In this section, we are interested in finding some conditions for a multifunctionF to have a selectionf with the same subdifferential.
Proposition 16.1 Let F:X→→Z be a convex continuous multifunction with Dom F = X and assume that Z+ has a compact base. Then F has a convex con-tinuous selection f:X →Z with ∂≤f(x) =∂≤F(x)for all x in X if and only if F(x)has an ideal minimum for all x∈X and there exists k in Z such that
∀x∈X, f(x) = IMIN(F(x)) +k.
Proof. IfF has a convex continuous selectionf, then from [90] Theorem 4.3 for anyxin X, ∂≤f(x) is nonempty and thus ∂≤F(x) is nonempty. This implies that IMIN(F(x))= ∅ for all x in X. Let us remark that if Z+ has a compact base then Z+ is normal. Since F is convex and continuous from Proposition 15.3, the mapping m:x )→ IMIN(F(x)) is continuous, convex and
∂≤IMIN(F(x)) =∂≤F(x), for all x∈X Thus, there exists k in Z such that for all x
f(x) = IMIN(F(x)) +k.
The converse implication is obtained thanks to the convexity and continuity of the selection x)→IMIN(F(x)) and the fact that ∂≤IMIN(F(x)) =∂≤F(x) for all x. $ Proposition 16.2 Under the assumptions of Proposition 16.1, we obtain a convex contin-uous selection f:X →Z of the multifunction F such that ∂≤F(x) =∂>f(x) for all x∈ X, if and only if for all x∈X we have ∂≤F(x) =∂>F(x), F(x)has an ideal minimum and
f(x) = IMIN(F(x)) +k.
Proof. Iff is a selection for the multifunctionF which has the required properties, then
∂>f(x) = ∅, for all x in X (f is continuous and convex and the main result of Zowe [90]
implies that ∂≤f(x)=∅, ∀x∈X and that ∂≤f(x)⊂∂>f(x)).
As a result,∂≤F(x)=∅ which implies that IMIN(F(x))=∅, ∀x∈X.
From the equality∂≤F(x) =∂>f(x), valid for every x∈X, we obtain that
z∗◦∂>f(x)⊆∂≤(z∗◦F)(x), ∀x∈X, ∀z∗ ∈Z+∗ (89).
For all z∗ ∈Z+∗ \ {0}, from [38] we have
∂≤(z∗◦f)(x)⊆z∗◦∂≤f(x)⊆z∗◦∂>f(x)⊆∂≤(y∗◦F)(x), ∀x∈X.
Hence we have
∂≤(z∗◦f)(x) =∂≤(z∗◦F)(x), ∀x∈ X.
Relation (97) implies that ∂∗f(x) ⊇ ∂>f(x). From Remark 15.4 we know that ∂≤f(x) =
∂∗f(x), ∀x ∈X and since∂≤f(x)⊆∂>f(x)∀x∈ X we get that ∂≤f(x) =∂>f(x), ∀x∈X.
Finally we have∂≤F(x) =∂≤f(x), ∀x∈Xwhich implies thatf(x) = IMIN(F(x))+k, ∀x∈ X. The converse implication is obvious, because the application f:x )→ IMIN(F(x)) +k is convex, continuous and∂≤F(x) =∂≤f(x) =∂>f(x), ∀x∈X. $ Remark 16.1 From proposition 4.6 we derive that if the ordering relation is not total, under the hypothesis of the precedent proposition, it does not exists a convex continuous selectionf for a convex continuous multifunctionF with ∂≤F(x) = ∂>f(x) for all x∈X.
We’ll say that a functionf:X →ZisZ+∗-continuous ifz∗◦f is a real continuous function, for all z∗ ∈Z+∗. It is ob vious that iff is continuous thenf isZ+∗-continuous. In [78] we meet the demicontinuous function which has the property that it isZ+∗-continuous but it may be not continuous.
Proposition 16.3 Let X, Z be two reflexive Banach spaces, the ordering coneZ+ having a compact base and F:X −−→−→ Z a convex, continuous multifunction with DomF =X. Under this assumptions, F has a convex, Z+∗-continuous selection f such that ∂∗F(x) =∂≤f(x)for all x∈X if and only if F(x) has Z+∗-minimum for all x∈X and there exist kz∗ ∈Z+ with z∗◦f(x) = IMIN(z∗◦F(x)) +kz∗, for all z∗ ∈Z+∗, x∈X.
Proof. If F has a selection f with the required properties, then for all x ∈ X, we have
∂≤f(x)=∅and thus∂∗F(x)=∅. This implies thatz∗◦F(x) has minimum denotedmz∗(x) for allz∗ ∈Z+∗ and x∈X. Since ∂∗F(x) =∂≤f(x) from [38] we have that
∂≤z∗ ◦f(x) =z∗◦∂≤f(x)⊆∂≤z∗◦F(x).
Now, from the convexity and from the continuity of F we obtain that z∗ ◦ F is a real convex continuous multifunction and thus mz∗ is a convex, continuous real function with
∂≤mz∗(x) = ∂≤z∗ ◦ F(x), for all x ∈ X and z∗ ∈ Z+∗. From the Rockafellar Theorem about the maximal monotonicity of the subdifferential for a real convex, continuous function, we obtain that ∂≤z∗ ◦f(x) = ∂≤z∗ ◦ F(x) for all x ∈ X and there exists kz∗ ∈ Z with z∗ ◦f(x) = IMIN(z∗ ◦F(x)) +kz∗, for all z∗ ∈ Z+∗. Since f(x) ∈ F(x), we obtain that kz∗ ∈Z+.
Converslly, if for all x ∈ X, z∗ ∈ Z+∗ we have that z∗ ◦F(x) has minimum and if there exists a selectionf andkz∗ ∈Z+ withz∗◦f(x) = IMIN(z∗◦F(x)) +kz∗ we shall obtain that
∂≤z∗◦F(x) = ∂≤z∗◦f(x) =z∗◦∂≤f(x).
Thus,
∂∗F(x) = {T ∈ L(X, Z)| ∀z∗ ∈Z+∗, z∗◦T ∈∂≤z∗◦F(x)}
= {T ∈ L(X, Z)| ∀z∗ ∈Z+∗, z∗◦T ∈∂≤z∗◦f(x)}
= ∂∗f(x) =∂≤f(x).
Since F is a convex, continuous multifunction, from the equality z∗◦f(x) = IMIN(z∗◦F(x)) +kz∗
true for all x ∈ X, z∗ ∈ Z+∗ we obtain that z∗◦f is a convex continuous function for all z∗ ∈ Z+∗. From Lemma 3.1, f is convex, and thus, F has a selection with the required
properties and the proof is complete. $
Proposition 16.4 Under the same hypothesis like in Proposition 16.3 we obtain a convex continuous selection f with the property that for all x ∈ X, ∂∗F(x) = ∂>f(x) if and only if for all x ∈ X, ∂≤f(x) =∂>f(x), F(x) has Z+∗-minimum and there exists kz∗ ∈ Z+ with z∗◦f(x) = IMIN(z∗◦F(x)) +kz∗, for all z∗ ∈Z+∗.
Proof.If there exists a selection with the required properties, we obtain that for allx∈X,
∂≤f(x)⊆∂∗F(x) which implies that ∂≤z∗◦f(x)⊆∂≤z∗◦F(x). Like in the previous proof, we obtain kz∗ ∈ Z+ with z∗◦f(x) = IMIN(z∗◦F(x)) +kz∗, for all z∗ ∈ Z+∗. This implies that ∂≤f(x) =∂∗F(x) =∂>f(x) and the conclusion follows.
The converse implication is obvious by using the previous proposition. $ Remark 16.2 If the order is not total, then it does not exists a convex continuous selection of a multifunction under the assumptions of the Proposition 16.4 such that∂∗F(x) =∂>f(x), for allx∈X.
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REZUMAT
In 1972, M. Valadier a prezentat ˆın articolul intitulat“Sous-diff´erentiabilit´e des fonctions convexes `a valeurs dans un espace vectoriel ordonn´e” unele generaliz˘ari ale teoremelor lui
Moreau privitoare la subdiferent¸iabilitatea funct¸iilor convexe ¸si continue. Studiul subdiferent¸ibilit˘at¸ii vectoriale mai fusese abordat pˆan˘a la Valadier doar de C. Raffin ˆın cadrul spat¸iilor IRI.
Rezultatele lui Valadier se refer˘a la subdiferent¸iala unei funct¸ii convexe ˆıntre dou˘a spat¸ii vectoriale topologice X et Y dat˘a de:
∂≤f(x0) ={T ∈ L(X, Y)| T(x−x0)≤f(x)−f(x0),∀x∈X}.
Teorema principal˘a a acestui articol generalizeaz˘a pentru funct¸iile cu valori ˆıntr-un spat¸iu vectorial topologic ordonat de un con normal ˆınzestrat cu o structur˘a de latice complet ordonat˘a, faptul ca subdiferent¸iala unei funct¸ii convexe ˆıntr-un punct de continuitate este o mult¸ime convex˘a, nevid˘a, echicontinu˘a ˆın L(X, Y). Subacelea¸si ipoteze, el stabile¸ste formulele de leg˘atur˘a (bine cunoscute ˆın cadrul real) ˆıntre subdiferent¸ial˘a si un tip de derivat˘a direct¸ional˘a, introdus˘a de asemenea ˆın acest articol:
f(x0, h) = max{T h, T ∈∂≤f(x0)}.
Unele propriet˘at¸i de echicontinuitate ce privesc subdiferent¸iala vectorial˘a au fost stabilite ˆın condit¸ii diverse pentru funct¸ie ¸si pentru conul de ordine. Renunt¸ˆand la continuitatea funct¸iei si la normalitatea conului, propriet˘at¸ile au fost verificate ˆın ipoteza c˘a funct¸ia este majorat˘a pe o vecin˘atate a punctului respectiv iar intervalele de ordine sunt m˘arginte.
Studiul subdiferent¸iabilit˘at¸ii vectoriale ˆın cadrul laticilor ordonate a fost continuat de nu-mero¸si matematicieni printre care amintim A.G. Kusraev, S.S. Kutateladze, A.M. Rubinov, N. Papageorgiu, etc. Acesta din urm˘a a dezvoltat reguli de calcul (generalizarea Teoremei Moreau-Rockafellar pentru suma funct¸iilor convexe continue, reguli de calcul pentru com-punerea ¸si supremumul de funct¸ii convexe) precum ¸si o teorie a dualit˘at¸ii vectoriale, similare aceleia existente ˆın cazul real. El a propus apoi o generalizare a subdiferent¸ialei lui Clarke pentru funct¸iile o-Lipschitz cu valori ˆıntr-o latice Banach complet ordonat˘a. Cu aceast˘a ocazie, el a introdus un tip de derivat˘a direct¸ional˘a f◦(x, h) definit˘a prin
f◦(x, h) = sup
Cu ajutorul acestei not¸iuni, Papageorgiu a generalizat rezultatele cunoscute pentru funct¸iile lipschitziene reale, a introdus reguli de calcul ¸si a formulat condit¸ii necesare pentru existent¸a subdiferent¸ialei vectoriale.
Tot ˆın cadrul laticilor ordonate, L. Thibault a propus o alt˘a subdiferent¸ial˘a introdus˘a de asemenea cu ajutorul unei derivate direct¸ionale f↑(x, h) defint˘a cu ajutorul conului tangent a lui Clarke; aceast˘a subdiferent¸ial˘a, definit˘a prin:
∂Tf(x) ={T ∈ L(X, Y)| T(h)≤ f↑(x, h),∀h∈X}
a fost utilizat˘a pentru studiul funct¸iilor “strict compact Lipschitz”, o clas˘a de funct¸ii care generalizeaz˘a funct¸iile strict diferent¸iabile ¸si care coincide cu clasa funct¸iilor lipschitziene pentru cazul spat¸iilor de dimensiune finit˘a.
J. Zowe a propus ˆın 1974, ˆın articolul ”Subdifferentiability of convex functions with values in an ordered vector space” o alt˘a manier˘a de studiu a subdiferent¸ialei vectoriale introdus˘a de Valadier, renunt¸ˆand la condit¸ia de latice dar impunˆand condit¸ii suplimentare asupra domeniului funct¸iei. El a demostrat c˘a rezultatul de existent¸˘a enunt¸at de Valadier pentru subdiferent¸iala convex˘a r˘amˆane adev˘arat dac˘aX este un spat¸iu Mackey ¸si interiorul conului dual pentru topologia Mackey este nevid. Acest rezultat este aplicat pentru cazul ˆın care X este un spat¸iu refelxiv iar Y un spat¸in semireflexiv ordonat de un con cu baz˘a slab compact˘a. Rezultatul lui J. Zowe a fost extins de M.M. Day, J. Jahn care trateaz˘a ¸si problema subdiferent¸iabilit˘at¸ii regulate pentru funct¸ii convexe.
Astfel se deschide o alt˘a direct¸ie de studiu a subdiferent¸iabilit˘at¸ii vectoriale prin s-calarizare. Urmarind acest punct de vedere, J.B. Hiriart Urruty ¸si L. Thibault au gen-eralizat subdiferent¸iala lui Clarke pentru funct¸iile local lipschitzienef :X →Y unde Xeste un spat¸iu Banach separabil iar Y un spat¸iu Banach reflexiv separabil. In acest cadru, ei au demonstrat existent¸a unei mult¸imi Γ(f, x0), convexe, ˆınchise maximale cu proprietatea c˘a
∂y∗◦f(x0) = y∗◦Γ(f, x0), pour touty∗ ∈Y∗. xn→x0 ¸si H este mult¸imea undef este diferent¸iabil˘a ˆın sens Hadamard.
Aceast˘a subdiferent¸ial˘a poate fi privit˘a ca generalizarea subdiferent¸ialei lui Clarke privit˘a ca ˆınf˘a¸sur˘atoarea convex˘a a limitelor gradient¸ilor.
Pentru cazul spat¸iilor de dimensiune finit˘a, F.H. Clarke a utilizat deja un tip de subdife-rent¸ial˘a definit ˆın aceast˘a manier˘a prin ˆınf˘a¸sur˘atoarea convex˘a a matricilor obt¸inute drept limite ale ¸sirurilor de tip (Df(xi))i, unde xi →x0 ¸si Df(xi) este matricea Jacobian˘a clasic˘a (presupunˆınd c˘a ea exist˘a ˆınxi). Este binecunoscut c˘a pentru cazul Y = IR, subdiferent¸iala unei funct¸ii unei funct¸ii convexe coincide cu subdiferent¸iala lui Clarke dar pentru spat¸iile de dimensiune mai mare ca 1, aceast˘a egalitate nu se p˘astreaz˘a dac˘a consider˘am ultimul tip de subdiferent¸ial˘a.
In anii ’80, un alt tip de subdiferent¸ial˘a vectorial˘a se impune, subdiferent¸iala de tip Pareto care permite studiul problemelor de optimizare Pareto ˆın leg˘atur˘a cu propriet˘at¸ile toplogice ale spat¸iului si ale conului de ordine. Rezultate privitoare la aceast˘a subdieferent¸ial˘a au fost obt¸inute de T. Tanino, Y. Sawaragi pentru cazul spat¸iilor de dimensiune finit˘a ¸si ulterior,
A.B. Nemeth, G. Isac, V. Postolic˘a au extins aceste rezultate pentru cazul spat¸iilor de dimensiune infinit˘a.
In anii ’90, B. Mordukhowich a introdus un alt “obiect” utilizat pentru studiul multifunct¸iilor cu grafic ˆınchis. Este vorba de coderivat˘a (care permite de asemenea introducerea unei noi subdiferent¸iale pentru funct¸iile reale), legat˘a de conul normal introdus de K. Kruger ¸si B.
Mordukhowich. Aceast˘a not¸iune permite studiul problemelor de analiz˘a neliniar˘a, control optimal, etc.
Reamintim printre numero¸sii matematicieni care au contribuit la studiul subdiefrent¸iabilit˘at¸ii vectoriale, J. Borwein, J.P. Penot, C. Malivert, M. Th´era, M. Volle, T. Reiland, H. Sweetser, etc.
In fat¸a numeroaselor tipuri de subdiferent¸iale vectoriale amintite, se degaj˘a ˆın mod natural o ˆıntrebare: este posibil de a g˘asi o procedur˘a general˘a de introducere a unei subdiferent¸iale vectoriale de manier˘a a reg˘asi subdiefernt¸ialele precedente drept cazuri par-ticulare?
Teza ˆı¸si propune o tratare general˘a a subdiferent¸iabilit˘at¸ii vectoriale, utilizˆınd cele doua direct¸ii principale de studiu, punctele eficiente ¸si scalarizarea.
Mult¸imile de puncte eficiente aproximative utilizate ˆın prima parte au permis introducerea unui nou tip de dominare ¸si rezultatele obt¸inute referitoare la existent¸a acestor puncte (ˆın special Teorema 1.3) vor fi utilizate ˆın mod sistematic ˆın Capitolul 3 pentru obt¸inerea de reguli de calcul pentru multifunct¸ii.
Aceste rezultate au permis de asemenea introducerea unui nou lagrangian ¸si Teorema 1.3 a fost utulizat˘a pentru generalizarea rezultatelor cunoscute ˆın teoria dualit˘at¸ii pentru spat¸iile de dimensiune finit˘a.
Privitor la multifunct¸ii, teorema lui Michael despre existent¸a unei select¸ii continue pentru o multifunct¸ie s.c.i. cu grafic ˆınchis a sugerat g˘asirea unei select¸ii avˆand aceea¸si subdiferent¸ial˘a ca ¸si multifunct¸ia considerat˘a; ultima parte a tezei este consacrat˘a acestei probleme.
Al doilea capitol trateaz˘a subdiferent¸iabilitatea cu ajutorul scalariz˘arii; noi tipuri de subdiferent¸iale vectoriale sunt introduse, au fost studiate propriet˘at¸ile lor principale precum
¸si leg˘atura cu subdiferent¸ialele vectoriale deja cunoscute.
Printre rezulatele generalizate reg˘asim teorema de medie a lui Zagrodny, teorema lui Correa-Jofr´e-Thibault privitoare la echivalent¸a ˆıntre monotonia subdiferent¸ialei ¸si convexi-tatea funct¸iei, precum ¸si unele rezulate de existent¸˘a, reguli de calcul ¸si propriet˘at¸i de opti-malitate.
In concluzie, putem spune c˘a rezultatele clasice pentru subdiferent¸ialele funct¸iilor reale se pot generaliza ˆın cadrul vectorial pentru subdiferent¸iale convenabile ˆın condit¸ii specifice fiec˘arui caz.
Subdiferent¸ialele de tip Pareto sunt dificil de utilizat din punct de vedere al calculelor dar au avantajul ca pot fi utilizate pentru studiul punctelor eficiente Pareto; r˘amˆane ˆın studiu explicitarea acestor rezultate pentru spat¸ii vectoriale particulare.