Quasi-Newton Method for Absolute Value Equation Based on Upper Uniform Smoothing Approximation Function

Abstract: Generally, absolute value equation (AVE), Ax - |x| = b, is an NP-hard problem. Especially, how to find all solutions of AVE with multi-solutions is actually a more difficult problem. In this paper, an upper uniform smooth approximation function of absolute value function is proposed, and some properties of uniform smooth approximation function are studied. Then, AVE, Ax - |x| = b, where A is a square matrix whose singular values exceed one, is transformed into smooth optimization problem by using the upper uniform smooth approximation function, and solved by quasi-Newton method. Numerical results in solving some given AVE problems demonstrated that our algorithm is valid and superior to that by lower uniform smooth approximation function.
Keywords: quasi-Newton method, absolute value equation, absolute value function, upper uniform smoothing approximation function, singular value
Author: Longquan Yong, Shouheng Tuo
Journal Code: jptkomputergg160158

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