This note has three goals: to define a generalized nonlinear complementarity-type problem of mathematical programming for arbitrary closed convex cone on Hausdorff topological vector space, to develop variational-type and complementarity-type problem in contractible space and the notion of variational-type and complementarity-type problem in Hadamard manifold.

generalized nonlinear complementarity problem, Hausdorff topological vector space, contractible space, Hadamard manifold, geodesic convex, generalized KKM-map.