Calculus of tangents and beyond Текст научной статьи по специальности «Математика»

Владикавказский математический журнал 2017, Том 19, Выпуск 4, С. 27-34

УДК 517.982.252

CALCULUS OF TANGENTS AND BEYOND1 A. G. Kusraev, S. S. Kutateladze

On the Sixtieth Anniversary of the Sobolev Institute

Optimization is the choice of what is most preferable. Geometry and local analysis of nonsmooth objects are needed for variational analysis which embraces optimization. These involved admissible directions and tangents as the limiting positions of the former. The calculus of tangents is one of the main techniques of optimization. Calculus reduces forecast to numbers, which is scalarization in modern parlance. Spontaneous solutions are often labile and rarely optimal. Thus, optimization as well as calculus of tangents deals with inequality, scalarization and stability. The purpose of this article is to give an overview of the modern approach to this range of questions based on non-standard models of set theory. A model of a mathematical theory is usually called nonstandard if the membership within the model has interpretation different from that of set theory. In the recent decades much research is done into the nonstandard methods located at the junctions of analysis and logic. This area requires the study of some new opportunities of modeling that open broad vistas for consideration and solution of various theoretical and applied problems.

Key words: Hadamard cone, Bouligand cone, Clarke cone, general position, operator inequality, Boolean valued analysis, nonstandard analysis.

Agenda. Optimization is the choice of what is most preferable. Geometry and local analysis of nonsmooth objects are needed for variational analysis which embraces optimization. These involved admissible directions and tangents as the limiting positions of the former. The calculus of tangents is one of the main techniques of optimization (cp. fl, 2]).

Calculus reduces forecast to numbers, which is scalarization in modern parlance. Spontaneous solutions are often labile and rarely optimal. Thus, optimization as well as calculus of tangents deals with inequality, scalarization and stability. Some aspects of the latter are revealed by the tools of nonstandard models to be touched sligtly in this talk (cp. [3-6]).

The best is divine. Leibniz wrote to Samuel Clarke (see [7, p. 54]; cp. [8]): "God can produce everything that is possible or whatever does not imply a contradiction, but he wills only to produce what is the best among things possible."

Enter the reals. Choosing the best, we use preferences. To optimize, we use infima and suprema for bounded sets which is practically the least upper bound property. So optimization needs ordered sets and primarily boundedly complete lattices.

To operate with preferences, we use group structure. To aggregate and scale, we use linear structure.

All these are happily provided by the reals R, a one-dimensional Dedekind complete vector lattice. A Dedekind complete vector lattice is a Kantorovich space.

© 2017 Kusraev A. G., Kutateladze S. S.

1 This article bases on a talk at the International Conference "Mathematics in teh Modern World," August 14-19, 2017, Novosibirsk.

Since each number is a measure of quantity, the idea of reducing to numbers is of a universal importance to mathematics. Model theory provides justification of the Kantorovich heuristic principle that the members of his spaces are numbers as well (cp. [9] and [10]).

Enter inequality and convexity. Life is inconceivable without numerous conflicting ends and interests to be harmonized. Thus the instances appear of multiple criteria decision making. It is impossible as a rule to distinguish some particular scalar target and ignore the rest of them. This leads to vector optimization problems, involving order compatible with linearity.

Linear inequality implies linearity and order. When combined, the two produce an ordered vector space. Each linear inequality in the simplest environment of the sort is some half-space. Simultaneity implies many instances and so leads to the intersections of half-spaces. These yield polyhedra as well as arbitrary convex sets, identifying the theory of linear inequalities with convexity.

Convexity, stemmimg from harpedonapters, reigns in optimization, feeding generation, separation, calculus, and approximation. Generation appears as duality; separation, as optimality; calculus, as representation; and approximation, as stability (cp. [11-13]).

Legendre in disguise. Assume that X is a vector space, E is an ordered vector space, E* is E with an adjoined top, f : X ^ E* is some operator, and C := dom(f) C X is a convex set. A vector program (C, f) is written as follows:

x e C, f (x) ^ inf.

(C, f)

f

f *(l) := sup (l(x) - f (x)), xex

with l e X# a linear functional over X. The epigraph of f * is a convex subset of X# and so f* is convex. Observe that —f*(0) is the value of (C,f).

Order omnipresent. A convex function is locally a positively homogeneous convex function, a sublinear functional. Recall that p : X ^ R is sublinear whenever

epip := {(x, t) e X x R | p(x) ^ t}

is a cone. Recall that a numeric function is uniquely determined from its epigraph. Given C C X, put

H(C) := {(x, t) e X x R+ | x e tC},

the Hormander transform of C. Now, C is convex if and only if H(C) is a cone. A space with ()

The order, the symmetry, the harmony enchant us ... (Leibniz). Thus, convexity and order are tightly intertwined.

Nonoblate cones. Consider cones Ki and K2 in a topological vector space X and put k := (Ki, K2). Given a pair k define the correspondence from X2 into X by the formula

:= {(ki,&2,x) e X3 : x = ki — k2, kl e Kj.

Clearly, is a cone or, in other words, a conic correspondence.

The pair k is nonoblate whenever is open at the zero. Since (V) = V n K — V n K2 for every V C X, the nonoblateness of k means that

kV := (V n Ki — V n K2) n (V n K2 — V n Ki)

is a zero neighborhood for every zero neighborhood V C X.

kV C V — V k

fact that the system of sets {kV} serves as a filterbase of zero neighborhoods while V ranges over some base of the same filter.

Let An : x ^ (x,..., x) be the embedding of X into the diagonal An(X) of Xn. A pair of cones k := (K1;K2) is nonoblate if and only if A := (K1 x K2, A2(X)) is nonoblate in X2.

Ki K2 $ C X x X2 defined as

$ := {(h,xi,X2) G X x X2 : x* + h G Kt (i := 1, 2)}

is open at the zero.

Ki K2 X

general position iff

(1) the algebraic span of K^d K2 is some subspace X0 C X; i. e., X0 = K1 — K2 = K2 — Ki

(2) the subspace X0 is complemented; i.e., there exists a continuous projection P : X ^ X such that P (X) = X0;

(3) K^d K2 constitute a nonoblate pair in X0.

General position of operators. Let an stand for the rearrangement of coordinates an : ((xi,yi),..., (xn,yn)) ^ ((xi,...,xn), (yi,...,yn))

which establishes an isomorphism between (X x Y)n and Xn x Yn.

Sublinear operators Pi,..., Pn : X ^ E U {+^} are in general position if so are the cones An(X) x En and an(epi(Pi) x ■ ■ ■ x epi(Pn)). KCX

nE(K) := {T G L(X, E) : Tk < 0 (k G K)}.

Clearly, (K) ^s a cone in L(X, E).

Theorem. Let Ki,..., Kn be cones in a topological vector space X and let E be a topological Kantorovich space. If Ki,..., Kn are in general position then

nE (Ki n ■ ■ ■ n Kn) = nE (Ki ) + ■ ■ ■ + nE (Kn).

XY space. Let B := B(Y) be the hase of Y, i.e., the complete Boolean algebras of positive projections in Y; and let m(Y) be the universal completion of Y. Denote by L(X, Y) the

X Y X Y X

by L(m) (X, Y) to mean the space of dominated, operators from X to Y. As usual, {T ^ 0} := {x G X | Tx < 0} ker(T) = T-i(0^r T : X ^ ^^so, P G Sub(X,Y) means that P is sublinear, while P G PSub(X, Y) means that P is polyhedral, i.e., finitely generated. The superscript (m) suggests domination.

Kantorovich's theorem. Find X satisfying

X^+W

Y

(1) (3 X) XA = B o ker(A) c ker(B).

(2) If W is ordered by W+ and A(X) - W+ = W+ - A(X) = W, then (cp. [2, p. 51])

(3 X ^ 0) XA = B o {A < 0} c {B < 0}.

The Farkas alternative. Let X be a Y-seminormed real vector space, with Y a Kantorovich space. Assume that Ai,..., AN and B belong to L(m) (X, Y).

Then one and only one of the following holds:

(1) There are x £ X and b, b' £ B such that 6' ^ b and

b'Bx > 0, bAix < 0,..., 6Anx < 0.

(2) There are positive orthomorphisms ai;..., aN £ Orth(m(Y))+ such that

N

B = ^ Afc. fe=i

Inhomogeneous inequalities.

X Y Y

Assume given some dominated operators A1 ,...,AN, B £ L(m)(X, Y) and elements ui,..., un, v £ Y. Tiie following are equivalent:

(1) For all b £ B the inhomogeneous operator inequality bBx ^ bv is a consequence of the consistent simultaneous inhomogeneous operator inequalities bA1 x ^ bui,..., 6Anx ^ 6un, i. e.,

{bB ^ bv} D {bAi ^ bui} n ■ ■ ■ n {bAN ^ buN}.

(2) There are positive orthomorphisms ai;..., aN £ Orth(m(Y)) satisfying

NN

B = ak Ak; v ^ ak uk. fe=i k=i

Boolean modeling. The above infinite-dimensional results appear as interpretations of one-dimensional predecessors on using model theory.

Cohen's final solution of the problem of the cardinality of the continuum within ZFC gave rise to the Boolean valued models by Scott, Solovay, and Vopenka (cp. [4]).

Takeuti coined the term "Boolean valued analysis" for applications of the models to analysis.

Scott's comments. Scott forecasted in 1969 (cp. [14]): "We must ask whether there is any interest in these nonstandard models aside from the independence proof; that is, do they have any mathematical interest? The answer must be yes, but we cannot yet give a really-good argument."

In 2009 Scott wrote2: "At the time, I was disappointed that no one took up my suggestion. And then I was very surprised much later to see the work of Takeuti and his associates. I think the point is that people have to be trained in Functional Analysis in order to understand these models. I think this is also obvious from your book and its references. Alas, I had no stu-

2Letter of April 29, 2009 to S. S. Kutateladze.

dents or collaborators with this kind of background, and so I was not able to generate any progress."

Art of invention. Leibniz wrote about his version of calculus that "the difference from Archimedes style is only in expressions which in our method are more straightforward and more applicable to the art of invention."

Nonstandard analysis has the two main advantages: it "kills quantifiers" and it produces the new notions that are impossible within a single model of set theory.

Let us turn to the nonstandard presentations of Kuratowski-Painleve limits of use in tangent calculus, and explore the variations of tangents.

Recall that the central concept of Leibniz was that of a monad (cp. [15]). In nonstandard analysis the monad ^(F) of a standard filter F is the intersection of all standard elements

Monadic limits. Let F C X x Y be an internal correspondence from a standard set X to a standard set Y. Assume given a standard filter N on X and a topology t on Y. Put

with * symbolizing standardization and y « y' standing for the infinite proxitity between y and y' in t, i. e. y' G ^(t(y)).

Call QiQ2(F) the Q^^limit of F (here Qk (k := 1, 2) is one of the quantifiers V or 3).

F

on some element of N and look at the 33-limit and the V3-limit. The former is the limit superior or upper limit; the latter is the limit inferior or lower limit of F along N. F

of F.

W (F) := *{y' : (Vx G ) n dom(F)) (Vy « y')(x,y) G F},

3V (F) := * {y' : (3 x G ju(N ) n dom(F )) (V y « y')(x,y) G F}, V3 (F) := * {y' : (Vx G ju(N) n dom(F)) (3 y « y')(x,y) G F}, 3 3 (F) := * {y' : (3 x G ju(N) n dom(F)) (3 y « y')(x,y) G F},

U

where N is the grill of a filter N on X, i. e., the family comprising all subsets of X meeting

MN).

Hadamard, Clarke, and Bouligand tangents.

Uer(x'), xeFnU, a 0<a^a'

xeF nu,

0<a^a'

T (X ), nu,

a' 0<a^a'

Bo (f,x'):= H U -

XeF nU

UGr(x'), xeFnU, a 0<a^a'

Fx

a

where, as usual, r(x') := x' + NT and NT, the zero neighborhood filterbase of the topology r. Obviously,

Ha (F, x') c Cl (F, x') c Bo (F, x'). Infinitesimal quantifiers. Agree on notation for a ZFC formula p and x' £ F :

(V'x) p := (Vx «T x') p := (Vx) (x £ F A x «T x') ^ p,

(V'h) p := (V h « h') p := (V h) (h £ X A h « h') ^ p, (V'a) p := (V a « 0) p := (V a) (a > 0 A a « 0) ^ p. The quantifiers 3'x, 3'h, 3'a are defined in the natural way by duality on assuming that

(3'x) p := (3 x «T x') p := (3 x) (x £ F A x «T x') A p,

(3'h) p := (3 h «T h') p := (3 h) (h £ X A h «T h') A p, (3'a) p := (3 a « 0) p := (3 a) (a > 0 A a « 0) A p.

333

h'

h' £ Bo(F, x') o (3'x) (3'a) (3'h) x + ah £ F.

VVV

Ha(F,x') = VVV (F, x'),

with the external set of positive infinitesimals.

VV3

Cl(F,x') = VV3 (F, x').

In more detail,

h' £ Cl(F, x') o (Vx) (V'a) (3'h) x + ah £ F.

Convexity is stable. Convexity of harpedonaptae was stable in the sense that no variation of stakes within the surrounding rope can ever spoil the convexity of the tract to be surveyed.

Stability is often tested by perturbation or introducing various epsilons in appropriate places, which geometrically means that tangents travel. One of the earliest excursions in this direction is connected with the classical Hyers-Ulam stability theorem for e-convex functions. Exact calculations with epsilons and sharp estimates are often bulky and slightly mysterious. Some alternatives are suggested by actual infinities, which is illustrated with the conception of infinitesimal optimality.

Enter epsilon. Assume given a convex operator / : X —» E* and a point x in the effective domain dom(f) := {x £ X | f (x) < of f.

Given e ^ 0 in the positive cone E+ of E, by the e-suhdifferential of / at x we mean the

set

def(x) := {T £ L(X,E) \ (Vir G X) (Tx - /(x) < Tx — f(x) + e)}.

Topological setting. The usual subdifferential df(x) is the intersection of e-subdifferen-tials:

df(x) := H 9ef(x).

£>0

In topological setting we use continuous operators, replacing L(X, E^th L(X, E).

e

Theorem. Let fi : X x Y ^ E 'an d f2 : Y x Z ^ E' be convex operators and 5, e £ E +. Suppose that the convolution f2Afa is 5-exact at some point (x, y, z); i. e., 5 + (f2Afi)(x, y) = fi (x, y)+f2 (y, z). If, moreover, the convex sets epi(fi, Z) and epi(X, f2) are in general position, then

de(f2Afi)(x,y)= U d£2f2(y,z) o d£l fi (x, y).

£l>0, £2 >0, £l+£2=£+5

Enter monad. Distinguish some downward-filtered subset E of E that is composed of positive elements. Assuming E and E standard, define the monad ^(E) of E as ME) := d{[0, e] | e £ °E}. The members of ^(E) are positive infinitesimals with respect to E. As usual, °E denotes the external set of all standard members of E, the standard part of E. Assume that the monad ^(E) is an external cone over °R and, moreover, ^(E) n °E = 0. EE

E

ei « e2 o ei — e2 £ ^(E) & e2 — ei £ ^(E). Infinitesimal subdifferential. Now

Df(x) := H 9ef(x) = |J def(x),

£€°E £€M(E)

which is the infinitesimal subdifferential of / at x. The elements of Df(x) are infinitesimal subgradients of / at x.

Infinitesimal solution. Assume that there exists a limited value e := infxeC f (x) of some program (C, f). A feasible point x0 is called an infinitesimal solution if f (x0) « e, i. e., if f (x0) ^ f (x) + e for every x £ C and every standard e £ E.

A point x0 £ X is an infinitesimal solution of the unconstrained problem f (x) ^ inf if and only if 0 £ Df (x0). Exeunt epsilon.

Theorem. Let fi : X x Y ^ E' and f2 : Y x Z ^ E 'be convex operators. Suppose that the convolution f2Afi is inhnitesimally exact at some point (x,y,z); i. e., (f2Afi)(x,y) « fi(x, y)+f2(y, z). If, moreover, the convex sets epi(fi, Z) and epi(X, f2) are in general position then

D(f2Afi)(x, y) = Df2(y, z) o Dfi (x, y). References

1. Clarke F. Nonsmooth Analysis in Systems and Control Theory // Encyclopedia of Complexity and Control Theory.—Berlin: Springer-Verlag, 2009,—P. 6271-6184.

2. Kusraev A. G. and Kutateladze S. S. Subdifferential Calculus: Theory and Applications.—Moscow: Nauka, 2007.

3. Bell J. L. Set Theory: Boolean Valued Models and Independence Proofs.—Oxford: Clarendon Press, 2005.

4. Kusraev A. G. and Kutateladze S. S. Introduction to Boolean Valued Analysis.—Moscow: Nauka, 2005.

5. Kanovei V. and Reeken M. Nonstandard Analysis: Axiomatically.—Berlin: Springer-Verlag, 2004.

6. Gordon E. I., Kusraev A. G., and Kutateladze S. S. Infinitesimal Analysis: Selected Topics.—Moscow: Nauka, 2011.

7. Anew R. G. W. Leibniz and Samuel Clarke Correspondence.—Indianopolis: Hackett Publ. Comp., 2000.

8. Ekeland I. The Best of All Possible Worlds: Mathematics and Destiny.—Chicago-London: The Univer. of Chicago Press, 2006.

9. Kusraev A. G. and Kutateladze S. S. Boolean Methods in Positivity j j J. Appl. Indust. Math.—2008.— Vol. 2, № 1,—P. 81-99.

10. Kutateladze S. S. Mathematics and Economics of Leonid Kantorovich j j Sib. Math. J.—2012,—Vol. 53, № 1,—P. 1-12.

11. Kutateladze S. S. Harpedonaptae and Abstract Convexity // J. Appl. Indust. Math.—2008.—Vol. 2, № l.-P. 215-221.

12. Kutateladze S. S. Boolean Trends in Linear Inequalities // J. Appl. Indust. Math.—2010.—Vol. 4, № 3.— P. 340-348.

13. Kutateladze S. S. Nonstandard tools of nonsmooth analysis // J. Appl. Indust. Math.—2012,—Vol. 6, № 3.—P. 332-338.

14. Scott D. Boolean Models and Nonstandard Analysis j j Applications of Model Theory to Algebra, Analysis, and Probability.—Holt, Rinehart, and Winston, 1969,—P. 87-92.

15. Kutateladze S. S. Leibnizian, Robinsonian, and Boolean Valued Monads j j J. Appl. Indust. Math.— 2011,—Vol. 5, № 3.-P. 365-373.

Received August 15, 2011 Kusraev Anatoly G.

Vladikavkaz Science Center of the RAS, Chairman 22 Markus Street, Vladikavkaz, 362027, Russia; North Ossetian State University, Head of the Department of Mathematical Analysis 44-46 Vatutin Street, Vladikavkaz, 362025, Russia E-mail: kusraev@smath.ru

Kutateladze Semen S. Sobolev Institute of Mathematics, Senior Staff Scientist

4 Acad. Koptyug avenue, Novosibirsk, 630090, Russia E-mail: sskut@math.nsc.ru

ИСЧИСЛЕНИЕ КАСАТЕЛЬНЫХ И ВОКРУГ Кусраев А. Г., Кутателадзе С. С.

Оптимизация — это выбор наиболее предпочтительного. Геометрия и локальный анализ негладких объектов необходимы для вариационного анализа, который включает оптимизацию. К ним относятся допустимые направления и касательные как предельные позиции первых. Исчисление касательных является одним из основных инструментов оптимизации. Исчисление сводит прогноз к числам, что на современном языке можно назвать скаляризацией. Спонтанные решения часто неустойчивы и редко оптимальны. Таким образом, оптимизация и исчисление касательных связаны с неравенствами, скаляризацией и устойчивостью. Цель настоящей статьи — дать обзор современного подхода к указанному кругу вопросов, основанного на применении нестандартных моделей. Модель математической теории обычно называется нестандартной, если отношение принадлежности в модели имеет интерпретацию, отличную от интерпретации теории множеств. В последние десятилетия во многих исследованиях используются нестандартные методы, расположенные на стыках анализа и логики. Эта область, дает некоторые новые возможности моделирования, открывающие широкие перспективы для рассмотрения и решения различных теоретических и прикладных задач.

Ключевые слова: конус Адамара, конус Булигана, конус Кларка, общее положение, операторное неравенство булевозначный анализ, нестандартный анализ.