A new continuation method for complementarity problems with uniformP-functions

The complementarity problem with a nonlinear continuous mappingf from the nonnegative orthantR₊ⁿ ofRⁿ intoRⁿ can be written as the system of equationsF(x, y) = 0 and(x, y) R₊²ⁿ, whereF denotes the mapping from the nonnegative orthantR₊²ⁿ ofR²ⁿ intoR₊ⁿ × Rⁿ defined byF(x, y) = (x₁y₁,,x_ny_n, f₁(x) – y₁,, f_n(x) – y_n) for every(x, y) R₊²ⁿ. Under the assumption thatf is a uniformP-function, this paper establishes that the mappingF is a homeomorphism ofR₊²ⁿ ontoR₊ⁿ × Rⁿ. This result provides a theoretical basis for a new continuation method of tracing the solution curve of the one parameter family of systems of equationsF(x, y) = tF(x⁰, y⁰) and(x, y) R₊²ⁿ from an arbitrary initial point(x⁰, y⁰) R₊²ⁿ witht = 1 until the parametert attains 0. This approach is an extension of the one used in the polynomially bounded algorithm recently given by Kojima, Mizuno and Yoshise for solving linear complementarity problems with positive semi-definite matrices.