex function has only a single argument. We then use this
insight to develop a second algorithm that solves the problem in a
finite number of iterations for an arbitrary number of arguments in
the convex function. A computational study illustrates the
efficiency of the algorithm and suggests that the averagecase
performance of these algorithms is a polynomial of low order.