I will focus on the envelope formula, an essential tool and a unifying framework for the first-order analysis of value functions in parameter-dependent optimization problems. In particular, I will discuss formal differentiation rules of value functions and derive a few familiar examples as particular cases of this technique, such as the Hamilton-Jacobi-Bellman PDE and the adjoint method. I will then discuss how to turn these formal differentiation rules into rigorous theorems via perturbation analysis of optimization problems. Finally, I will apply these ideas to parameter identification problems based on optimal transportation distances and variational analysis of mean-field games.
I am currently an Assistant Adjunct Professor at the Department of Mathematics at UCLA. I previously held postdoctoral and visiting positions at McGill University, King Abdullah University of Science and Technology, National Academy of Sciences of Armenia, and Technical University of Lisbon. I have also been a Senior Fellow at the Institute for Pure and Applied Mathematics (IPAM) at UCLA for its Spring 2020 Program on High Dimensional Hamilton-Jacobi PDEs and a Simons CRM Scholar at the Centre de Recherches Mathématiques (CRM) at the University of Montreal for its Spring 2019 Program on Data Assimilation: Theory, Algorithms, and Applications. I obtained my Ph.D. in the framework of UT Austin -- Portugal CoLab under the supervision of Professors Diogo Gomes and Alessio Figalli. My research interests include calculus of variations, optimal control theory, mean-field games, partial differential equations, mathematics applied to machine learning, dynamical systems, and shape optimization problems