Abstract
We investigate a kind of generalized equations involving absolute values of variables as |A|x-|B||x|=b, where A in R^{ntimes n} is a symmetric matrix, B in R^{ntimes n} is a diagonal matrix, and bin 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.
Highlights
The absolute value equation of the typeAx + B|x| = b (1.1)was investigated in [14, 22, 25,26,27,28]
Rohn et al [30] gave the sufficient conditions for unique solvability of AVE (1.2) and an iterative method to solve it
4 Numerical experiments we present numerical results to show the efficiency of the modified HS conjugate gradient method (Algorithm 3.1)
Summary
We propose a new generalized absolute value equation (GAVE) problem of the form Mangasarian et al [17] gave the existence and nonexistence results of (1.2) and proved the equivalence relations between (1.2) and the generalized linear complementarity problem. We propose a modified HS conjugate gradient method to compute the solution of GAVE (1.3).
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