A novel adaptive quasi-Newton-type update and its global convergence without Lipschitz condition for constrained system of nonlinear monotone equations.

Abstract

This paper presents a new iterative method with a restart feature for solving constrained system of nonlinear monotone equations. The scheme, which is a double-parameter method, was initiated by considering a positive-definite adaptation of the quasi-Newton update proposed by Andrei (J. Comput. Appl. Math. 332, 26-44 (2018)) for unconstrained optimization. The two scaling parameters of the method are obtained by employing the measure function by Byrd and Nocedal (SIAM J. Numer. Anal. 26, 727-739 (1989)), which ensures that the condition number of the update is minimized. Another important attribute of the method is that its global convergence analysis is conducted without the Lipschitz assumption, which is a strong condition. Furthermore, the two parameters embedded in the scheme help in maintaining a balance in the distribution of the eigenvalues of its update matrix. The method converges globally regardless of the line search procedure employed. Numerical experiments with some methods in the literature show that the scheme is effective.