Modified HS conjugate gradient method for solving generalized absolute value equations

Abstract

We investigate a kind of generalized equations involving absolute values of variables as |A|x−|B||x|=b$|A|x-|B||x|=b$, where A∈Rn×n$A \\in R^{n\\times n}$ is a symmetric matrix, B∈Rn×n$B \\in R^{n\\times n}$ is a diagonal matrix, and b∈Rn$b\\in R^{n}$. A sufficient condition for unique solvability of the proposed generalized absolute value equations is also given. By utilizing an equivalence relation to the unconstrained optimization problem, we propose a modified HS conjugate gradient method to solve the transformed unconstrained optimization problem. Only under mild conditions, the global convergence of the given method is also established. Finally, the numerical results show the efficiency of the proposed method.