Nonemptiness of Approximate Subdifferentials of Midpoint δ-Convex Functions

ABSTRACT

For some given positive δ, a function f:D ⊆ X → ℝ is called midpoint δ-convex if it satisfies the Jensen inequality f[(x ₀ + x ₁)/2] ≤ [f(x ₀) + f(x ₁)]/2 for all x ₀, x ₁ ∈ D satisfying ‖x ₁ − x ₀‖ ≥ δ (Hu, Klee, and Larman, SIAM J. Control Optimiz. Vol. 27, 1989). In this paper, we show that, under some assumptions, the approximate subdifferentials of midpoint δ-convex functions are nonempty.

Keywords:

• Approximate subdifferential
• δ-Convexity
• Generalized convexity

AMS Classification 1991:

• 26B25
• 26A51
• 52A01

ACKNOWLEDGMENT

This work was done within the framework of the Mathematics Research Fellowship of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.