[Paper Review] A note on the isotonic vector-valued convex functions

Constantin P. Niculescu, Octav Olteanu|arXiv (Cornell University)|May 3, 2020

Optimization and Variational Analysis29 references4 citations

TL;DR

This paper characterizes isotonicity of continuous convex vector-valued functions on ordered Banach spaces using subdifferentials. It establishes that a convex function is isotone (order-preserving) if and only if its subdifferential at each point is a positive operator, providing a variational criterion for isotonicity in infinite-dimensional ordered spaces with applications to Choquet integrals and sublinear operators.

ABSTRACT

The property of isotonicity of a continuous convex function defined on the entire space or only on the positive cone is characterized via subdifferentials. Numerous examples illustrating the obtained results are included.

Motivation & Objective

• To characterize isotonicity of continuous convex functions with values in ordered Banach spaces.
• To establish a variational criterion for isotonicity using subdifferentials.
• To extend results from scalar to vector-valued convex functions in ordered topological vector spaces.
• To provide examples and applications, including to Choquet integrals and sublinear operators.
• To clarify the relationship between order structure, convexity, and continuity in infinite-dimensional settings.

Proposed method

• Uses subdifferentials of convex functions as the primary tool to analyze isotonicity in ordered Banach spaces.
• Applies the concept of positive linear operators and their continuity via the Banach-Steinhaus theorem.
• Employs the Hahn-Banach separation theorem to characterize order relations via dual cone elements.
• Utilizes the structure of regularly ordered Banach spaces, where norms are compatible with the order cone.
• Analyzes the Choquet integral as a continuous, sublinear, and isotone functional on continuous functions.
• Constructs vector-valued operators via multiple submodular capacities to generalize scalar results to R^n-valued settings.

Experimental results

Research questions

• RQ1 Under what conditions is a continuous convex function between ordered Banach spaces isotone (order-preserving)?
• RQ2 How can the isotonicity of a convex function be characterized via its subdifferential?
• RQ3 What role do positive linear operators and subdifferentials play in determining isotonicity in infinite-dimensional spaces?
• RQ4 How do Choquet integrals relate to sublinear and isotone functionals in the context of vector-valued convex functions?
• RQ5 Can the least upper bound property of submodular capacities be established via positive, regular, and σ-additive vector measures?

Key findings

• A continuous convex function Φ between ordered Banach spaces is isotone if and only if its subdifferential at each point consists of positive linear operators.
• Every positive linear operator between ordered Banach spaces is necessarily continuous, as shown via the Banach-Steinhaus theorem and sequence arguments.
• The dual space of an ordered Banach space, when equipped with the dual cone, is itself a regularly ordered Banach space.
• The Choquet integral defines a continuous, sublinear, and isotone functional on C(X), and extends naturally to vector-valued settings via multiple capacities.
• For every non-zero continuous function h, there exists a positive linear operator T dominated by the vector-valued Choquet operator P such that T(h) = P(h).
• Every submodular capacity μ: B(K) → R^n is the least upper bound of all positive, regular, and σ-additive vector measures dominated by μ.