A Simplicial Algorithm for the Nonlinear Stationary Point Problem on an Unbounded Polyhedron

A path following algorithm is proposed for finding a solution to the nonlinear stationary point problem on an unbounded, convex and pointed polyhedron. The algorithm can start at an arbitrary point of the polyhedron. The path to be foliowed by the algorithm is described as the path of zeros of some piecewise continuously differentiable function defined on an appropriate subdivided manifold. This manifold is induced by a generalized primal-dual pair of subdivided manifolds. The path of zeros can be approximately foliowed by dividing the polyhedron into simplices and replacing the original function by its piecewise linear approximation with respect to this subdivision. The piecewise linear path of this function can be generated by alternating replacement steps and linear programming pivot steps. We also state a condition under which the path of zeros converges to a solution and we describe how the algorithm operates when the problem is linear or when the polyhedron is the Cartesian product of a polytope and an unbounded polyhedron. 1 Institute of Socio-Economic Planning, University of Tsukuba, Tsukuba, Ibaraki 305, Japan. 2 Department of Econometrics, Vrije Universiteit, P.O.Box 7161, 1007 MC Amsterdam, The Netherlands. 3 Department of Econometrics, Tilburg University, P.O.Box 90153, 5000 LE Tilburg, The Netherlands.