Archive 2025-2026

SPOT 102 – Lundi 12 janvier 2026

14h – Joseph Morlier (ISAE-SUPAERO) – Embedding Sustainability in Design Optimization: Which Mathematical Recipes?

This work presents mathematical methods to embed sustainability criteria, such as CO₂ footprint and Life Cycle Assessment (LCA), directly into multidisciplinary design optimization (MDO). The main challenge lies in handling environmental data coming from discrete material databases. Two classes of problems are addressed: P1, with fixed or imposed topology, and P2, with free topology. For P1, continuous relaxation techniques allow eco-material selection to be integrated into the global MDO loop. Applications include a solar-powered HALE aircraft minimizing CO₂ emissions through coupled aerodynamic, structural, and energy disciplines. Variational Autoencoders are introduced to map discrete material properties into a continuous latent space, enabling gradient-based multi-objective optimization. For P2, SIMP-based topology optimization is extended with environmental and manufacturing considerations. The approach is successfully applied to metallic and composite structures, relying on surrogate models to reduce computational cost.

15h – Cheik Traoré (Toulouse School of Economics) – Stochastic proximal methods and variance reduction

Stochastic algorithms, particularly stochastic gradient descent (SGD), have become the preferred methods in data science and machine learning. SGD is indeed efficient for large-scale problems. However, due to its variance, its convergence properties are unsatisfactory. This issue has been addressed by variance reduction techniques such as SVRG and SAGA. Recently, the stochastic proximal point algorithm (SPPA) emerged as an alternative and was shown to be more robust than SGD with respect to step size settings. In this talk, we will examine the SPPA algorithm. Specifically, we will demonstrate how variance reduction techniques can improve the convergence rates of stochastic proximal point methods, as has already been demonstrated for SGD.

SPOT 100-101 – Monday 13 October 2025

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A special event to celebrate the 100th edition of SPOT, with 4 talks during a whole optimization afternoon !

Curiosities and counterexamples in smooth convex optimization

Edouard Pauwels (Toulouse School of Economics (TSE))

13/10/2025 14:00

We present a list of counterexamples to conjectures in smooth convex coercive optimization. We will detail two extensions of the gradient descent method, of interest in machine learning: gradient descent with exact line search, and Bregman descent (also known as mirror descent). We show that both are non convergent in general. These examples are based on general smooth convex interpolation results. Given a decreasing sequence of convex compact sets in the plane, whose boundaries are Ck curves (k > 1, arbitrary) with positive curvature, there exists a Ck convex function for which each set of the sequence is a sublevel set. The talk will provide proof arguments for this results and detail how it can be used to construct the announced counterexamples.

Conjectures in Real Algebra and Polynomial Optimization through High Precision Semidefinite Programming

Michal Kočvara (University of Birmingham)

13/10/2025 14:50

We study degree bounds for the denominator-free Positivstellens »atze in real algebra, based on sums of squares (SOS), or equivalently the convergence rate for the moment-sums of squares hierarchy in polynomial optimization, from a numerical point of view. As standard semidefinite programming (SDP) solvers do not provide reliable answers in many important instances, we use a new high-precision SDP solver, Loraine.jl, to support our investigation. We study small instances (low-degree, small number of variables) of one-parameter families of examples, and propose several conjectures for the asymptotic behavior of the degree bounds. Our objective is twofold: first, to raise awareness on the bad performance of standard SDP solvers in such examples, and then to guide future research on the Eﬀective Positivstellensätze. Joint work with Lorenzo Baldi.

Weak optimal transport with moment constraints: constraint qualification, dual attainment and entropic regularization

Guillaume Carlier (Université Paris Dauphine)

13/10/2025 16:10

Weak optimal transport is a nonlinear version of the classical mass transport of Monge and Kantorovich which has received a lot of attention since its introduction by Gozlan Roberto, Samson and Tetali, ten years ago. In this talk, I will address weak optimal problems (possibly entropically penalized) incorporating both soft and hard (including the case of the martingale condition) moment constraints. Even in the special case of the martingale optimal transport problem, existence of Lagrange multipliers corresponding to the martingale constraint is notoriously hard (and may fail unless some specific additional assumptions are made). We identify a condition of qualification of the hard moment constraints (which in the martingale case is implied by well-known conditions in the literature) under which general dual attainment results are established. We also analyze the convergence of entropically regularized schemes combined with penalization of the moment constraint and illustrate our theoretical findings by numerically solving in dimension one, the Brenier-Strassen problem of Gozlan and Juillet and a family of problems which interpolates between monotone transport and left-curtain martingale coupling of Beiglböck and Juillet. This talk is based on a recent joint work with Hugo Malamut and Maxime Sylvestre.

The Moment-SOS hierarchy for computing: I: Mixtures of Gaussians closest (in W_2-Wasserstein distance) to a given measure. II: The total variation distance between two given probability measures

Jean-Bernard Lasserre (CNRS)

13/10/2025 17:00

We present two recent applications of the Moment-SOS hierarchy.
I. We first consider an optimal transport formulation for computing
mixtures of Gaussians that minimize the W_2-Wasserstein distance to a given measure.
II. We next consider the problem of computing the total variation between two given measures.

For each problem we provide an associated hierarchy of semidefinite relaxations that converges to
the desired result. Importantly, in both cases the support of the input measures is not assumed to be compact.
Finally, the approach for problem II can also be used to solve problem I with the TV distance rather than the
W_2 Wasserstein distance.