A dynamic gradient approach to Pareto optimization with nonsmooth convex objective functions

In a general Hilbert framework, we consider continuous gradient-like dynamical systems for constrained multiobjective optimization involving non-smooth convex objective functions. Our approach is in the line of a previous work where was considered the case of convex di erentiable objective functions. Based on the Yosida regularization of the subdi erential operators involved in the system, we obtain the existence of strong global trajectories. We prove a descent property for each objective function, and the convergence of trajectories to weak Pareto minima. This approach provides a dynamical endogenous weighting of the objective functions. Applications are given to cooperative games, inverse problems, and numerical multiobjective optimization

Possible physical universes

The purpose of this paper is to discuss the various types of physical universe which could exist according to modern mathematical physics. The paper begins with an introduction that approaches the question from the viewpoint of ontic structural realism. Section 2 takes the case of the 'multiverse' of spatially homogeneous universes, and analyses the famous Collins-Hawking argument, which purports to show that our own universe is a very special member of this collection. Section 3 considers the multiverse of all solutions to the Einstein field equations, and continues the discussion of whether the notions of special and typical can be defined within such a collection

Carath\'eodory Theory and A Priori Estimates for Continuity Inclusions in the Space of Probability Measures

In this article, we extend the foundations of the theory of differential inclusions in the space of probability measures with compact support, laid down recently in one of our previous work, to the setting of general Wasserstein spaces. Anchoring our analysis on novel estimates for solutions of continuity equations, we propose a new existence result ``\`a la Peano'' for this class of dynamics, under mere Carath\'eodory regularity assumptions. The latter is based on a set-valued generalisation of the semi-discrete Euler scheme proposed by Filippov to study ordinary differential equations with measurable right-hand sides. We also bring substantial improvements to the earlier versions of the Filippov theorem, compactness and relaxation properties of the solution sets of continuity inclusions which are derived in the more restrictive Cauchy-Lipschitz setting

A nonlinear structured population model: Global existence and structural stability of measure-valued solutions

This paper is devoted to the study of the global existence and structural stability of measure-valued solutions to a nonlinear structured population model given in the form of a nonlocal first-order hyperbolic problem on positive real numbers. In distinction to previous studies, where the L^1 norm was used, we apply the flat metric, similar to the Wasserstein W^1 distance. We argue that stability using this metric, in addition to mathematical advantages, is consistent with intuitive understanding of empirical data. Structural stability and the uniqueness of the weak solutions are shown under the assumption about the Lipschitz continuity of the kinetic functions. The stability result is based on the duality formula and the Gronwall-type argument. Using a framework of mutational equations, existence of solutions to the equations of the model is also shown under weaker assumptions, i.e., without assuming Lipschitz continuity of the kinetic functions