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Convex Analysisの二,三の進展について

The purpose of this self-contained work is to provide a reference on the topic and to provide a unified collection of a number of results which are currently scattered throughout the literature. The first chapter of the book treats the classical implicit function theorem in a way that will be useful for students and teachers of undergraduate calculus. The remaining part becomes gradually more advanced, and considers implicit mappings defined by relations other than equations, e.g., variational problems. Applications to numerical analysis and optimization are also provided.

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Implicit Functions and Solution Mappings

This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Set-valued analysis

Functions Defined Implicitly by Equations.- Implicit Function Theorems for Variational Problems.- Regularity properties of set-valued solution mappings.- Regularity Properties Through Generalized Derivatives.- Regularity in infinite dimensions.- Applications in Numerical Variational Analysis.

Implicit Functions and Solution Mappings: A View from Variational Analysis

Recently Clarke, Stern and Wolenski characterized, in a Hilbert space, the closed subsets C for which the distance function dC is continuously differentiable everywhere on an open “tube” of uniform thickness around C Here a corresponding local theory is developed for the property of dC being continuously differentiable outside of C on some neighborhood of a point x ∈ C This is shown to be equivalent to the prox-regularity of C at x, which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation Additional characterizations are provided in terms of dC being locally of class C 1+ or such that dC + σ| · |2 is convex around x for some σ > 0 Prox-regularity of C at x corresponds further to the normal cone mapping NC having a hypomonotone truncation around x, and leads to a formula for PC by way of NC  The local theory also yields new insights on the global level of the Clarke-Stern-Wolenski results, and on a property of sets introduced by Shapiro, as well as on the concept of sets with positive reach considered by Federer in the finite dimensional setting

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Local differentiability of distance functions

We introduce and study decompositions of finite sets as well as coverings of their convex hulls, and use these objects to develop various estimates of and formulas for the “hull-volume” of the sets (i.e., the volume of their convex hull). We apply our results to the convergence analysis of the “iterate-sets” associated with each iteration of a reduce-or-retreat optimization method (including pattern-search methods like Nelder–Mead as well as model-based methods).

Hull-Volume with Applications to Convergence Analysis

S INCE 1963 when first performed in humans, liver transplantations have made spectacular developments due to improved liver preservation techniques, surgical techniques, and drug therapy. Because of its complex metabolism the liver can cause a complex pathophysiology and for liver transplantations this requires the use of adjunct perioperative drugs that have to be carefully balanced with each other to avoid negative side effects. As one of the largest of all abdominal operations, liver transplantations are a challenge for surgeons and anesthesiologists alike. This article will provide an overview of adjunct drugs used in this procedure.

Adjunct drugs in liver transplantation

Four particular examples of methods within our framework are explored in detail, including generalized line-search and trust-region methods, as well as a general pattern-search method and the classic Nelder–Mead method. In each case, we consider descent, and develop conditions ensuring lower-diminishing target-gaps and an approach to stationarity.

Particular Reduce-or-Retreat Methods

We introduce our framework, including all of the elements generated at each iteration, as well as the major components of reduction and retreat. The property of nonincreasing function values inherent in the framework is strengthened via the definition of a descent method, and lower-diminishing target-gaps are explored as the result of a successful run of a method within the framework. Finally, we develop our extended notion of approaching stationarity and link this property both to stationary points and to lower-diminishing target-gaps.

A Framework for Reduce-or-Retreat Minimization

We review some basic concepts from classical dynamical systems, including equilibria, phase space, trajectories, basins of attraction, stability, and asymptotic stability. We also highlight some recently introduced concepts of basin size and basin entropy. We present subgradient systems as generalizations of gradient systems and connect the equilibria of these to the minimizers of the objective function underlying the system.

Adam B. Levy

Papers

Hull-Volume with Applications to Convergence Analysis

Adjunct drugs in liver transplantation

Particular Reduce-or-Retreat Methods

A Framework for Reduce-or-Retreat Minimization

Equilibria in Dynamical Systems