Alexander Ioffe 教授（イスラエル工科大学）の来日について
凸解析・変分学の専門家 A. Ioffe 教授が慶應義塾大学経済学部特別招聘教授として来日され、いろいろな形で研究活動に参加されると同時に連続講義もして下さいます。また教授の業績に対し慶應義塾大学は名誉博士の学位を授与することも決まっております。今日までに決まっているスケジュールをご紹介致します。

連続講義Ⅰ
慶應義塾大学大学院経済研究科における講義□講義題目
Introduction to Modern Variational Analysis:
Stability, Regularity and Critical Points□開催日時
４月５日（水）、１２日（水）、１９日（水）、２６日（水）水曜日 １０時４５分〜１２時１５分□場所

５月８日（月）月曜日 １５時〜１７時３０分
慶應義塾大学三田キャンパス
水曜日 ４２１ 教室
月曜日 ４５６-Ｂ 教室□講義内容
The course will be devoted to mathematical theories related to two problems that can beloosely formulated as follows:(a) what is a regular behavior of a system;(b) how the behavior of a system is affected by perturbations (of the system) .
Modern regularity theory, unlike its classical predecessor, is basically quantitative,
so the main attention in the lectures will be paid to development, calculation andestimation of quantitative measures of regularity. Most of the results will be accompaniedby explanations rather than rigorous proofs. But several proofs which either contain
important ideas or are widely applicable within and outside the theory will be given in fulldetails.
Lecture 1. INTRODUCTION. LINEAR SYSTEMS.
Examples of regular behavior. Banach open mapping theorem (quantitative version).
Perturbations of regular (surjective) linear operators. Fredholm operators. Stability ofindex.
Lecture 2. DIFFERENTIABLE MAPPINGS.
Frechet differentiability. Theorems of Lusternik and Graves. Applications: general inverse
and implicit functions theorems; exact penalties and necessary optimality conditions insmooth constraint optimization. Critical values of smooth maps. Sard-Smale theorem.
Transversality.
Lecture 3. GENERAL THEORY: SET-VALUED MAPPINGS IN METRIC SPACES.
Basic concepts of regularity theory: openness, Lipschitz stability, metric regularity.
Equivalence theorem. Variational principle of Ekeland. Error bounds Criteria ofregularity and estimates for regularity constants. Additive perturbations and radius ofregularity. The finite dimensional theory.
Lecture 4. CONVEX SET-VALUED MAPPINGS
Main concepts of convex analysis (separation, Minkowski and Fenchel duality,
subdifferential calculus, convex processes, constraint systems). Set-valued mappings
with convex graphs, their derivatives and coderivatives. Theorem of Robinson-Ursescu.
Calculation of regularity rates. Application: distance to infeasibility; constraint systems
of convex and convex semiinfinite programming. Error bounds for convex functions andconstraint systems. Stability of solutions and well-posedness of convex optimization
problems.
Lecture 5. GENERICITY PROPERTIES
Critical values of set-valued mappings. Small sets. Set-valued mappings with stratifiedgraphs. Basic notions and facts from the theory of o-minimal structures (monotonicity
theorem, uniform finiteness theorem, cell decomposition theorem, definable choicetheorem). Main examples (semilinear functions, semialgebraic functions, globally subanalytic functions). Optimization of definable functions: gradient descent and Newton
method. Morse-Sard theorem for critical and asymptotically critical values of definable
set-valued mappings and set-valued mappings admitting smooth stratification.

連続講義Ⅱ
慶應義塾大学大学院理工学研究科における講義□講義題目
Variational Analysis and Non-Smooth Regularity□開催日時
４月２７日（木）、４月２８日（金）、５月１１日（木）１６時３０分〜１８時□場
所
慶應義塾大学矢上台キャンパス
４月２７日（木）、５月１１日（木）１４棟２０３教室
４月２８日（金） 未定□講義内容
Suppose we have a family of problems depending on a certain parameter. What will
happen with solutions when parameter changes? How it is possible to know whether thebehavior of solutions is sufficiently regular in one or another sense or to guarantee such
behavior? The classical analysis has developed a variety of tools to deal with suchproblems: implicit function theorem, Lusternik-Graves theorem, Sard's theorem, Thom'stransversality theorem. Modern variational analysis is concerned with extensions of such
results and basic ideas behind them to ̀̀bad'' maps and functions. The main stimulus tothese studies comes from optimization theory and calculus of variations.
Lecture 1. METRIC REGULARITY THEORY.
Introduction to the subject. Ekeland's variational principle and error bounds. Definitionand various characterizations of regular points of mappings between metric spaces.
Lecture 2. SUBDIFFERENTIAL CALCULUS AND REGULARITY IN Banach SPACES.
Brief introduction to the theory of subdifferentials of non-differentiable functions onBanach spaces, subdifferential characterization of regular and critical behavior,
non-smooth extensions of the Lusternik-Graves and implicit function theorems.
Lecture 3. SARD THEOREM FOR STRATIFIED AND DEFINABLE MAPPINGS.
A brief account of the theory of definable set in o-minimal structures and a sketch of theproof of an extension of Sard's theorem to non-smooth definable mappings and, moregenerally, to set-valued mappings whose graphs admit stratifications of a slightly weaker
type than Whitney stratification.

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