This presentation explores recent developments in the theory of duality and gap functions for equilibrium problems ([1,2,3,4]). We investigate the construction of gap functions derived from conjugate duality principles in convex optimization, providing a unified framework for analyzing equilibrium problems ([1]). The presentation concludes with a comprehensive review of recent literature and promising research directions, emphasizing potential applications in variational inequalities and Nash equilibrium problems. This work contributes to the broader understanding of duality theory in equilibrium problems and provides valuable tools for both theoretical analysis and practical implementations ([5]).

References

1. Altangerel, L., Bot, R. I. and Wanka, G. On Gap Functions for Equilibrium Problems via Fenchel Duality, Pacific Journal of Optimization 2 (3), 667-678, 2006.
2. Altangerel, L. On Gap Functions for Quasi-Equilibrium Problems Via Duality, Journal of Mathematical Sciences, 2024.
3. Konnov, I. V., Schaible, S. Duality for Equilibrium Problems under Generalized Monotonicity, Journal of Optimization Theory and Applications 104, 395-408.
4. Mastroeni, G. Gap functions for equilibrium problems, J. Glob. Optim. 27(4):411-426, 2003.
5. Pappalardo, M., Mastroeni, G. and Passacantando, M. Merit Functions: a Bridge Between Optimization and Equilibria, 4OR-Q J Oper Res 12, 1-33, 2014.

In this talk, we explore étale groupoids $G$ with a locally compact Hausdorff unit space $X$, where $G$ itself may not be globally Hausdorff. For such groupoids, the essential $C^*$-algebra $C_{\operatorname{ess}}^*(G)$ offers a more suitable framework than the reduced $C^*$-algebra $C_r^*(G)$, as it captures additional structural nuances. Specifically, $C_{\operatorname{ess}}^*(G)$ arises as a proper quotient of $C_r^*(G)$.

We introduce the concept of essential amenability for groupoids, a condition that is strictly weaker than (topological) amenability yet sufficient to guarantee the nuclearity of $C_{\operatorname{ess}}^*(G)$. To establish this, we define a maximal version of the essential $C^*$-algebra and show that any function with dense cosupport must be supported within the set of "dangerous arrows”, that is, arrows that cannot be topologically separated.

This essential amenability framework characterizes the nuclearity of $C_{\operatorname{ess}}^*(G)$ and establishes its isomorphism to the maximal essential $C^*$-algebra. Our results offer new insights into the interplay between groupoid structure and operator algebras, extending the utility of $C_{\operatorname{ess}}^*(G)$ in non-Hausdorff settings. This is based on joint work with Diego Martinez.