Calculus of tangents and beyond

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 (ср. [1, 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.

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 Lop. f : X ^ E* is some operator. and C := dom(f) С X is a, convex set. A vector program (C,f ) is written as follows:

x E C, f (x) ^ inf .

( C, f )

Legendre transform or Young-Fenchel transform of f:

f*(l) := sup (l(x) - f (x)), x∈X with l E X# a, linear functional over X. The epigraph of f * is a conve.e subset of X# arid so f * is com’ex. C)bserve that —f *(0) is I lie 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) 6 t} is a cone. Recall that a numeric function is uniquely determined from its epigraph.

Given C С X. pul,

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 a cone is a (pre) ordered vector space.

The order, the symmetry, the harmony enchant us .. . (Leibniz).

Thus, convexity and order are tightly intertwined.

Nonoblate cones. Consider cones K1 and K₂ in a topologic al vector space X and put к := (K1 , K₂). Given a pair к define I,lie correspondenee Ф_к fix mi X² in to X by I,lie formula

Фк := {(ki,k2,x) E X3 : x = ki — k2, ki E Ki}.

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

The pair к is nonoblatc whenever Фк is open al, ille zero. Since Фк(V) = VП Ki — VП K2 for every V С X. the nonoblateness of к means I,ha I, kV := (V П Ki — V П K) П (V П K — V П Ki)

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

Open correspondences. Since k V С V — V, the nonoblateness of к is equivalent to the fact that the system of sets { k V } serves as a filterbase of zего neighborhoods while V ranges over some base of the same filter.

Let An : x ^ (x,..., x) be the embedding of X into the diagonal A_n(X) 0f Xⁿ. A pair of cones к := (K_i, K₂) is nonoblate if and only if A := (K1 x K2, A2(X )) is 110110blate in X².

Cones Ki and K2 constitute a nonoblate pair if and only if the conic correspondence Ф С X x X2 defined as

Ф := {(h, xi,X2) E X x X2 : x, + h E K (1 := 1, 2)} is open at the zero.

General position of cones. Cones K_i and K₂ in a topologica 1 vector space X are m general position iff

• (1)    the algebraic span of K_i and K₂ is some subspace Xo С X; i. e., Xo = K_i — K₂ = K — Ki

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

• (3)    K_i aiid K₂ constitute a 110noblate pair in Xo.

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

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

Sublinear operators P_i,..., P_n : X ^ E U{ + to} are in general position if so are the cones An(X) x Eⁿ aiid an(epi(Pi) x • • • x epi(Pn)).

Given a cone K С X. pul, пе(K) := {T E L(X, E) : Tk 6 0 (k E K)}.

Clearly, пе (K) is a cL(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 пе(Ki П • • • П Kn) = пе(Ki) +-----+ пе(Kn).

Environment for inequality. Assume that X is a real vector space, Y is a Kantorovich space. Let B := B(Y) be the base of Y, i.e., the complete Boolean algebras of positive projections in Y: arid lei, m(Y) be the tmiversa 1 completion of Y. Denote by L(X, Y) the XY   X         Y      X by L(m) (X, Y) we mean the space oj deaminated operators from X to Y. As 1 isual. {T 6 0} : = {x E X | Tx 6 0}: ker(T) = T-i(0) fc>r T : X ^ Y. Also. P E Sub(X,Y) means that P is sublinear, while P E PSub(X, Y) means that P is polyhedral, i.e., finitely generated. The superscript (m) suggests domination.

Kantorovich’s theorem. Find X satisfying

X -^-WW

X t Y

• (1)    (3X) XA = B о ker(A) C ker(B).

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

(3 X > 0) XA = B о {A 6 0} C {B 6 0}.

The Farkas alternative. Let X be a Y-seminormed real vector space, with Y a Kantorovich space. Assume that A1,..., An a nd B belong to L^(m)(X, Y ).

Then one and only one of the following holds:

• (1)    There are x G X and b, b⁰ 6 B such that b⁰ 6 b and

b⁰Bx > 0, bA1x 6 0,..., bANx 6 0.

• (2)    There are positive orthomorphisms a1, ..., aN G Orth(m(Y))+ such that

N

B = £ak Ak.

k=1

Inhomogeneous inequalities.

Theorem. Let X be a Y-seminormed real vector space, with Y a Kantorovich space. Assume given some dominated operators A1 ,...,An, B G L^(m)(X, Y) and elements ui,..., un ,v G Y. The following are equivalent:

• (1)    For all b G B the inhomogeneous operator inequality bBx 6 bv is a consequence of the consistent simultaneous inhomogeneous operator inequalities bA1 x 6 bu1, ..., bANx 6 buN, i. e.,

{bB 6 bv} D {bA1 6 bu1} П • • • П {bAN 6 buN }.

• (2)    There are positive orthomorphisms a1,..., aN G Orth(m(Y)) satisfying

  N

  B = ^ak Ak i k =i

  
  

  N

  v >  ^akuk .

  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 of F.

Monadic limits. Let F С X x Y be an internal correspondence from a standard set X l,o a standard set Y. Assume given ;r standard filter N c >n X and a, topology т on Y. Pul,

W(F) := *{y0 : (Vx G ^(N) П dom(F)) (Vy « y0) (x, y) g

3 V (F) := * {y0 : (3 x G ^(N) П dom(F)) (Vy ^ y) (x,y) G

V3 (F) := *{y0 : (Vx G ^(N) П dom(F)) (3y ^ y0) (x,y) G

33 (F) := *{y0 : (3x G ^(N) П dom(F)) (3y « y0) (x,y) G with * symbolizing staiidardization and y « y' standing for the uijiiute proxitity between y and y0 iii т. i. e. y0 G ^(r(y)).

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

Kuratowski-Painleve limits. Assume for instance that F is a standard correspondence on some element of N and loo к al, the 33-lirriit arid I,lie V3-liiiiit. The former is I,lie limit superior or upper limit' the latter is the limit inferior or lower limit of F along N.

Theorem. If F is a standard correspondence then

33 (F)= \ cl( [ F (x)); U ∈ N    x ∈ U

V3(F) = П clf J F(x) UcN vxeU where N is the grill of a filler N cin X. i. e.. the family comprising all subsets XX mooting h(N )•

Hadamard, Clarke, and Bouligand tangents.

Ha (F.x⁰) := [ rnl_T \ F-x

UCt (x0),      xCFnU, a0        0

Cl (F,x⁰) := \ U П (F^ + V

VCNT UCt(x⁰), xCFnU, ^V _a⁰    00

Bo (f,x0) := Pl

UCt (x⁰), α⁰

τ x∈F ∩U, 0

F

— x

α

where, as usual, т(x⁰) := X + NT aiid NT- the zero neighborhood filtorbase of the topology t. Obviously,

Ha (F, x') C Cl (F, X) C Bo (F, X).

Infinitesimal quantifiers. Agree on notation for a ZFC formula y and X E F :

(V^x) y := (Vx ^_T x⁰) y := (Vx) (x E F A x ^_T x⁰) to- y,

(V^h) y := (V h %_T h⁰) y := (V h) (h E X A h ^_T h⁰) to y, (V^a) y := (V a « 0) y := (V a) (a > 0 A a « 0) to y.

∃^•x ∃^•h ∃^•α

(^•x) y := (3 x ~_T x⁰) y := (3 x) (x E F A x ~_T x⁰) A y,

(3^h) y := (3 h ra_T h⁰) y := (3 h) (h E X A h ^_T h⁰) A y,

(3^a) y := (3 a « 0) y := (3 a) (a > 0 A a « 0) A y.

Infinitesimal representations. The Bouligand cone is the standardization of the 333-eoiie: i. e.. if h' is standard then h' E Bo(F, x0) о (34) (Уa) (У h) x + ah E F.

The Hadamard cone is the standardization of the VVV-cone:

Ha(F,x0) = VVV (F, x0), with ^(R+) the external set of positive infinitesimals.

The Clarke cone is the standardization of the VV3-cone: i. e.,

Cl(F,x⁰) = VV3 (F,x⁰).

In more detail, h0 E Cl(F, x0) о (Vx) (Va) (У h) x + ah E 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

ε

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 f : X to E^ and a point X in the effective domain dom(f) := {x E X | f (x) < +^} оf f.

Given e > 0 in the positivo cone E+ оf E. liy the E-subdiJferential of f al, X we mean the set df(X) := {T E L(X, E) | (Vx E X) (Tx - f (x) 6 Tx - f (x) + e)}.

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

df (x) := \ def (x).

e>o

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

e-optimality.

Theorem. Let f1 : X x Y ^ E * and f2 : Y x Z ^ E •be convex o;aerators and 5,e E E +. Suppose that the convolution f2Af1 is д-exact at some point (x,y,z) i. e., д+ (f2Af1)(x, y) = f1(x, y)+f2(y, z) If, moreover, the convex sets epi(fi, Z) and epi(X, f2) are in general position, then de(f2Af1)(x,y) =    [    dS2 f2(y,z) ° dei fi(x,y).

ei>o, £2>0, £1+e2=e+5

Enter monad. Distinguish some downward-filtered subset E оf E that is composed of positive elements. Assuming E and E standard, define the monad p(E) оf E as ^(E) := Q{[0,e] | e E °E}. The members of p(E) are positive inJinitesimals with respect I,о E. As usual. °E denotes the external set оf all standard members of E. the standiird part of E.

Assume that the monad ^(E) is an extern al cone over ° R and, moreover, ^(E) П °E = 0. In application, E is usually the filter of order-units of E. The relation of inJinite proximity or infinite closeness between the members of E is introduced as follows:

ei ~ e2 ^ ei — e2 E y(E) & e2 — ei E y(E).

Infinitesimal subdifferential. Now

Df(x) := \ def(X) = [ def(x), eC°E           eepJE)

which is the infinitesimal subdifferential of f at X. The elements of Df (x) are infinitesimal subgradients of f al, X.

Infinitesimal solution. Assume that there exists a limited value e := inf_xGc f (x) of some program (C,f). A feasilale point xo is called an injinitesimal solution if f (xo) ~ e. i- e.. If f (xo) 6 f (x) + e for every x E C and every standard e E E.

A point xo E X is an infinitesimal solution of the unconstrained problem f (x) ^ inf if and only if 0 E Df (xo).

Exeunt epsilon.

Theorem. Let f1 : X x Y ^ E* and f₂ : Y x Z ^ E* be convex operators. Suppose that the convolui ion f₂Af1 is iiifiniiesinially e.vaci at some point (x,y,z) i. e.. (f₂Af1)(x,y) ^ f1(x, y)+f₂(y, z). If, moreover, the convex sets epi(f1, Z) and epi(X, f₂) are in general position then

^D(f2^Afi^)(x,y⁾ = D^f2⁽y^,z) ° D^fi^(x,y⁾.