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A smoothing Newton method for the second-order cone complementarity problem. (English) Zbl 1274.90268
The paper introduces a new variant of smoothing Newton method for the second-order cone complementarity problem based on a new type of smoothing of Fischer-Burmeister NCP function. The authors propose to use a smoothing function $\varphi (\mu, x, y) = (\cos \mu + \sin \mu )(x + y) - \sqrt{(\cos \mu - \sin \mu )^2 (x-y)^2 + 4\mu ^2 e},$ where $$e$$ is the identity element with respect to Jordan product. Basic results of Euclidean Jordan algebra are summarized and the Jacobian of $$\varphi$$ is computed.
The proposed variant of the Newton method is accompanied with analysis of global convergence, and the authors establish the local quadratic convergence of the algorithm without the strict complementarity condition. The paper is concluded with reports on the numerical performance of the proposed method in comparison with the interior point method.
Currently, the smoothing Newton methods for second-order cone complementarity problems attract a lot of attention and there is already a considerable number of papers with similar algorithms based on another smoothings of Fisher-Burmeister NCP function, cf. eg. [X. D. Chen, D. Sun and J. Sun, Comput. Optim. Appl. 25, No. 1–3, 39–56 (2003; Zbl 1038.90084)] or [Y. Narushima, N. Sagara and H. Ogasawara, J. Optim. Theory Appl. 149, No. 1, 79–101 (2011; Zbl 1221.90085)].

##### MSC:
 90C25 Convex programming 90C30 Nonlinear programming 90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
##### Citations:
Zbl 1038.90084; Zbl 1221.90085
SDPT3
Full Text:
##### References:
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