Outer Γ-Convexity in Vector Spaces
Hoàng Xuân Phú · Numerical Functional Analysis and Optimization · 2008
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.