A hybrid method for solving systems of nonsmooth equations with box constraints
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A hybrid method for solving systems of nonsmooth equations with box constraints Yigui Ou1 · Haichan Lin1 Received: 31 May 2019 / Accepted: 14 February 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper proposes a hybrid method for solving systems of nonsmooth equations with box constraints, which combines the idea of Levenberg–Marquard-like method with the nonmonotone strategy and the smoothing approximation technique. Under mild assumptions, the proposed method is proven to possess global and local superlinear convergence. Preliminary numerical results are reported to show the efficiency of this proposed method in practical computation. Keywords Nonsmooth equations with box constraints · Smooth approximation · Nonmonotone technique · Levenberg–Marquard method · Convergence analysis
1 Introduction In this paper, the system of nonsmooth equations with box constraints is considered as follows F(x) = 0, x ∈ X = {x ∈ R n |l ≤ x ≤ u},
(1)
where F : X → R m is locally Lipschitz continuous but not necessarily differentiable, T and u = (u , u , . . . , u )T with l ∈ R , l , . . . , l ) {+∞} and u i ∈ l = (l n 1 2 n i 1 2 R {+∞} satisfying li < u i for all i = 1, 2, . . . , n. The constrained nonlinear system of equations are closely related to various optimization problems, such as nonlinear programming, the semi-infinite programming, the maximal monotone operator problems, the nonlinear complementarity problems and the variational inequality problems, etc., see Refs. [1–3] and references therein.
This work is partially supported by NNSF of China (Nos. 11961018, 11261015) and NSF of Hainan Province (No. 2016CXTD004).
B 1
Yigui Ou [email protected] Department of Mathematics, Hainan University, Haikou 570228, China
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Y. Ou, H. Liu
As a result, such this kind of problem has attracted much more attention in both nonlinear optimization and numerical analysis, and many numerical methods have been proposed. Up to now, several main methods, namely Newton-type methods (see Ref. [4] for instance), trust-region-type methods (see Refs. [5,6] for instance), Levenberg– Marquard-type methods (see Ref. [7] for instance) and smoothing methods (see Refs. [8–10] for instance), have been developed to solve systems of nonsmooth equations. However, as pointed out in [6,9], there are some drawbacks in the first and second type of methods. For the third kind of methods, a trial step is obtained at each iteration by solving linear system of equations, which is in general a nontrivial task. In addition, difficulties arise in the first three types of methods in that computing the subgradient of F(x) is not easy for a general nonsmooth function. Smoothing methods for nonlinear optimization problems have been studied for decades. Extensive numerical experiments show that smoothing methods are efficient both for smooth optimization problems (see Ref. [11] for instance) and for nonsmooth problems (see Refs. [7–10] for instance). The main feature of smoothing methods is that they convert the original nons
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