Subdifferentials: Theory and Applications

407 Pages·2.212 MB·English

by  A.G. Kusraev, S.S. Kutateladze

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Contents
Preface vii
Chapter 1. Convex Correspondences and Operators
1. Convex Sets .................................................. 2
2. Convex Correspondences ..................................... 12
3. Convex Operators ............................................ 22
4. Fans and Linear Operators ................................... 35
5. Systems of Convex Objects ................................... 47
6. Comments ................................................... 58
Chapter 2. Geometry of Subdifferentials
1. The Canonical Operator Method ............................. 62
2. The Extremal Structure of Subdifferentials ................... 78
3. Subdifferentials of Operators Acting in Modules .............. 92
4. The Intrinsic Structure of Subdifferentials .................... 108
5. Caps and Faces .............................................. 123
6. Comments ................................................... 134
Chapter 3. Convexity and Openness
1. Openness of Convex Correspondences ......................... 137
2. The Method of General Position .............................. 151
3. Calculus of Polars ............................................ 164
4. Dual Characterization of Openness ........................... 177
5. Openness and Completeness .................................. 187
6. Comments ................................................... 196
vi
Chapter 4. The Apparatus of Subdifferential Calculus
1. The Young-Fenchel Transform ................................ 200
2. Formulas for Subdifferentiation ............................... 212
3. Semicontinuity ............................................... 221
4. Maharam Operators .......................................... 233
5. Disintegration ................................................ 244
6. Infinitesimal Subdifferentials .................................. 255
7. Comments ................................................... 266
Chapter 5. Convex Extremal Problems
1. Vector Programs. Optimality ................................. 269
2. The Lagrange Principle ...................................... 274
3. Conditions for Optimality and Approximate Optimality ....... 282
4. Conditions for Infinitesimal Optimality ....................... 290
5. Existence of Generalized Solutions ............................ 293
6. Comments ................................................... 305
Chapter 6. Local Convex Approximations
1. Classification of Local Approximations ........................ 308
2. Kuratowski and Rockafellar Limits ........................... 320
3. Approximations Determined by a Set of Infinitesimals ......... 330
4. Approximation to the Composition of Sets .................... 342
5. Subdifferentials of Nonsmooth Operators ..................... 348
6. Comments ................................................... 358
References 362
Author Index 390
Subject Index 393
Symbol Index 398
Preface
The subject of the present book is subdifferential calculus. The main source of
this branch of functional analysis is the theory of extremal problems. For a start,
we explicate the origin and statement of the principal problems of subdifferential
calculus. To this end, consider an abstract minimization problem formulated as
follows:
x ∈ X, f(x) → inf.
HereX isavectorspaceandf : X → ℝisanumericfunctiontakingpossiblyinfinite
values. In these circumstances, we are usually interested in the quantity inf f(x),
the value of the problem, and in a solution or an optimum plan of the problem (i.e.,
such an x¯ that f(x¯) = inff(X)), if the latter exists. It is a rare occurrence to solve
an arbitrary problem explicitly, i.e. to exhibit the value of the problem and one of
its solutions. In this respect it becomes necessary to simplify the initial problem by
reducing ittosomewhatmoremanageablemodificationsformulatedwiththedetails
of the structure of the objective function taken in due account. The conventional
hypothesis presumed in attempts at theoretically approaching the sought reduction
is as follows. Introducing an auxiliary function l, one considers the next problem:
x ∈ X, f(x)−l(x) → inf.
Furthermore, the new problem is assumed to be as complicated as the initial prob-
lem provided that l is a linear functional over X, i.e., an element of the algebraic
#
dual X . In other words, inanalysis of the minimizationproblem for f, we consider
as known the mapping f∗ : X# → ℝ that is given by the relation
f∗(l) := sup (l(x)−f(x)).
x∈X
viii Preface
∗
The f thus introduced is called the Young-Fenchel transform of the function f.
Observe thatthequantity−f∗(0)presents thevalueoftheinitialextremalproblem.
The above-described procedure reduces the problem that we are interested in
to that of change-of-variable in the Young-Fenchel transform, i.e., to calculation of
the aggregate (f ∘G), where G : Y → X is some operator acting from Y to X. We
∗
emphasize that f is a convex function of the variable l. The very circumstance by
itself prompts us to await the most complete results in the key case of convexity of
the initial function. Indeed, defining in this event the subdifferential of f at a point
x¯, we can conclude as follows. A point x¯ is a solution to the initial minimization
problem if and only if the next Fermat optimality criterion holds:
0 ∈ ∂f(x¯).
It is worth noting that the stated Fermat criterion isof littleavail if we lack effective
tools for calculating the subdifferential ∂f(x¯). Putting it otherwise, we arrive at
the question of deriving rules for calculation of the subdifferential of a composite
mapping ∂(f ∘G)(y¯). Furthermore, the adequate understanding of G as a convex
mapping requires that some structure of an ordered vector space be present in X.
(For instance, the presentation of the sum of convex functions as composition of
a linear operator and a convex operator presumes the introduction into ℝ2 the
coordinatewise comparison of vectors.)
Thus, we are driven with necessity to studying operators that act in ordered
vector spaces. Among the problems encountered on the way indicated, the central
places are occupied by those of finding out explicit rules for calculation of the
Young-Fenchel transform or the subdifferential of a composite mapping. Solving
the problems constitutes the main topic of subdifferential calculus.
Now the case of convex operators, which is of profound import, appears so
thoroughly elaborated that one might speak of the completion of a definite stage of
the theory of subdifferentials.
Research of the present days is conducted mainly in the directions related
to finding appropriate local approximations to arbitrary not necessarily convex
operators. Most principal here is the technique based on the F. Clarke tangent
cone which was extended by R. T. Rockafellar to general mappings. However, the
stageof perfection isfar from being obtainedyet. It isworth nonetheless to mention
that key technical tricks in this direction lean heavily on subdifferentials of convex
mappings.
Preface ix
In this respect we confine the bulk of exposition to the convex case, leaving
the vast territory of nonsmooth analysis practically uncharted. The resulting gaps
transpire. A slight reassuring apology for us is a pile of excellent recent books and
surveys treating raw spots of nonsmooth analysis. The tool-kit of subdifferential
theory is quite full. It contains the principles of classical functional analysis, meth-
ods of convex analysis, methods of the theory of ordered vector spaces, measure
theory, etc.
Many problems of subdifferential theory and nonsmooth analysis were recently
solvedon using nonstandard methods ofmathematicalanalysis(ininfinitesimal and
Boolean-valued versions). In writing the book, we bear in mind the intention of
(and the demand for) making new ideas and tools of the theory more available for
awiderreadership. Thelimitsofeverybook(thisoneinclusively)aretoonarrowfor
leaving an ample room for self-contained and independent exposition of all needed
facts from the above-listed disciplines.
We therefore choose a compromising way of partial explanations. In their
selection we make use of our decade experience from lecture courses delivered in
Novosibirsk and Vladikavkaz (North Ossetian) State Universities.
One more point deserves straightforward clarification, namely, the word “ap-
plications” in the title of the book. Formally speaking, it encompasses many appli-
cations of subdifferential theory. To list a few, we mention the calculation of the
Young-Fenchel transform, justification of the Lagrange principle and derivation of
optimality criteria for vector optimization problems. However, much more is left
intact and the title to a greater extent reflects our initial intentions and fantasies
as well as a challenge to further research.
The first Russian edition of this book appeared in 1987 under the title “Sub-
differential Calculus” soon after L. V. Kantorovich and G. P. Akilov passed away.
To the memory of the outstanding scholars who taught us functional analysis we
dedicate this book with eternal gratitude.
A. G. Kusraev
S. S. Kutateladze
Chapter 1
Convex Correspondences and Operators
The concept of convexity is among those most important for contemporary func-
tional analysis. It is hardly puzzling because the fundamental notion of the indi-
cated discipline, that of continuous linear functional, is inseparable from convexity.
Indeed, the presence of such a nonzero functional is ensured if and only if the
space under consideration contains nonempty open convex sets other than the en-
tire space.
Convex sets appear in many ways and sustain numerous transformations with-
out loosing their defining property. Among the most typical should be ranked the
operation of intersection and various instances of set transformations by means of
affine mappings. Specific properties are characteristic of convex sets lying in the
product of vector spaces. Such sets are referred to as convex correspondences. All
linear operators are particular instances of convex correspondences. The impor-
tance of convex correspondences increased notably in the last decades due to their
interpretation as models of production.
Among convex correspondences locatedin the product of a vector space and an
ordered vector space, a rather especial role is played by the epigraphs of mappings.
Such a mapping, a function with convex epigraph, is called a convex operator.
Among convex operators, positive homogeneous ones are distinguished, entitled
sublinear operators and presenting the least class of correspondences that includes
all linear operators and is closed under the taking of pointwise suprema. Some
formal justification and even exact statement of the preceding claim require the
specification of assumptions on the ordered vector spaces under consideration. It is
worth stressing that all the concepts of convex analysis are tightly interwoven with
2 Chapter 1
various constructions of the theory of ordered vector spaces. Furthermore, the
central place is occupied by the most qualified spaces, Kantorovich spaces or K-
spaces for short, which are vector lattices whose every above-bounded subset has
a least upper bound. The immanent interrelation between K-spaces and convexity
is one of the most important themes of the present chapter. An ample space is
also allotted to describing in detail the technique of constructing convex operators,
correspondences and sets from the already-given ingredients. An attractive feature
of convexity theory is an opportunity to provide various convenient descriptions
for one and the same class of objects. The general study of convex classes of
convex objects constitutes a specific direction of research, global convex analysis,
which falls beyond the limits of the present book. Here we restrict ourselves to
discussing the simplest methods and necessary constructions that are connected
with the introduction of the Minkowski duality and related algebraic systems of
convex objects.
1.1. Convex Sets
This section is devoted to the basic algebraic notions and constructions con-
nected with convexity in real vector spaces.
1.1.1. Fix a set (cid:3) ⊂ ℝ2. A subset C of a vector space X is called a (cid:3)-set if
withany two elements x,y ∈ C it contains each linear combination αx+βy withthe
coefficients determined by the pair (α,β) ∈ (cid:3). The family of all (cid:3)-sets in a vector
space X is denoted by P (X). Hence C ∈ P (X) if and only if for every (α,β) ∈ (cid:3)
(cid:2) (cid:2)
the inclusion holds αC + βC ⊂ C (here and henceforth αC := {αx : x ∈ C} and
C +D := {x+y : x ∈ C,y ∈ D}). We now list some simple properties of (cid:3)-sets.
(1) The intersection of each family of (cid:3)-sets in a vector space is a (cid:3)-set.
(2) The union of every upward-filtered family of (cid:3)-sets in a vector space is
a (cid:3)-set.
⊲ Let E ⊂ P (X) be some family of (cid:3)-sets. Put D := ∪E. Take x,y ∈ D
(cid:2)
and (α,β) ∈ (cid:3). By the definition of D there are x ∈ A and y ∈ B for suitable A
and B of E. By assumption there is a subset C ∈ E such that A ⊂ C and B ⊂ C.
Consequently, the elements x and y belong to C. Since C is a (cid:3)-set it follows that
αx+βy ∈ C ⊂ D. ⊳
(3) Assume that for every index ξ ∈ (cid:7) a vector space X and a set C ⊂ X
ξ ξ ξ
Convex Correspondences and Operators 3
∏ ∏
are given. Put C := C and X := X . Then C ∈ P (X) if and only if
ξ∈(cid:4) ξ ξ∈(cid:4) ξ (cid:2)
C ∈ P (X ) for all ξ ∈ (cid:7).
ξ (cid:2) ξ
⊲ Take x,y ∈ C and (α,β) ∈ (cid:3). As it is easily seen, αx+βy ∈ C means that
αx +βy ∈ C , for all ξ ∈ (cid:7), whence the claim follows immediately. ⊳
ξ ξ ξ
(4) If C and D are (cid:3)-sets in a vector space and λ ∈ ℝ, then the sets λC and
C +D are (cid:3)-sets too.
1.1.2. We now introduce the main types of (cid:3)-sets used in the sequel.
(1) If (cid:3) := ℝ2 then nonempty (cid:3)-sets in X are vector subspaces of X.
(2) Let (cid:3) := {(α,β) ∈ ℝ2 : α+β = 1}. Then nonempty (cid:3)-sets are called affine
subspaces, affine varieties, or flats. If X0 is a subspace of X and x ∈ X then the
translation x + X0 := {x} + X0 is an affine subspace parallel to X0. Conversely,
every affine subspace L defines the unique subspace L−x := L+(−x), where x ∈ L,
from which is obtained by a suitable translation.
(3) If (cid:3) := ℝ+×ℝ+, then nonempty (cid:3)-sets are called cones or, more precisely,
convex cones. In other words, a nonempty subset K ⊂ X is said to be a cone if
K+K ⊂ K andαK ⊂ K forallα ∈ ℝ+. (Hereandhenceforthℝ+ := t ∈ ℝ : t ≥ 0.)
(4) Take (cid:3) := {(α,0) ∈ ℝ2 : ∣α∣ ≤ 1}. The corresponding (cid:3)-sets are called
balanced or equilibrated.
(5) Let (cid:3) := {(α,β) ∈ ℝ2 : α ≥ 0,β ≥ 0,α + β = 1}. In this case (cid:3)-sets
are called convex. Clearly, linear subspaces and flats are convex. As it might be
expected, (convex!) cones are included into the class of convex sets.
(6) If (cid:3) := {(α,β) ∈ ℝ2 : α ≥ 0,β ≥ 0,α+β ≤ 1}, then a nonempty (cid:3)-set is
called a conic segment or slice. A set is a conic segment if and only if it is convex
and contains zero.
(7) Let (cid:3) := {(α,β) ∈ ℝ2 : ∣α∣+ ∣β∣ ≤ 1}. A nonempty (cid:3)-set in this case is
called absolutely convex. An absolutely convex set is both convex and balanced.
(8) If (cid:3) := {(−1,0)}, then (cid:3)-sets are said to be symmetric. The symmetry of
a set M obviously means that M = −M. Subspaces and absolutely convex sets are
symmetric.
1.1.3. Let P(X) := P∅(X) be the set of all subsets of X. For every M ∈
P(X) put ∩{ }
H (M) := C ∈ P (X) :C ⊃ M .
(cid:2) (cid:2)