Optimal correction of the absolute value equations

Abstract

In this paper, we study the optimum correction of the absolute value equations through making minimal changes in the coefficient matrix and the right hand side vector and using spectral norm. This problem can be formulated as a non-differentiable, non-convex and unconstrained fractional quadratic programming problem. The regularized least squares is applied for stabilizing the solution of the fractional problem. The regularized problem is reduced to a unimodal single variable minimization problem and to solve it a bisection algorithm is proposed. The main difficulty of the algorithm is a complicated constraint optimization problem, for which two novel methods are suggested. We also present optimality conditions and bounds for the norm of the optimal solutions. Numerical experiments are given to demonstrate the effectiveness of suggested methods.