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In this paper, as an extension of the isotone projection cone, we consider the isotonicity of the metric projection operator with respect to the mutually dual orders induced by the cone and its dual cone in Hilbert spaces. We first discuss the relation between the isotonicity of the projection onto the cone and its dual cone. Some sufficient conditions for the isotonicity of the projection with respect to the mutually dual orders onto general cones are then proved. By using the self-dual cone in the hyperplane, we establish some specific isotone projection cones. Some properties and representations of the projection onto these cones are studied. We also prove some heredity and expansivity for the isotonicity of the projection onto these cones. By using the isotonicity characterizations of the projection with respect to the mutually dual orders, some solvability and approximation theorems for the complementarity and conic optimization problems are obtained. In Theorems 5.1–5.4, if the order relation does not satisfy the regularity, to guarantee convergence of isotone Picard iterations, three different order methods are used, respectively. Our results generalize those about the isotone projection cone and methods to establish isotone iterations for complementarity problems.

Isotonicity of the metric projection with respect to the mutually dual orders and complementarity problems