An R-Linearly Convergent Nonmonotone Derivative-Free Method for Symmetric Cone Complementarity Problems

This paper extends the derivative-free descent method [18] for the nonlinear complementarity problem to the symmetric cone complementarity problem (SCCP). The algorithm is based on the unconstrained implicit Lagrangian reformulation of the SCCP, but uses a convex combination of the negative partial gradients of the implicit Lagrangian function ψα, i.e. the vector of the form −θ∇xψα −(1 −θ)∇yψα for θ ∈ [0, 1], as the search direction, and a nonmonotone line search rule to seek a desirable stepsize. We show that the derivative-free algorithm converges in terms of the implicit Lagrangian value for a large class of SCCPs that may even not be monotone. If θ is restricted to be less than a threshold θ̄ ∈ (0, 1) and the SCCP is strongly monotone, the sequence generated converges globally to the solution of SCCP at a R-linear rate.