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

Abstract

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.

Keywords:

• Cone
• dual cone
• projection
• isotonicity
• complementarity problem

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 41A65
• 47H07
• 06B30
• 47J20
• 47H10

Acknowledgements

The authors would like to thank the referee for his/her very important comments that improve the results and the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The authors were supported financially by the National Natural Science Foundation of China (Grant Nos. 11871302 and 71773067), Key R&D Program of Shandong Province (Grant Nos. 2019GGX101024 and 2019GGX101061) and the support from the Australian Research Council for the research is also acknowledged.