[Paper Review] Bronsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces

M. Marques Alves, B. F. Svaiter|ArXiv.org|Feb 13, 2008

Optimization and Variational Analysis17 references15 citations

TL;DR

This paper establishes that in non-reflexive Banach spaces, a lower semicontinuous convex function $ h: X \times X^* \to \overline{\mathbb{R}} $ satisfying $ h(x,x^*) \geq \langle x,x^* \rangle $ and $ h^*(x^*,x^{**}) \geq \langle x^*,x^{**} \rangle $ for all $ (x,x^*), (x^*,x^{**}) $ generates a maximal monotone operator via its sublevel set. The key contribution is proving that such operators satisfy a strict Brønsted-Rockafellar property, extending known results from reflexive to non-reflexive settings.

ABSTRACT

In this work we are concerned with maximality of monotone operators representable by certain convex functions in non-reflexive Banach spaces. We also prove that these maximal monotone operators satisfy a Bronsted-Rockafellar type property. We show that if a function in XxX^* and its conjugate are above the duality product in their respective domains, then this function represents a maximal monotone operator.

Motivation & Objective

• To establish sufficient conditions for maximal monotonicity of operators representable by convex functions in non-reflexive Banach spaces.
• To extend the Brønsted-Rockafellar property to non-reflexive settings using convex representations.
• To characterize the Fitzpatrick function and its conjugate in relation to the duality product in $ X \times X^* $ and $ X^* \times X^{**} $.
• To resolve the question of whether the generalized condition (4) is sufficient for maximal monotonicity in non-reflexive spaces.

Proposed method

• Define a convex function $ h $ on $ X \times X^* $ such that $ h(x,x^*) \geq \langle x,x^* \rangle $ and its conjugate satisfies $ h^*(x^*,x^{**}) \geq \langle x^*,x^{**} \rangle $ for all $ x^{**} \in X^{**} $.
• Construct the operator $ T = \{(x,x^*) \in X \times X^* \mid h^*(x^*,x) = \langle x,x^* \rangle \} $, showing it is maximal monotone.
• Use the Fitzpatrick function $ \varphi_T $ as the minimal convex representation of $ T $, and prove $ \varphi_T^*(x^*,x^{**}) \geq \langle x^*,x^{**} \rangle $.
• Apply the strict Brønsted-Rockafellar property: for any $ \eta > \varepsilon > 0 $, if $ x^* \in T^\varepsilon(x) $, then there exists a sequence approximating $ (x,x^*) $ in $ T $ with controlled error.
• Leverage lower semicontinuity and convergence arguments to show $ (z,z^*) \in T $, proving closedness and maximality.
• Use the fact that the convex closure of $ h $ preserves the conjugate and majorization, allowing removal of lower semicontinuity in some results.

Experimental results

Research questions

• RQ1 Is the condition $ h(x,x^*) \geq \langle x,x^* \rangle $ and $ h^*(x^*,x^{**}) \geq \langle x^*,x^{**} \rangle $ sufficient for maximal monotonicity in non-reflexive Banach spaces?
• RQ2 Does the Fitzpatrick function of a maximal monotone operator in non-reflexive spaces satisfy $ \varphi_T^*(x^*,x^{**}) \geq \langle x^*,x^{**} \rangle $ for all $ x^{**} \in X^{**} $?
• RQ3 Can the Brønsted-Rockafellar property be strengthened in non-reflexive spaces to allow approximation of $ \varepsilon $-enlargements by elements in the graph of $ T $?
• RQ4 Is the operator $ T = \{(x,x^*) \mid h^*(x^*,x) = \langle x,x^* \rangle \} $ maximal monotone under the generalized condition (4)?
• RQ5 Does the conjugate of the Fitzpatrick function majorize the duality product in $ X^* \times X^{**} $ for such operators?

Key findings

• The operator $ T = \{(x,x^*) \in X \times X^* \mid h^*(x^*,x) = \langle x,x^* \rangle \} $ is maximal monotone if $ h $ is proper, convex, lower semicontinuous, and satisfies $ h(x,x^*) \geq \langle x,x^* \rangle $ and $ h^*(x^*,x^{**}) \geq \langle x^*,x^{**} \rangle $ for all $ x^{**} \in X^{**} $.
• The Fitzpatrick function $ \varphi_T $ of such $ T $ satisfies $ \varphi_T^*(x^*,x^{**}) \geq \langle x^*,x^{**} \rangle $ for all $ x^{**} \in X^{**} $, extending the duality product majorization to the bidual.
• The maximal monotone operator $ T $ satisfies the strict Brønsted-Rockafellar property: for any $ \eta > \varepsilon > 0 $, if $ x^* \in T^\varepsilon(x) $, then for every $ \lambda > 0 $, there exists $ (\bar{x}_\lambda, \bar{x}^*_\lambda) \in T $ such that $ \|x - \bar{x}_\lambda\| < \lambda $ and $ \|x^* - \bar{x}^*_\lambda\| < \eta / \lambda $.
• The result holds even if $ h $ is not lower semicontinuous, as the convex closure of $ h $ preserves the conjugate and majorization properties.
• The proof relies on constructing a sequence $ (u_k, u_k^*) \to 0 $ such that $ \langle u_k, u_k^* \rangle \to 0 $, and using lower semicontinuity of $ h $ to conclude $ (z,z^*) \in T $, proving closedness and maximality.
• The class of functions satisfying the generalized condition (4) fully characterizes maximal monotone operators in non-reflexive Banach spaces, extending the known characterization from reflexive settings.