Least Deviation Decomposition with Respect to a Pair of Convex Sets

Let

and

K_2 K 2

be two nonempty closed convex sets in some normed space

(H,\|\cdot\|) ( H , ∥ ⋅ ∥ )

. This paper is concerned with the question of finding a "good" decomposition, with respect to

K_1 K 1

and

K_2 K 2

, of a given element of the Minkowski sum

K_1+K_2 K 1 + K 2

. We introduce and discuss the concept of least deviation decomposition. This concept is an extension of the Moreau orthogonal decomposition with respect to a pair of mutually polar cones. Techniques of convex analysis are applied to obtain some sensitivity and duality results related to the decomposition problem.