Outer Γ-Convexity in Vector Spaces

Abstract

A subset S of some vector space X is said to be outer Γ-convex w.r.t. some given balanced subset Γ ⊂ X if for all x ₀, x ₁ ∊ S there exists a closed subset Λ ⊂ [0,1] such that {x _λ | λ ∊ Λ} ⊂ S and [x ₀, x ₁] ⊂ {x _λ | λ ∊ Λ} + 0.5 Γ, where x _λ: = (1 − λ)x ₀ + λ x ₁. A real-valued function f:D → ℝ defined on some convex D ⊂ X is called outer Γ-convex if for all x ₀, x ₁ ∊ D there exists a closed subset Λ ⊂ [0,1] such that [x ₀, x ₁] ⊂ {x _λ | λ ∊ Λ} + 0.5 Γ and f(x _λ) ≤ (1 − λ)f(x ₀) + λ f(x ₁) holds for all λ ∊ Λ. Outer Γ-convex functions possess some similar optimization properties as these of convex functions, e.g., lower level sets of outer Γ-convex functions are outer Γ-convex and Γ-local minimizers are global minimizers. Some properties of outer Γ-convex sets and functions are presented, among others a simplex property of outer Γ-convex sets, which is applied for establishing a separation theorem and for proving the existence of modified subgradients of outer Γ-convex functions.

Keywords:

• Generalized convexity
• Global minimizer
• Γ-local minimizer
• Nonconvexity measure
• Outer Γ-convexity
• Rough convexity
• Self-Jung constant
• Separation theorem
• Simplex property
• Subgradients of nonconvex functions

AMS Subject Classification:

• 52A01
• 52A30
• 52A41
• 49K27
• 90C26

ACKNOWLEDGMENT

The author would like to express his sincere gratitude to Prof. Dr. Dr. h.c. Eberhard Zeidler for inviting him to the Max Planck Institute for Mathematics in the Sciences where parts of this work were done.