A smoothing quasi-Newton method for solving general second-order cone complementarity problems
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A smoothing quasi-Newton method for solving general second-order cone complementarity problems Jingyong Tang1 · Jinchuan Zhou2 Received: 24 June 2019 / Accepted: 14 November 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Recently, there are much interests in studying smoothing Newton method for solving montone second-order cone complementarity problem (SOCCP) or SOCCPs with Cartesian P/P0 property. In this paper, we propose a smoothing quasi-Newon method for solving general SOCCP. We show that the proposed method is well-defined without any additional assumption and has global convergence under standard conditions. Moreover, under the Jacobian nonsingularity assumption, the method is shown to have local superlinear or quadratic convergence rate. Our preliminary numerical experiments show the method could be very effective for solving SOCCPs. Keywords Second-order cone complementarity problem · Smoothing function · Quasi-Newton method · Superlinear/Quadratical convergence
1 Introduction In recent years, optimization problems with second-order cone constraints have received considerable attention from researchers for their broad applications in a wide range of fields (e.g., [1,6,16,25]). The second-order cone in Rn is also called Lorentz cone or ice-cream cone, defined as T T Kn := {(x1 , x2:n ) ∈ R × Rn−1 : x1 ≥ x2:n },
where · denotes the Euclidean norm, i.e., 2-norm. In this paper we consider the secondorder cone complementarity problem (SOCCP), which finds x ∈ Rn such that (SOCCP) x ∈ K, F(x) ∈ K, x, F(x) = 0,
B
(1.1)
Jingyong Tang [email protected] Jinchuan Zhou [email protected]
1
School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China
2
Department of Statistics, School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China
123
Journal of Global Optimization
where ·, · denotes the inner product associated with the Euclidean norm · , F : Rn → Rn n is the Cartesian product of secondis a continuously differentiable function and K ⊂ R n n r 1 order cones, that is, K = K × · · · × K with n = ri=1 n i and n i ≥ 1 for i = 1, . . . , r . When n i = 1 for all i = 1, . . . , r , SOCCP in (1.1) will reduce to the well-known Nonlinear Complementarity Problem (NCP). Moreover, the KKT conditions for any Second-Order Cone Programming (SOCP) with continuously differentiable functions can be also written in the form of the SOCCP (see, [17, Section 6]). For the Cartesian structure of K, in the rest of this paper, we partition x = (x1 , . . . , xr ) and F = (F1 (x), . . . , Fr (x)) with xi ∈ Rn i and Fi (x) : Rn → Rn i . There have been many methods proposed for solving SOCCP in the literature, such as the merit function approach [4], the descent method [5], the damped Gauss-Newton Method [29], the regularization method [30] and the smoothing Newton methods [6,7,9,12,17,18,33,34]. Among them, the smoothing Newton methods are considered to be a class of most effective methods for solving SOCCP, which in
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