A Low-Cost Optimization Approach for Solving Minimum Norm Linear Systems and Linear Least-Squares Problems (Journal of Computational Mathematics, 2024)
Recently, the authors proposed a low-cost approach, named Optimization Approach forLinear Systems (OPALS) for solving any kind of a consistent linear system regarding thestructure, characteristics, and dimension of the coefficient matrix $A.$ The results obtainedby this approach for matrices with no structure and with indefinite symmetric part wereencouraging when compare with other recent and well-known techniques. In this work,we proposed to extend the OPALS approach for solving the Linear Least-Squares Problem(LLSP) and the Minimum Norm Linear System Problem (MNLSP) using any iterative low-cost gradient-type method, avoiding the construction of the matrices $A^TA$ or $AA^T,$ andtaking full advantage of the structure and form of the gradient of the proposed nonlinearobjective function in the gradient direction. The combination of those conditions togetherwith the choice of the initial iterate allow us to produce a novel and efficient low-costnumerical scheme for solving both problems. Moreover, the scheme presented in thiswork can also be used and extended for the weighted minimum norm linear systems andminimum norm linear least-squares problems. We include encouraging numerical resultsto illustrate the practical behavior of the proposed schemes.