Alexandru Ioan Cuza University of Ia si [610096]

"Alexandru Ioan Cuza" University of Ia¸ si
Faculty of Mathematics
Habilitation Thesis
Optimality conditions in vector optimization
A view through scalarization methods
and metric regularity of mappings
Author: Marius DUREA
Ia¸ si, 2012

To my family

Contents
I Abstract 4
1 Abstract –English version 5
2 Rezumat –versiunea în limba român ¼a 7
II Scienti…c achievements and evolution plan 9
3 Scienti…c achievements 10
3.1 Necessary optimality conditions for (weak) Pareto e¢ ciency . . . . . . . . . 17
3.1.1 Solid set-valued optimization . . . . . . . . . . . . . . . . . . . . . . . 17
3.1.2 On the behavior of Lagrange multipliers set . . . . . . . . . . . . . . 31
3.2 Necessary optimality conditions for special types of solutions . . . . . . . . . 46
3.2.1 Sharp solutions: single-valued case . . . . . . . . . . . . . . . . . . . 46
3.2.2 Sharp solutions: set-valued case . . . . . . . . . . . . . . . . . . . . . 56
3.2.3 Lagrange claims for set-valued maps . . . . . . . . . . . . . . . . . . 70
3.2.4 Approximate solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3 Openness for set-valued maps and its applications in set-valued optimization 79
3.3.1 Implicit multifunction theorems and applications . . . . . . . . . . . 79
3.3.2 Chain rules for openness . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.3.3 Conditions for openness and optimality conditions: general method . 108
3.3.4 Other conditions: strong slope approach and primal space approach . 123
3.4 Stability issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.4.1 Stability of approximate solutions sets . . . . . . . . . . . . . . . . . 135
3.4.2 Pointwise well-posedness in vector optimization . . . . . . . . . . . . 143
3.4.3 Regularization of set-valued maps . . . . . . . . . . . . . . . . . . . . 150
4 Further possible developments 160
4.1 Research directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
4.2 Didactic activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5 Bibliography 166
3

Part I
Abstract
4

Chapter 1
Abstract –English version
The main purpose of this thesis is to give an informal account on the scienti…c achievements
and the intended evolution plan of the author’ s academic career. Therefore, the present work
includes the most important results obtained by the author, alone or in collaboration, after
its PhD thesis defence in 2004 as well as a plan for further career possible developments.
The topic of this thesis is part of the general domain of nonlinear programming with
applications in Economics, a very important …eld from theoretical as well as from practical
point of view. We split the author’ s scienti…c achievements into three major directions which
are uni…ed by convergence of objectives and methods. The objectives concern the investiga-
tion of vector optimization problems with nonsmooth data from point of view of optimality
conditions, and stability analysis, while the methods are those of nonlinear and variational
analysis. In fact, we initiate and develop several methods of study for the proposed problems
and we show that, in the process, several important related issues come into attention and
should be deeply investigated.
A short motivation of the research themes presented here is that to optimize general
models arising from practice was always a goal of modern mathematics and besides scalar
programming which was initially considered, the multicriteria programming or vectorial pro-
gramming is known for a long time in Economics. The growing interest for vectorial pro-
gramming of the Optimization mathematical community is proved by the active research in
this …eld of some well internationally recognized mathematicians among which we mention
here J. Borwein, J. Jahn, B. Jimenez, D. T. Luc, B. S. Mordukhovich, V. Novo, C. Tammer,
L.Thibault, C. Z ¼alinescu. In the last …fty years it was noticed that the degree of generality of
the models should grow in order to cover practical demands and the replacement of objective
functions by multifunctions is necessary. Among other necessary tools for this newly consid-
ered framework, the question of introducing di¤erentiability notions for set-valued maps has
appeared. There are several concepts in this respect, but probably the best construction in
terms of theory is the Mordukhovich coderivative and the related generalized di¤erentiation
objects.
Several works of Robinson in 1970s and early 1980s made clear the idea that there is
a tight connection between some already classical results in theory of stability in scalar
optimization and some conditions naturally arising from the study of metric regularity of
the constraint systems. In turn, the equivalent properties of metric regularity and openness
5

at linear rate of set-valued maps have an important history being nowadays the part of the
general topic in nonsmooth analysis called variational analysis. A landmark of this research
area was the famous Robinson-Ursescu Theorem and a series of works of Milyutin in 1970s
concerning the preservation of regularity under functional perturbation. Since 1980s to our
time many mathematicians have participated into a joint e¤ort on the development and
understanding of these problems. We mention here only a few, having major contributions
to the …eld: J. P. Aubin, A. Dontchev, H. Frankowska, A. Io¤e, B. S. Mordukhovich, J.-P.
Penot, S. M. Robinson, R. T. Rockafellar, C. Ursescu.
In the last ten years, the author of this thesis was involved in the research areas mentioned
above (i.e. vector optimization and the study of openness at linear rate of set-valued maps
by the use variational methods in nonsmooth analysis) and the results of the present thesis
are in dialog with several issues which are recurrent in the works of the above mentioned
authors.
Therefore, the Scienti…c achievements chapter (i.e. Chapter 3) follows the author’ s most
important works in a sequence where the …rst criterion is not the chronology, but the devel-
opments of the leading ideas.
Having this principle in mind, we divided this main chapter into four sections entitled as
follows: Necessary optimality conditions for (weak) Pareto e¢ ciency (Section 3.1), Neces-
sary optimality conditions for special types of solutions (Section 3.2), Openness for set-valued
maps and its applications in set-valued optimization (Section 3.3), and Stability issues (Sec-
tion 3.4). Notice that, before the start of the …rst section, we have presented the framework
and the main notations used in the sequel. Section 3.1 contains, on one hand, results on solid
vector optimization (especially governed by set-valued maps) obtained by means of scalar-
ization methods and, on the other hand, the roots for the developments in the subsequent
sections. Namely, the limitations of the study of the classical types of solutions exclusively
by scalarization become apparent and this leads to consider special types of solutions (Sec-
tion 3.2), to develop new methods in non solid optimization (Section 3.3) and to identify
several stability issues (Section 3.4). Every one of these sections are in turn divided follow-
ing the speci…c needs of the study therein; we list here the most representative questions
which are extensively investigated: solid set-valued optimization (Subsection 3.1.1), sharp
solutions (Subsections 3.2.1, and 3.2.2), implicit multifunctions theorems (Subsection 3.3.1),
openness and optimality conditions (Subsection 3.3.3), well-posedness in vector optimiza-
tion (Subsection 3.4.2). Throughout, the main tools are various generalized di¤erentiation
objects for sets (tangent and normal cones), single-valued and set-valued maps (subdi¤eren-
tials and derivatives, coderivatives, respectively) and related calculus rules on appropriate
classes of Banach spaces. Besides these questions, we study as well several related aspects
of the theory: behavior of Lagrange multipliers sets in smooth and nonsmooth vector opti-
mization (Subsection 3.1.2), Lagrange claims for set-valued maps (Subsection 3.2.3), chain
rules for openness at linear rate for multifunctions (Subsection 3.3.2), a lower semicontinuous
regularization method for set-valued maps (Subsection 3.4.3).
Chapter 4 contains some ideas on further developments of author’ s academic and scienti…c
career and the main considerations concern the publishing, didactic and guiding of young
people activities. An extended bibliography which consist of 137 titles ends the thesis.
6

Chapter 2
Rezumat –versiunea în limba român ¼a
Principalul scop al acestei teze este acela de a prezenta într-o manier ¼a concis ¼a cele mai
importante realiz ¼ari ¸ stiin¸ ti…ce ale autorului precum ¸ si un plan al evolu¸ tiei ulterioare a carierei
academice a acestuia. Drept urmare, lucrarea de fa¸ t ¼a include cele mai importante rezultate
ob¸ tinute de autor, singur sau în colaborare, dup ¼a momentul sus¸ tinerii tezei de doctorat în
2004 precum ¸ si un plan al posibilelor dezvolt ¼ari ale carierei.
Tematica tezei este parte a domeniului mai general al program ¼arii neliniare cu aplica¸ tii în
economie, un domeniu de cercetare foarte important, atât din punct de vedere teoretic cât ¸ si
practic. Am împ ¼ar¸ tit realiz ¼arile ¸ stiin¸ ti…ce ale autorului în trei direc¸ tii majore care sunt unite
prin convergen¸ ta obiectivelor ¸ si a metodelor de studiu. Obiectivele se refer ¼a la investigarea
problemelor de optimizare vectorial ¼a cu date nenetede din punct de vedere al condi¸ tiilor de
optimalitate ¸ si al stabilit ¼a¸ tii, în timp ce metodele sunt proprii analizei neliniare ¸ si analizei
varia¸ tionale. De fapt, ini¸ tiem ¸ si dezvolt ¼am mai multe metode de studiu a problemelor propuse
¸ si, pe parcurs, detect ¼am mai multe probleme importante care trebuie analizate în profunzime.
O scurt ¼a motivare a temelor de cercetare prezentate aici este dat ¼a de faptul c ¼a a opti-
miza modelele generale ce se ivesc în practic ¼a a fost mereu un scop important al matematicii
moderne, iar, pe lâng ¼a programarea scalar ¼a care a fost considerat ¼a ini¸ tial, programarea mul-
ticriterial ¼a sau vectorial ¼a este de mult timp cunoscut ¼a în economie. Interesul în cre¸ stere al
matematicienilor care se ocup ¼a de optimizare pentru programarea vectorial ¼a este probat de
activitatea de cercetare în acest domeniu a unor matematicieni bine cunoscu¸ ti printre care
men¸ tion ¼am pe J. Borwein, J. Jahn, B. Jimenez, D. T. Luc, B. S. Mordukhovich, V. Novo,
C. Tammer, L.Thibault, C. Z ¼alinescu. În ultimii cincizeci de ani s-a constatat c ¼a gradul de
generalitate a modelelor trebuie s ¼a creasc ¼a pentru a acoperi necesit ¼a¸ tile practice ¸ si de aseme-
nea este important ¼a înlocuirea func¸ tiilor obiectiv prin multifunc¸ tii. Toate acestea necesit ¼a
noi unelte ¸ si, printre altele, a ap ¼arut problema introdurerii no¸ tiunilor de diferen¸ tiabilitate
pentru multifunc¸ tii. Ast ¼azi, exist ¼a mai multe concepte de acest tip, dar probabil c ¼a cel mai
bun din punct de vedere teoretic este coderivata Mordukhovich înpreun ¼a cu celelalte obiecte
înrudite legate de teoria diferen¸ tierii generalizate.
Mai multe articole ale lui Robinson din anii 1970 ¸ si începutul anilor 1980 au eviden¸ tiat
idea unei strânse conexiuni între unele rezultate clasice ale teoriei stabilit ¼a¸ tii din optimizarea
scalar ¼a ¸ si unele condi¸ tii care apar în mod natural în studiul regularit ¼a¸ tii metrice a sistemelor
de constrângeri. La rândul lor, propriet ¼a¸ tile echivalente de regularitate metric ¼a ¸ si deschidere
7

cu rat ¼a liniar ¼a a multifunc¸ tiilor au o istorie deja imporant ¼a, devenind parte integrant ¼a a
analizei varia¸ tionale. Rezultatele fundamentale pentru aceast ¼a direc¸ tie de cercetare au fost
faimoasa Teorem ¼a Robinson-Ursescu ¸ si o serie de lucr ¼ari ale lui Milyutin din anii 1970 pri-
vitoare la p ¼astrarea regularit ¼a¸ tii la perturb ¼ari func¸ tionale. Din anii 1980 pân ¼a acum, mul¸ ti
matematicieni au participat într-un efort comun de dezvoltare ¸ si în¸ telegere a acestor proble-
me. Men¸ tion ¼am aici doar c⸠tiva dintre cei care au avut contribu¸ tii majore în acest domeniu:
J. P. Aubin, A. Dontchev, H. Frankowska, A. Io¤e, B. S. Mordukhovich, J.-P. Penot, S. M.
Robinson, R. T. Rockafellar, C. Ursescu.
În ultimii zece ani, autorul acestei teze a fost implicat in domeniile de cercetare men-
¸ tionate anterior (i.e. optimizarea vectorial ¼a ¸ si studiul deschiderii liniare cu rat ¼a liniar ¼a a
multifunc¸ tiilor prin intermediul metodelor varia¸ tionale din analiza neliniar ¼a) iar rezultatele
din teza de fa¸ t ¼a sunt în dialog cu mai multe probleme care apar cu insisten¸ t ¼a în lucr ¼arile
autorilor mai sus men¸ tiona¸ ti. Prin urmare, capitolul privitor la Realiz ¼arile ¸ stiin¸ ti…ce (i.e.
Capitolul 3) urmeaz ¼a cele mai importante lucr ¼ari ale autorului într-o succesiune în care
primul criteriu nu este cronologia, ci dezvoltatea ideilor fundamentale.
Având în vedere acest principiu, am împ ¼ar¸ tit acest capitol principal în patru sec¸ tiuni inti-
tulate dup ¼a cum urmeaz ¼a:Condi¸ tii necesare de optimalitate pentru e…cienta (slab ¼a) Pareto
(Sec¸ tiunea 3.1), Condi¸ tii necesare de optimalitate pentru tipuri speciale de solu¸ tii (Sec¸ ti-
unea 3.2), Deschiderea aplica¸ tiilor multivoce ¸ si aplica¸ tii în optimizare (Sec¸ tiunea 3.3), and
Probleme de stabilitate (Sec¸ tiunea 3.4). Înaintea startului primei sec¸ tiuni men¸ tionate sunt
prezentate cadrul ¸ si principalele nota¸ tii utilizate în continuare. Sec¸ tiunea 3.1 con¸ tine, pe
de o parte, rezultate de optimiziare vectoril ¼a solid ¼a (mai ales pentru probleme guvernate
de multifunc¸ tii) ob¸ tinute folosind metode de scalarizare ¸ si, pe de alt ¼a parte, r ¼ad¼acinile dez-
volt¼arilor din sec¸ tiunile ulterioare. Mai exact, limit ¼arile studiului solu¸ tiilor clasice exclusiv
prin scalarizare devin evidente ¸ si acestea conduc la considerarea altor tipuri de solu¸ tii (Sec¸ ti-
unea 3.2), la dezvoltarea unor noi metode în optimizarea nesolid ¼a (Sec¸ tiunea 3.3) ¸ si la iden-
ti…carea mai multor probleme de stabilitate (Sec¸ tiunea 3.4). Fiecare din aceste sec¸ tiuni este,
la rândul s ¼au, împ ¼ar¸ tit¼a conform studiului speci…c corespunz ¼ator; prezent ¼am aici cele mai
reprezentative probleme investigate: optimizare solid ¼a cu multifunc¸ tii (Subsec¸ tiunea 3.1.1),
solu¸ tii exacte (Subsec¸ tiunile 3.2.1 ¸ si 3.2.2), teoreme implicite pentru multifunc¸ tii (Subsec¸ ti-
unea 3.3.1), deschidere ¸ si condi¸ tii de optimalitate (Subsec¸ tiunea 3.3.3), probleme vectoriale
bine puse (Subsec¸ tiunea 3.4.2). Pe parcursul lucr ¼arii, principalele unelte sunt mai multe
tipuri de obicte de diferen¸ tiabilitate generalizat ¼a pentru mul¸ timi (conuri tangente ¸ si nor-
male), func¸ tii ¸ si multifunc¸ tii (subdiferen¸ tiale ¸ si respectiv, derivate ¸ si coderivate) precum ¸ si
reguli de calcul asociate pe diverse clase de spa¸ tii Banach. În afar ¼a de aceste chestiuni prin-
cipale, studiem de asemenea mai multe chestiuni înrudite din cadrul teoriei: comportarea
mul¸ timilor de multiplicatori Lagrange în optimizarea vectorial ¼a neted ¼a ¸ si neneted ¼a (Sub-
sec¸ tiunea 3.1.2), aser¸ tiuni Lagrange pentru multifunc¸ tii (Subsec¸ tiunea 3.2.3), reguli de tip
lan¸ t pentru deschiderea cu rat ¼a liniar ¼a a multifunc¸ tiilor (Subsec¸ tiunea 3.3.2), o metod ¼a de
regularizare inferior semicontinu ¼a a aplica¸ tiilor multivoce (Subsec¸ tiunea 3.4.3).
Capitolul 4 con¸ tine câteva idei asupra dezvolt ¼arii ulterioarea a carierei ¸ stiin¸ ti…ce ¸ si acade-
mice a autorului, iar principalele considera¸ tii se refer ¼a la activitatea publicistic ¼a, activitatea
didactic ¼a ¸ si la activitatea de coordonare a tinerilor cu aptitudini pentru cercetare. O bibli-
ogra…e extins ¼a ce cuprinde 137 de titluri încheie teza.
8