HOMOTOPY CONTINUATION METHODS FOR NONLINEAR COMPLEMENTARITY PROBLEMS.

HOMOTOPY theory; NONLINEAR theories; STOCHASTIC convergence; CONTINUATION methods; NONLINEAR statistical models; LINEAR complementarity problem; LINEAR programming; ALGORITHMS; EQUATIONS; MATHEMATICAL programming

A complementarily problem with a continuous mapping f from Rn into itself can be written as the system of equations F(x, y) = 0 and (x, y) ⩾ 0. Here F is the mapping from R2n into itself defined by F(x, y) = (x1y1, x2y2, …, xnyn, y - f(x)). Under the assumption that the mapping f is a P0-function, we study various aspects of homotopy continuation methods that trace a trajectory consisting of solutions of the family of systems of equations F(x, y) = t(a, b) and (x, y) ⩾ 0 until the parameter t ⩾ o attains 0. Here (a, b) denotes a 2n-dimensional constant positive vector. We establish the existence of a trajectory which leads to a solution of the problem, and then present a numerical method for tracing the trajectory. We also discuss the global and local convergence of the method. [ABSTRACT FROM AUTHOR]

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