[Paper Review] Vector duality via conditional extension of dual pairs

Samuel Drapeau, Asgar Jamneshan+1 more|arXiv (Cornell University)|Aug 31, 2016

Optimization and Variational Analysis16 references3 citations

TL;DR

This paper establishes a Fenchel-Moreau type duality for proper convex functions $ f: X \to \overline{L^0} $, where $ X $ is a Banach space and $ \overline{L^0} $ is the space of extended real-valued measurable functions. By conditionally extending the dual pair $ (X,Y,\langle\cdot,\cdot\rangle) $ to Bochner spaces $ L^0(X) \times L^0(Y) $, the authors prove that lower semi-continuity of $ f $ is equivalent to the dual representation $ f(x) = \operatorname{ess\,sup}_{y \in L^0(Y)} \{ \langle x,y \rangle - f^*(y) \} $, generalizing classical duality to vector-valued settings via conditional set theory and Bochner space extensions.

ABSTRACT

A Fenchel-Moreau type duality for proper convex and lower semi-continuous functions $f\colon X o \overline{L^0}$ is established where $(X,Y,\langle \cdot,\cdot angle)$ is a dual pair of Banach spaces and $\overline{L^0}$ is the set of all extended real-valued measurable functions. We provide a concept of lower semi-continuity which is shown to be equivalent to the existence of a dual representation in terms of elements in the Bochner space $L^0(Y)$. To derive the duality result, several conditional completions and extensions are constructed. This is an earlier version of arXiv e-print 1708.03127, where the main results were formulated in an abstract setting of conditional completions, conditional extensions and conditional real numbers.

Motivation & Objective

• To establish a Fenchel-Moreau type duality for convex functions $ f: X \to \overline{L^0} $, where $ X $ is a Banach space and $ \overline{L^0} $ is the space of extended real-valued measurable functions.
• To define a notion of lower semi-continuity for such functions that is equivalent to the existence of a dual representation in terms of elements in $ L^0(Y) $, the Bochner space of $ Y $-valued measurable functions.
• To develop a conditional extension framework for dual pairs of Banach spaces, enabling the application of conditional Fenchel-Moreau duality in the original setting.
• To generalize classical duality results to vector-valued and module-based settings where standard scalarization techniques fail due to $ \operatorname{int}(L^0_+) = \emptyset $ and $ (L^0)^* = \{0\} $.

Proposed method

• Construct conditional completions of Banach spaces $ X $ and $ Y $, proving they are isometrically isomorphic to the Bochner spaces $ L^0(X) $ and $ L^0(Y) $, respectively.
• Extend the duality pairing $ \langle \cdot, \cdot \rangle $ on $ X \times Y $ to a conditional duality pairing on $ L^0(X) \times L^0(Y) $, defined via almost everywhere limits of simple functions.
• Define a conditionally lower semi-continuous extension $ f_c $ of $ f $ from $ X $ to $ L^0(X) $, with values in $ \overline{L^0} $, using the conditional structure of the space.
• Apply a conditional version of the Fenchel-Moreau theorem in the context of conditional set theory to derive a dual representation of $ f_c $ in terms of elements in $ L^0(Y) $.
• Translate the conditional dual representation back to the original space by evaluating constant elements, yielding the main duality formula: $ f(x) = \operatorname{ess\,sup}_{y \in L^0(Y)} \{ \langle x,y \rangle - f^*(y) \} $.
• Use the stable function structure on $ L^0(X) $ and $ L^0(Y) $, where operations are defined pointwise almost everywhere, to ensure compatibility with the conditional topology and norm.

Experimental results

Research questions

• RQ1 What is the appropriate notion of lower semi-continuity for functions $ f: X \to \overline{L^0} $ that ensures a Fenchel-Moreau type dual representation?
• RQ2 Can the classical Fenchel-Moreau duality be extended to vector-valued functions with values in $ \overline{L^0} $, especially when standard scalarization fails?
• RQ3 How can the duality pairing and convex conjugate be conditionally extended from $ X \times Y $ to $ L^0(X) \times L^0(Y) $ in a way that preserves duality and completeness?
• RQ4 What role does conditional set theory play in enabling the construction of dual representations for functions on $ L^0 $-valued domains?
• RQ5 Under what conditions does the dual representation $ f(x) = \operatorname{ess\,sup}_{y \in L^0(Y)} \{ \langle x,y \rangle - f^*(y) \} $ hold, and when does it fail?

Key findings

• A new notion of lower semi-continuity for functions $ f: X \to \overline{L^0} $ is introduced, defined via convergence of conditional duality pairings: $ \langle x_\alpha, y \rangle_c \to \langle x, y \rangle_c $ implies $ f(x) \leq \operatorname{ess\,liminf} f_s(x_\alpha) $ for all $ y \in Y_c $ and stable nets $ (x_\alpha) $ in $ X_s $.
• The conditional completion of $ X $ is isometrically isomorphic to $ L^0(X) $, and similarly for $ Y $, establishing that $ L^0(X) $ and $ L^0(Y) $ are the conditional completions of $ X $ and $ Y $, respectively.
• The duality pairing $ \langle \cdot, \cdot \rangle $ on $ X \times Y $ extends to $ L^0(X) \times L^0(Y) $ via almost everywhere limits of simple functions, and satisfies $ \langle x, y \rangle = \langle x, y \rangle_c $ under the identification of $ X_c $ with $ L^0(X) $.
• The main duality result is established: a proper convex function $ f: X \to \overline{L^0} $ is lower semi-continuous if and only if $ f(x) = \operatorname{ess\,sup}_{y \in L^0(Y)} \{ \langle x, y \rangle - f^*(y) \} $ for all $ x \in X $, where $ f^*(y) = \operatorname{ess\,sup}_{x \in X} \{ \langle x, y \rangle - f(x) \} $.
• The identity map $ I: L^\infty \to L^0 $ is not lower semi-continuous, and does not admit a Fenchel-Moreau representation of the form (8), demonstrating the necessity of the lower semi-continuity condition.
• The framework applies to arbitrary complete Boolean algebras, not just measure algebras, showing that the results are robust beyond classical measure-theoretic settings.