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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

The Basics of Practical Optimization, Second Edition

Multisets are collections of elements which may appear with multiplicity. We review the multiplicity function and cardinality, and we introduce the notion of filter to extract the distinct elements in a multiset. We define a notion of outer limit for sequences of multisets, and we define and develop “multiset mappings” that take multisets to multisets. We introduce a generalized continuity property for multiset mappings (called calmness) that relies on choices of pre-distance functions, and we investigate how this property generalizes a similar property for set-valued mappings. We define a generalized derivative for multiset mappings and use it to characterize calmness.

Multisets and Multiset Mappings

We illustrate simulated basins of attraction for four example objective functions applying the same four different numerical methods to each (coordinate-search, and steepest-descent with three different line-searches). We define basin size and basin entropy (quantifying basin complexity) and compute each for our examples. We apply the Nelder–Mead method to all four example functions in a separate section because its iterate-multisets are non-singleton, which requires more complicated illustrations. We use the same tools to investigate the practical significance of two well-known counterexamples to good convergence behavior in numerical minimization: the canoe function with coordinate-search and McKinnon’s function (McKinnon, SIAM J. Optim. 9, 148–158 (1998)) with Nelder–Mead. We use our notions of basin size and basin entropy to quantify the extent to which initial data are likely to lead to undesirable consequences.

Adam B. Levy

Papers

The Basics of Practical Optimization, Second Edition

Multisets and Multiset Mappings

Basin Analysis via Simulation