New Preconditioners Applied to Linear Programming and the Compressive Sensing Problems

Abstract

In this paper, we present new preconditioners based on the incomplete Cholesky factorization and on the splitting preconditioner. In the first approach, we consider the interior point methods that are very efficient for solving linear programming problems. The results of the numerical tests for this problem present satisfactory results in relation to the time and number of iterations. In the second approach, we apply a new preconditioner in compressive sensing (CS) problems, which is an efficient technique to acquire and reconstruct signal. An approach for solving this problem is the primal-dual Newton conjugate gradients. We present a new preconditioner, in the construction of which we exploited the fact that close to a solution we can split the variables into two groups and the matrices satisfy certain properties, as demonstrated in a method known from the literature (Fountoulakis 2015). Once the preconditioner exploiting these features has been computed, we apply an incomplete Cholesky factorization on it, and use the factor found as the true preconditioner. Therefore, the new preconditioner is the result of the combination of two preconditioners. The results obtained are satisfactory in relation to the time and the quality of the reconstructed image.