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A linearly convergent derivative-free descent method for the second-order cone complementarity problem. (English) Zbl 1229.90239
Summary: We consider a class of derivative-free descent methods for solving the second-order cone complementarity problem (SOCCP). The algorithm is based on the Fischer-Burmeister (FB) unconstrained minimization reformulation of the SOCCP, and utilizes a convex combination of the negative partial gradients of the FB merit function \(\psi _{\text{FB}}\) as the search direction. We establish the global convergence results of the algorithm under monotonicity and the uniform Jordan \(P\)-property, and show that under strong monotonicity the merit function value sequence generated converges at a linear rate to zero. Particularly, the rate of convergence is dependent on the structure of second-order cones. Numerical comparisons are also made with the limited BFGS method used by J.-S. Chen and P. Tseng [Math. Program. 104, No. 2–3 (B), 293–327 (2005; Zbl 1093.90063)], which confirm the theoretical results and the effectiveness of the algorithm.

MSC:
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C56 Derivative-free methods and methods using generalized derivatives
Citations:
Zbl 1093.90063
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