Under what cases of cone $K$ is $x = \text{proj}_K(x) + \text{proj}_{-K^*}(x)$ possible for all $x$?

Question

Here, $K^*$ is the dual cone of $K$:

$K^* = \{x \mid x^Ty \geq 0 \forall y\in K\}.$

The property is true if $K$ is the nonnegative cone or the positive semidefinite cone. Does a more general property apply? Is there an intuitive proof?

thanks!

Answer 1

Ok so I found the answer: basically it's all $K$ closed convex cones (woah!), and is a consequence of one of Moreau's theorems.

Here's a link that summarizes it: http://www.convexoptimization.com/wikimization/index.php/Moreau's_decomposition_theorem

The original source is

J.-J. Moreau, Decomposition orthogonale d’un espace hilbertien selon deux cones mutuellement polaires, Comptes Rendus de l’Acad´emie des Sciences de Paris S´erie A, vol. 255, pp. 238–240, 1962.