A simple closure condition for the normal cone intersection formula

Abstract

Abstract. In this paper it is shown that if C and D are two closed convex subsets of a Banach space X and, is weak* closed, where σ C and N C are the support function and the normal cone of the set C respectively. This closure condition is shown to be weaker than the standard interior-pointlike conditions and the bounded linear regularity condition.

On the convergence rate improvement of a splitting method for finding the resolvent of the sum of maximal monotone operators

“…On the other hand, ( 12) is closely related to the averaged alternating modified reflections algorithm in [3]. Aragón Artacho and Campoy considered the following iterative scheme: (13) where v 0 ∈ H, γ > 0 and 13) is equivalently written as…”

“…Remark 5.3. (i) Burachik and Jeyakumar[13] showed that the normal cone intersection formulaN C∩D (x) = N C (x) + N D (x) (∀x ∈ C ∩ D)holds whenever epiσ C + epiσ D is weakly closed [13, Theorem 3.1]. Furthermore, it was shown that 0 ∈ sri(C − D) implies epiσ C + epiσ D is weakly closed [13, Proposition 3.1].…”

Closedness Type Regularity Conditions in Convex Optimization and Beyond

“…However, the latter is a priori violated for many classes of problems where the constraint cone has an empty interior, and several generalizations were proposed for it, the socalled interiority type regularity conditions, that involve notions of generalized interior of a set. But even these fail for large classes of problems and, inspired by Precupanu's pioneering work [1], Burachik, Jeyakumar and their coauthors, and on the other hand Boţ, Wanka and their coauthors have proposed in [2][3][4][5][6][7][8][9] a new class of regularity conditions, the closedness type ones. They have proven first to be sufficient conditions for guaranteeing duality statements in optimization and subdifferential formulae in convex analysis, delivering in the meantime results and formulae in some related research fields as well.…”

“…As mentioned above, the first papers dealing with closedness type regularity conditions for convex optimization problems were [2][3][4][5][6][7], having as a starting point earlier statements from Precupanu [1]. Afterwards, it was noticed that other regularity conditions from the literature such as the Basic Constraint Qualification (see for instance, [11][12][13]) or the Farkas-Minkowski Constraint Qualification (cf.…”

Boundary Half-Strips and the Strong CHIP

“…The converse to Moreau's theorem obviously holds for sets C and D such that every linear and continuous map bounded above on any one of the sets C ∩ D, C, or D necessarily achieve their maximums on this set. On this ground, a first partial converse of the Moreau result has recently been proved by Bauschke, Borwein, and Li for Hilbert spaces (see [1,Proposition 6.4]); the result was extended to the setting of Banach spaces by Burachik and Jeyakumar [3,Proposition 4.2]. Their result states that, if C and D is a pair of closed and convex cones with the strong CHIP, then the inf-convolution of their support functionals is always exact.…”