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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 a new class of multifunctions whose graphs under certain "kernel inverting" matrices, are locally equal to the graphs of Lipschitzian (single-valued) mappings. We characterize the existence of Lipschitzian localizations of these multifunctions in terms of a natural condition on a generalized Jacobian mapping. One corollary to our main result is a Lipschitzian inverse mapping theorem for the broad class of "max hypomonotone" multifunctions. We apply our theoretical results to the sensitivity analysis of solution mappings associated with parameterized optimization problems. In particular, we obtain new characterizations of the Lipschitzian stability of stationary points and Karush-Kuhn-Tucker pairs associated with parameterized nonlinear programs.

Lipschitzian Multifunctions and a Lipschitzian Inverse Mapping Theorem

Purpose
To report the case of a laparoscopic donor nephrectomy in which the preoperative evaluation of the patient gave no indication of the surgical difficulties that were encountered intraoperatively, resulting in substantial bleeding, a suspected gas embolism, and emergency conversion of the procedure from laparoscopic to open donor nephrectomy.

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Unexpected surgical difficulties leading to hemorrhage and gas embolus during laparoscopic donor nephrectomy: a case report.

Calmness is a restricted form of local Lipschitz continuity where one point of comparison is fixed. We study the calmness of solutions to parameterized optimization problems of the form $$ \min \{f(x,w)\} \hbox{ over all } x \in\reals^{n}, $$ where the extended real-valued objective function f is continuously prox-regular in x with compatible parameterization in w. This model covers most finite-dimensional optimization problems, though we focus particular attention here on the case of parameterized nonlinear programming. We give a second-order sufficient condition for there to exist unique optimal solutions that are calm with respect to the parameter. We also characterize a slightly stronger stability property in terms of the same second-order condition, thus clarifying the gap between our sufficient condition and the calmness property. In the case of nonlinear programming, our results complement a long study of the stability properties of optimal solutions: for instance, one consequence of our results is that the Mangasarian--Fromovitz constraint qualification when paired with a new (and relatively weak) second-order condition ensures the calmness of solutions to parameterized nonlinear programs.

Calm Minima in Parameterized Finite-Dimensional Optimization

Primal lower-nice functions defined on Hilbert spaces provide examples of functions that are “integrable” (i.e. of functions that are determined up to an additive constant by their subgradients). The class of primal lowernice functions contains all convex and lower-C2 functions. In finite dimensions the class of primal lower-nice functions also contains the composition of a convex function with a C2 mapping under a constraint qualification. In Banach spaces certain convex composite functions were known to be primal lower-nice (e.g. a convex function had to be continuous relative to its domain). In this paper we weaken the assumptions and provide new examples of convex composite functions defined on a Banach space with the primal lower-nice property. One consequence of our results is the identification of new examples of integrable functions on Hilbert spaces.

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Convex composite functions in Banach spaces and the primal lower-nice property

BACKGROUND Dietary supplementation with a fish oil and arginine-enriched immunoenhancing diet (Impact; Sandoz Nutrition, Minneapolis, MN) in a rat cardiac allograft model using donor-specific transfusion (DST) and cyclosporin (CsA) resulted in significant prolongation of cardiac allograft survival with many animals developing long-term tolerance. This study was done to determine whether arginine or fish oil was the active ingredient. METHODS A standard AIN-76A diet was modified to include either 10% fish oil, 2% arginine, or 5% arginine with or without fish oil. Diets were fed to Lewis strain rats that received Ax C9935 Irish (ACI) heterotopic cardiac allografts beginning on day 1 and continuing indefinitely. A DST (1.0 mL ACI whole blood) was given with 10 mg/kg CsA on day 1 relative to transplant and 2.5 mg/kg/d on days 0 to 6. Groups of animals receiving AIN-76A diet fortified with 2% glycine and animals receiving a DST or DST/CsA and regular laboratory chow served as controls. RESULTS Mean survival times +/- SEM in days were as follows: untreated, 7.1 +/- 0.4; CsA/2% glycine, 8.5 +/- 0.6; DST only, 9.6 +/- 1.1; DST/CsA, 26.6 +/- 6.4; CsA/2% arginine, 25.5 +/- 3.9; DST/CsA/2% arginine, 68.7 +/- 8.9; DST/CsA/5% arginine, 90.1 +/- 31.1; CsA/fish oil, 73.6 +/- 26.1; and DST/CsA/fish oil/5% arginine, 90.1 +/- 31.1. The effect of arginine was slightly dose dependent and was seen best in combination with DST, but the effect of fish oil was not enhanced by DST. CONCLUSIONS Both fish oil and arginine dietary supplementation significantly improved allograft survival but through different mechanisms (DST vs non-DST dependent).

Adam B. Levy

Papers

Lipschitzian Multifunctions and a Lipschitzian Inverse Mapping Theorem

Unexpected surgical difficulties leading to hemorrhage and gas embolus during laparoscopic donor nephrectomy: a case report.

Calm Minima in Parameterized Finite-Dimensional Optimization

Convex composite functions in Banach spaces and the primal lower-nice property

Arginine, Fish Oil, and Donor‐Specific Transfusions Independently Improve Cardiac Allograft Survival in Rats Given Subtherapeutic Doses of Cyclosporin