Greedy approximation in convex optimization
This work addresses the need for efficient sparse solutions in engineering applications, representing an incremental advancement by applying existing techniques to a new context.
The paper tackles the problem of finding sparse approximate solutions to convex optimization problems by adapting greedy approximation techniques from nonlinear approximation theory, showing that the same methods can be used for both approximation and optimization tasks.
We 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. The problem of approximation of a given element of a Banach space by linear combinations of elements from a given system (dictionary) is well studied in nonlinear approximation theory. At a first glance the settings of approximation and optimization problems are very different. In the approximation problem an element is given and our task is to find a sparse approximation of it. In optimization theory an energy function is given and we should find an approximate sparse solution to the minimization problem. 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 approximation technique can be adjusted for finding a sparse solution of an optimization problem.