Moreau-Type Characterizations of Polar Cones

2018/10/09 by Valeriu Soltan, Soltan, Valeriu
Computer Science · Mathematics · #15A39 #52A20 #90C25 #Advanced Banach Space Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Metric Geometry (math.MG) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1810.04206

openalex publication_date 2018/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A theorem of Moreau (1962) states that given a closed convex cone C and its (negative) polar cone C^∘ in a real Hilbert space H, vectors y ∈ C and z ∈ C^∘ are metric projections of a vector u ∈ H on C and C^∘, respectively, if and only if they satisfy the following conditions: y and z are orthogonal and u = y + z. We show that these conditions provide characteristic properties of polar cones C and C^∘ in the family of pairs of convex subsets of H or ℝⁿ. A related result on separation of C a face of C^∘ in ℝⁿ is proved.

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