We review basic properties of the moment-LP and moment-SOS hierarchies for polynomial optimization and compare them. We also illustrate how to use such a methodology in two applications outside optimization. Namely :
- for approximating (as claosely as desired in a strong sens) set defined with quantifiers of the form
$R_1 =\{ x\in B : f(x,y)\leq 0 $ for all $y$ such that $(x,y) \in K \}$.
$D_1 =\{ x\in B : f(x,y)\leq 0 $ for some $y$ such that $(x,y) \in K \}$.
by a hierarchy of inner sublevel set approximations
$\Theta_k = \left \{ x\in B : J_k(x)\leq 0 \right \}\subset R_f$.
or outer sublevel set approximations
$\Theta_k = \left \{ x\in B : J_k(x)\leq 0 \right \}\supset D_f$.
for some polynomiales $(J_k)$ of increasing degree :
- for computing convex polynomial underestimators of a given polynomial $f$ on a box $B \subset R^n$.[-]

Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and computing, and the potential applications to economics and social sciences are numerous.
In the limit when $n \to +\infty$, a given agent feels the presence of the others through the statistical distribution of the states. Assuming that the perturbations of a single agent's strategy does not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation.
The latter system of PDEs has closed form solutions in very few cases only. Therefore, numerical simulation are crucial in order to address applications. The present mini-course will be devoted to numerical methods that can be used to approximate the systems of PDEs.
The numerical schemes that will be presented rely basically on monotone approximations of the Hamiltonian and on a suitable weak formulation of the Fokker-Planck equation.
These schemes have several important features:

- The discrete problem has the same structure as the continous one, so existence, energy estimates, and possibly uniqueness can be obtained with the same kind of arguments

- Monotonicity guarantees the stability of the scheme: it is robust in the deterministic limit

- convergence to classical or weak solutions can be proved

Finally, there are particular cases named variational MFGS in which the system of PDEs can be seen as the optimality conditions of some optimal control problem driven by a PDE. In such cases, augmented Lagrangian methods can be used for solving the discrete nonlinear system. The mini-course will be orgamized as follows

1. Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions.

2. Monotone finite difference schemes

3. Examples of applications

4. Variational MFG and related algorithms for solving the discrete system of nonlinear equations[-]

Recently, an important research activity on mean field games (MFGs for short) has been initiated by the pioneering works of Lasry and Lions: it aims at studying the asymptotic behavior of stochastic differential games (Nash equilibria) as the number $n$ of agents tends to infinity. The field is now rapidly growing in several directions, including stochastic optimal control, analysis of PDEs, calculus of variations, numerical analysis and computing, and the potential applications to economics and social sciences are numerous.
In the limit when $n \to +\infty$, a given agent feels the presence of the others through the statistical distribution of the states. Assuming that the perturbations of a single agent's strategy does not influence the statistical states distribution, the latter acts as a parameter in the control problem to be solved by each agent. When the dynamics of the agents are independent stochastic processes, MFGs naturally lead to a coupled system of two partial differential equations (PDEs for short), a forward Fokker-Planck equation and a backward Hamilton-Jacobi-Bellman equation.
The latter system of PDEs has closed form solutions in very few cases only. Therefore, numerical simulation are crucial in order to address applications. The present mini-course will be devoted to numerical methods that can be used to approximate the systems of PDEs.
The numerical schemes that will be presented rely basically on monotone approximations of the Hamiltonian and on a suitable weak formulation of the Fokker-Planck equation.
These schemes have several important features:

- The discrete problem has the same structure as the continous one, so existence, energy estimates, and possibly uniqueness can be obtained with the same kind of arguments

- Monotonicity guarantees the stability of the scheme: it is robust in the deterministic limit

- convergence to classical or weak solutions can be proved

Finally, there are particular cases named variational MFGS in which the system of PDEs can be seen as the optimality conditions of some optimal control problem driven by a PDE. In such cases, augmented Lagrangian methods can be used for solving the discrete nonlinear system. The mini-course will be orgamized as follows

1. Introduction to the system of PDEs and its interpretation. Uniqueness of classical solutions.

2. Monotone finite difference schemes

3. Examples of applications

4. Variational MFG and related algorithms for solving the discrete system of nonlinear equations[-]

This talk will cover two recent advancements in the theory of online algorithms for dynamic matching markets. The first set of results concern a stochastic model of matching with Poisson arrivals and memoryless departures over edge-weighted graphs. The second set of results focus on the incorporation of serial correlation properties in classical online stochastic matching models. We develop new mathematical programming relaxations and correlated rounding schemes, yielding the first constant-factor performance guarantees in such settings.[-]

We propose a PDE framework for modeling the distribution shift of a strategic population interacting with a learning algorithm. We consider two particular settings one, where the objective of the algorithm and population are aligned, and two, where the algorithm and population have opposite goals. We present convergence analysis for both settings, including three different timescales for the opposing-goal objective dynamics. We illustrate how our framework can accurately model real-world data and show via synthetic examples how it captures sophisticated distribution changes which cannot be modeled with simpler methods.[-]

We discuss the natural Lagrangian and Eulerian formulations of multi-agent deterministic optimal control problems, analyzing their relations with a novel Kantorovich formulation. We exhibit some equivalence results among the various representations and compare the respective value functions, by combining techniques and ideas from optimal transport, control theory, Young measures and evolution equations in Banach spaces. We further exploit the connections among Lagrangian and Eulerian descriptions to derive consistency results as the number of particles/agents tends to infinity. (In collaboration with Giulia Cavagnari, Stefano Lisini and Carlo Orrieri)[-]

We will present a variation of the unbalanced optimal transport model and Wasserstein Fisher-Rao metric on positive measures, in which one imposes additional affine integral equality constraints. This is motivated by multiple examples from mathematics and applied mathematics that naturally involve comparing and interpolating between two measures in particular subspaces or in which one enforces some constraints on the interpolating path itself. Building from the dynamic formulation of the Wasserstein Fisher-Rao metric, we introduce a class of constrained problems where the interpolating measure at each time must satisfy a given stationary or time-dependent constraint in measure space. We then specifically derive general conditions under which the existence of minimizing paths can be guaranteed, and then examine some of the properties of the resulting models and the metrics that are induced on measures. We will further hint at the potential of this approach in various specific situations such as the comparison of measures with prescribed moments, the unbalanced optimal transport under global mass evolution or obstacle constraints, and emphasize some connections with the construction of Riemannian metrics on the space of all convex shapes in an Euclidean space. We shall conclude with a few remaining unsolved/open questions.[-]

Everything is under control: mathematics optimize everyday life.
In an empirical way we are able to do many things with more or less efficiency or success. When one wants to achieve a parallel parking, consequences may sometimes be ridiculous... But when one wants to launch a rocket or plan interplanetary missions, better is to be sure of what we do.
Control theory is a branch of mathematics that allows to control, optimize and guide systems on which one can act by means of a control, like for example a car, a robot, a space shuttle, a chemical reaction or in more general a process that one aims at steering to some desired target state.
Emmanuel Trélat will overview the range of applications of that theory through several examples, sometimes funny, but also historical. He will show you that the study of simple cases of our everyday life, far from insignificant, allow to approach problems like the orbit transfer or interplanetary mission design.[-]