A new smoothing-type algorithm for nonlinear weighted complementarity problem
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A new smoothing-type algorithm for nonlinear weighted complementarity problem Ziyu Liu1 · Jingyong Tang1 Received: 23 January 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract In this paper we study the nonlinear weighted complementarity problem (denoted by NWCP). We first introduce a smoothing function which can be used to reformulate the NWCP as a system of smooth nonlinear equations. Then we propose a new smoothingtype algorithm to solve the NWCP which adopts a nonmonotone line search scheme. In each iteration, our algorithm solves one linear system of equations and performs one line search. Under suitable assumptions, we prove that the proposed algorithm is globally and locally quadratically convergent. Some numerical results are reported. Keywords Nonlinear weighted complementarity problem · Smoothing Newton algorithm · Nonmonotone line search · Quadratical convergence Mathematics Subject Classification 90C25 · 90C30
1 Introduction The weighted complementarity problem (WCP) was introduced by Potra [8]. Let (V, ◦) be an Euclidean Jordan algebra and K = {x ◦ x : x ∈ V} be the symmetric cone. Given a vector w ∈ K, the WCP is the problem of finding (x, s, y) ∈ V×V×Rm such that x ∈ K, s ∈ K, F(x, s, y) = 0, x ◦ s = w, where F : V × V × Rm → V × Rm is a continuously differentiable nonlinear map. The WCP significantly extends the scope of general complementarity problems and can model a wider class of problems arising from real applications [8]. Although Potra presented the notion of the WCP in Euclidean Jordan algebra setting, they only studied
B 1
Jingyong Tang [email protected] School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
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Z. Liu, J. Tang
the case of a linear WCP over the nonnegative orthant (denoted by LWCP) which finds (x, s, y) ∈ Rn × Rn × Rm such that x ≥ 0, s ≥ 0, P x + Qs + Ry = a, xs = w.
(1.1)
Here P ∈ R(n+m)×n , Q ∈ R(n+m)×n , R ∈ R(n+m)×m are given matrices, a ∈ Rn+m is a given vector and w ≥ 0 is a given weight vector (the data of the problem), xs denotes the vector with components xi si . In [8], Potra proposed two interior-point methods to solve the LWCP and established their computational complexity. In [9], Potra gave a characterization of sufficient LWCP and proposed a corrector-predictor interior-point method for its numerical solution. Recently, some authors studied smoothing Newton algorithms for solving the LWCP [12,13]. Lately, Chi, Gowda and Tao [3] studied the weighted horizontal linear complementarity problem in the setting of Euclidean Jordan algebras and established some existence and uniqueness results. Gowda [4] considered weighted LCPs and interior point systems for copositive linear transformations on Euclidean Jordan algebras. In this paper, we study the nonlinear weighted complementarity problem (NWCP), which is to find (x, s, y) ∈ Rn × Rn × Rm such that x ≥ 0, s ≥ 0, F(x, s, y) = a, xs = ω,
(1.2)
where F(x, s, y) : Rn+n+m −→ Rn+m is a continuously differentiable function. Obviously,
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