How to project onto an isotone projection cone

The solution of the complementarity problem defined by a mapping f : R-n -> R-n and a cone K subset of R-n consists of finding the fixed points of the operator P-K o (I - f), where P-K is the projection onto the cone K and I stands for the identity mapping. For the class of isotone projection cones (cones admitting projections isotone with respect to the order relation they generate) and f satisfying certain monotonicity properties, the solution can be obtained by iterative processes (see G. Isac, A.B. Nemeth, Projection methods, isotone projection cones, and the complementarity problem,]. Math. Anal. Appl. 153(1) (1990) 258-275 and S.Z. Nemeth, Iterative methods for nonlinear complementarity problems on isotone projection cones, J. Math. Anal. Appl. 350(1) (2009) 340-347). These algorithms require computing at each step the projection onto the cone K. In general, computing the projection mapping onto a cone K is a difficult and computationally expensive problem. In this note it is shown that the projection of an arbitrary point onto an isotone projection cone in R-n can be obtained by projecting recursively at most n - 1 times into subspaces of decreasing dimension. This emphasizes the efficiency of the algorithms mentioned above and furnishes a handy tool for some problems involving special isotone projection cones, as for example the non-negative monotone cones occurring in reconstruction problems (see e.g. Section 5.13 in J. Dattorro, Convex Optimization and Euclidean Distance Geometry, Meboo, 2005, v2009.04.11). (C) 2010 Elsevier Inc. All rights reserved.