  This paper is a follow up to the previous author's paper on convex
optimization. In that paper we began the process of adjusting greedy-type
algorithms from nonlinear approximation for finding sparse solutions of convex
optimization problems. We modified there three the most popular in nonlinear
approximation in Banach spaces greedy algorithms -- Weak Chebyshev Greedy
Algorithm, Weak Greedy Algorithm with Free Relaxation and Weak Relaxed Greedy
Algorithm -- for solving convex optimization problems. We continue to study
sparse approximate solutions to convex optimization problems. It is known that
in many engineering applications researchers are interested in an approximate
solution of an optimization problem as a linear combination of elements from a
given system of elements. There is an increasing interest in building such
sparse approximate solutions using different greedy-type algorithms. In this
paper we concentrate on greedy algorithms that provide expansions, which means
that the approximant at the $m$th iteration is equal to the sum of the
approximant from the previous iteration ($(m-1)$th iteration) and one element
from the dictionary with an appropriate coefficient. The problem of greedy
expansions of elements of a Banach space is well studied in nonlinear
approximation theory. At a first glance the setting of a problem of expansion
of a given element and the setting of the problem of expansion in an
optimization problem are very different. However, it turns out that the same
technique can be used for solving both problems. We show how the technique
developed in nonlinear approximation theory, in particular, the greedy
expansions technique can be adjusted for finding a sparse solution of an
optimization problem given by an expansion with respect to a given dictionary.
