Integration of Fenchel subdifferentials revisited

We obtain a simple integration formula for the Fenchel subdifferentials on Euclidean spaces and analyze some of its consequences. For functions defined on locally convex spaces, we present a similar result in terms of ε-subdifferentials

Vector subdifferentials and tangent cones

Following the Rockafellar's definition for the subdifferential of a real map we define a vector subdifferential using the normal cone to the epigraph of the function. For several kinds of normal cones we have different subdifferentials; we give properties, links between them, links with addapted directional derivatives and a genaralization for the Correa Joffré Thibault and for Zagrodny theorem from the real case

Introduction to Nonsmooth Analysis and Optimization

This book aims to give an introduction to generalized derivative concepts useful in deriving necessary optimality conditions and numerical algorithms for infinite-dimensional nondifferentiable optimization problems that arise in inverse problems, imaging, and PDE-constrained optimization. They cover convex subdifferentials, Fenchel duality, monotone operators and resolvents, Moreau--Yosida regularization as well as Clarke and (briefly) limiting subdifferentials. Both first-order (proximal point and splitting) methods and second-order (semismooth Newton) methods are treated. In addition, differentiation of set-valued mapping is discussed and used for deriving second-order optimality conditions for as well as Lipschitz stability properties of minimizers. The required background from functional analysis and calculus of variations is also briefly summarized.Comment: arXiv admin note: substantial text overlap with arXiv:1708.0418

The Geometry and Calculus of Losses

Statistical decision problems lie at the heart of statistical machine learning. The simplest problems are binary and multiclass classification and class probability estimation. Central to their definition is the choice of loss function, which is the means by which the quality of a solution is evaluated. In this paper we systematically develop the theory of loss functions for such problems from a novel perspective whose basic ingredients are convex sets with a particular structure. The loss function is defined as the subgradient of the support function of the convex set. It is consequently automatically proper (calibrated for probability estimation). This perspective provides three novel opportunities. It enables the development of a fundamental relationship between losses and (anti)-norms that appears to have not been noticed before. Second, it enables the development of a calculus of losses induced by the calculus of convex sets which allows the interpolation between different losses, and thus is a potential useful design tool for tailoring losses to particular problems. In doing this we build upon, and considerably extend existing results on $M$-sums of convex sets. Third, the perspective leads to a natural theory of ``polar'' loss functions, which are derived from the polar dual of the convex set defining the loss, and which form a natural universal substitution function for Vovk's aggregating algorithm.Comment: 65 pages, 17 figure

Orlicz-type Function Spaces and Generalized Gradient Flows with Degenerate Dissipation Potentials in Non-Reflexive Banach Spaces: Theory and Application

This thesis explores two important areas in the mathematical analysis of nonlinear partial differential equations: Generalized gradient flows and vector valued Orlicz spaces. The first part deals with the existence of strong solutions for generalized gradient flows, overcoming challenges such as non-coercive and infinity-valued dissipation potentials and non-monotone subdifferential operators on non-reflexive Banach spaces. The second part focuses on the study of Banach-valued Orlicz spaces, a flexible class of Banach spaces for quantifying the growth of nonlinear functions. Besides improving many known results by imposing minimal assumptions, we extend the theory by handling infinity-valued Orlicz integrands and arbitrary Banach-values in the duality theory. The combination of these results offers a powerful tool for analyzing differential equations involving functions of arbitrary growth rates and leads to a significant improvement over previous results, demonstrated through the existence of weak solutions for a doubly nonlinear initial-boundary value problem of Allen-Cahn-Gurtin type.Comment: Doctoral thesi