December 6, 2023December 6, 2023 by alkhwarizmi

Mouhcine Assouli

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$\begin{quote} \begin{center} \textbf{Deep learning for mean field games with non-separable hamiltonians} \end{center} \medskip This paper introduces a new method based on Deep Galerkin Methods (DGMs) for solving high-dimensional stochastic Mean Field Games (MFGs). We achieve this by using two neural networks to approximate the unknown solutions of the MFG system and forward-backward conditions. Our method is efficient, even with a small number of iterations, and is capable of handling up to 300 dimensions with a single layer, which makes it faster than other approaches. In contrast, methods based on Generative Adversarial Networks (GANs) cannot solve MFGs with non-separable Hamiltonians. We demonstrate the effectiveness of our approach by applying it to a traffic flow problem, which was previously solved using the Newton iteration method only in the deterministic case. We compare the results of our method to analytical solutions and previous approaches, showing its efficiency. We also prove the convergence of our neural network approximation with a single hidden layer using the universal approximation theorem. \end{quote}$

December 6, 2023December 6, 2023 by alkhwarizmi

Noureddine Lamsahel

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$\begin{quote} \begin{center} \textbf{Eigenvalues distribution and control theory} \end{center} \medskip This work deals with the isogeometric Galerkin discretization of the eigenvalues problem related to the Laplace operator subject to homogeneous Dirichlet boundary conditions on bounded intervals. The main objective is the use of GLT theory to study the behavior of the gap of discrete spectra toward the uniform gap condition needed for the uniform boundary observability/controllability problems. The analysis refers to a regular B-spline basis and concave or convex reparametrizations. Under suitable assumptions on the reparametrization transformation, we prove that structure emerges within the distribution of the eigenvalues once we reframe the problem into GLT-symbol analysis. We also demonstrate numerically, that the necessary average gap condition proposed in [1] is not equivalent to the uniform gap condition. However, by improving the result in [2] we construct sufficient criteria that guarantee the uniform gap property. \smallskip \begin{enumerate} \item[] [1] D. Bianchi and S. Serra-Capizzano. Spectral analysis of finite-dimensional approximations of 1d waves in non-uniform grids. Calcolo, 55(4):1-28, 2018. \smallskip \item[] [2] D. Bianchi. Analysis of the spectral symbol associated to discretization schemes of linear self-adjoint differential operators. Calcolo, 58(3):38, 2021. \end{enumerate} \end{quote}$

December 6, 2023December 6, 2023 by alkhwarizmi

Ilias Ftouhi

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$\begin{quote} \begin{center} \textbf{Shape optimization and estimation of the fundamental frequency} \end{center} \medskip Spectral theory has been interesting various communities in the last century and still is. We are particularly interested in the spectrum of the Laplace operator with Dirichlet boundary conditions on $\partial\Omega$ where $\Omega\subset\mathbb{R}^d$, and more precisely its first eigenvalue also known as the fundamental frequency ($\lambda_1(\Omega)$). Unfortunately, for almost all given sets $\Omega$, there is no explicit formula for $\lambda_1(\Omega)$. This motivates to look for estimates via other functionals, which are much easier to compute (for example: Perimeter $P(\Omega)$ and measure $|\Omega|$). First, we give a brief introduction on shape optimization and spectral theory, then we introduce the following set of points, which can be called the Blaschke-Santalo diagram of the triplet $(\lambda_1,P,|\cdot|)$: $$\mathcal{C}_{\mathcal{F}_{ad}}:=\big\{ \big(P(\Omega),\lambda_1(\Omega)\big)\ |\ \Omega \in \mathcal{F}_{ad}\ \text{and}\ |\Omega|=1 \big\},$$ where $\mathcal{F}_{ad}$ is a given class of subsets of $\mathbb{R}^d$. Notice that the characterization of such diagram is equivalent to finding all the possible scale-invariant inequalities between the three involved quantities ($P$, $\lambda_1$ and $|.|$ in our case). We are able to completely describe the diagram for open sets in $\mathbb{R}^d$. It shows that there is no other inequality than the Faber-Krahn and the isoperimetric ones. This motivates us to investigate what happens for other classes of sets like the simply-connected and convex ones. We will give an advanced description of the Blaschke-Santalo diagram for convex sets in the plane. This work is in collaboration with Jimmy Lamboley. \end{quote}$

December 6, 2023December 6, 2023 by alkhwarizmi

Marouane Ibn Brahim

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$\begin{quote} \begin{center} \textbf{Maximum Load Assortment Optimization} \end{center} \medskip Motivated by modern-day applications such as Attended Home Delivery and Preference- based Group Scheduling, where decision makers wish to steer a large number of customers toward choosing the exact same alternative, we introduce a novel class of assortment opti- mization problems, referred to as Maximum Load Assortment Optimization. In such settings, given a universe of substitutable products, we are facing a stream of customers, each choosing between either selecting a product out of an offered assortment or opting to leave without making a selection. Assuming that these decisions are governed by the Multinomial Logit choice model, we define the random load of any underlying product as the total number of customers who select it. Our objective is to offer an assortment of products to each customer so that the expected maximum load across all products is maximized. We consider both static and dynamic formulations of the maximum load assortment optimization problem. In the static setting, a single offer set is carried throughout the entire process of customer arrivals, whereas in the dynamic setting, the decision maker offers a personalized assortment to each customer, based on the entire information available at that time. As can only be expected, both formulations present a wide range of computational challenges and analytical questions. The main contribution of this paper resides in proposing efficient algorithmic approaches for computing near-optimal static and dynamic assortment policies. In particular, we develop a polynomial-time approximation scheme (PTAS) for the static problem formulation. Additionally, we demonstrate that an elegant policy utilizing weight-ordered assortments yields a 1/2-approximation. Concurrently, we prove that such policies are sufficiently strong to provide a 1/4-approximation with respect to the dynamic formulation, establishing a constant-factor bound on its adaptivity gap. Finally, we design an adaptive policy whose expected maximum load is within factor O(1) of optimal, admitting a quasi-polynomial time implementation. \end{quote}$

December 6, 2023December 6, 2023 by alkhwarizmi

Zakaria Mzaouali

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$\begin{quote} \begin{center} \textbf{Quantum Computing to Solve Optimization Problems} \end{center} \medskip Quantum computers stands out as a promising tool to tackle complex optimization problems that remain intractable for classical machines. This talk delves into the revolutionary potential of quantum computing in reshaping the landscape of decision-making and optimization. We'll begin by introducing the fundamental principles of quantum mechanics that allow for a computational advantages. Next, we'll explore the development of quantum algorithms (e.g. QAOA), with a special focus on quantum annealers implemented by D-Wave systems to illustrate their potential in solving NP-hard problems more efficiently than their classical counterparts. Real-world applications, ranging from logistics and supply chain optimization to financial modeling, will be highlighted. Finally, we'll discuss current challenges in quantum computing and the road ahead for physicists and mathematicians. \end{quote}$

December 6, 2023December 6, 2023 by alkhwarizmi

Anas Abdelhakmi

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$\begin{quote} \begin{center} \textbf{A Multi-Period Black-Litterman Model with Desynchronized Expert Views } \end{center} \medskip The Black-Litterman model is a framework for incorporating forward-looking expert views in a portfolio optimization problem. Existing work focuses almost exclusively on single-period problems where the horizon of expert forecasts matches that of the investor. We consider a multi-period generalization of the Black- Litterman model where the investor trades dynamically, and the horizon of expert views may differ from that of the investor. By exploiting an underlying graphical structure relating the asset price process and the forward-looking expert views, we derive the conditional distribution of asset returns when the price process is assumed to be geometric Brownian motion. The new price process is an affine factor model where an adjustment of the unconditional log-price process is a factor, and is closely related to the smoothed estimates of future asset prices. We derive an explicit expression for the optimal dynamic investment policy and analyze the hedging demand associated with the new covariate. More generally, this application shows that smoothed estimates of underlying random processes appear naturally as state variables when there are forward looking forecasts. \end{quote}$

December 6, 2023December 6, 2023 by alkhwarizmi

Mohamed El Machkouri

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$\begin{quote} \begin{center} \textbf{On some variations of the Elephant Random Walk} \end{center} \medskip The Elephant Random Walk (ERW) was introduced by Schütz and Trimper in 2004 with a view to study memory effects in a one-dimensional discrete-time nearest-neighbor walk on $\mathbb{Z}$ with a complete memory of its whole past. The name of the model is inspired by the traditional saying that elephants can always remember anywhere they have been. The memory of the walker is measured in terms of a parameter p between zero and one and the model exhibits three regimes: diffusive regime $(0 < p < 3/4)$, critical regime $(p = 3/4)$ and superdiffusive regime $(3/4 < p < 1)$. The ERW has drawn a lot of attention in the last years and several results (law of large numbers, central limit theorem, law of the iterated logarithm,..) have been established for each of the three regimes. In 2022, Gut and Stadtmüller introduced an extension of the ERW model allowing the memory of the walker to gradually increase in time. For this new model, we establish central limit theorems in the three regimes and we show how to estimate the memory parameter p of the model. Finaly, we introduce another new variation of the ERW as a rule for the sequential allocation of drugs in a toxicity-response study. \end{quote}$

December 6, 2023December 6, 2023 by alkhwarizmi

Fatima ezzahra Saissi

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$\begin{quote} \begin{center} \textbf{Optimality conditions for bilevel optimization problems with extremal value function} \end{center} \medskip We provide necessary and sufficient optimality conditions for the bilevel programming problem with extremal value function. Our approach is based on the use of Fenchel lagrange duality. This latter is applied after a decomposition of the problem into a family of convex subproblems. The optimality conditions are expressed in terms of subdifferentials and normal cones in the sense of convex analysis and the obtained results are new in the literature of bilevel programming. \end{quote}$

December 6, 2023December 6, 2023 by alkhwarizmi

Trung Hieu Vu

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$\begin{quote} \begin{center} \textbf{Sum of Squares Decompositions of Polynomials over their Gradient Ideals with Rational Coefficients} \end{center} \medskip Assessing non-negativity of multivariate polynomials over the reals, through the computation of certificates of non-negativity, is a topical issue in polynomial optimization. This is usually tackled through the computation of sums-of-squares decompositions which rely on efficient numerical solvers for semi-definite programming. This method faces two difficulties. The first one is that the certificates obtained this way are approximate and then non-exact. The second one is due to the fact that not all non-negative polynomials are sums-of-squares. In this paper, we build on previous works by Parrilo, Nie, Demmel, and Sturmfels who introduced certificates of non-negativity modulo gradient ideals. We prove that, actually, such certificates can be obtained exactly, over the rationals if the polynomial under consideration has rational coefficients and we provide exact algorithms to compute them. We analyze the bit complexity of these algorithms and deduce bit size bounds of such certificates. The talk is based on joint work with Victor Magron and Mohab Safey El Din. \end{quote}$

December 6, 2023December 6, 2023 by alkhwarizmi

Zaïneb Bel Afia

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$\begin{quote} \begin{center} \textbf{Chebyshev varieties} \end{center} Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials. They arise when solving polynomial equations expressed in the Chebyshev basis. Chebyshev polynomials are widely used in optimization and numerical analysis, and their analytic aspects have been studied intensively. On the other hand, their algebraic properties remain relatively unexplored. This talk aims to bridge this gap. I will discuss defining equations of Chebyshev varieties and give some key properties. Through examples, I will motivate their use in effective computations. This is ongoing work with Chiara Meroni (Harvard University) and Simon Telen (MPI MiS). \end{quote}$