Convex Analysis

Convex Analysis

Copyright Date: 1970
Pages: 472
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  • Book Info
    Convex Analysis
    Book Description:

    Available for the first time in paperback, R. Tyrrell Rockafellar's classic study presents readers with a coherent branch of nonlinear mathematical analysis that is especially suited to the study of optimization problems. Rockafellar's theory differs from classical analysis in that differentiability assumptions are replaced by convexity assumptions. The topics treated in this volume include: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle- functions.

    This book has firmly established a new and vital area not only for pure mathematics but also for applications to economics and engineering. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book. There is also a guide for the reader who may be using the book as an introduction, indicating which parts are essential and which may be skipped on a first reading.

    eISBN: 978-1-4008-7317-3
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Preface
    (pp. vii-viii)
    R. T. R.
  3. Table of Contents
    (pp. ix-x)
  4. Introductory Remarks: A Guide for the Reader
    (pp. xi-xviii)

    This book is not really meant to be read from cover to cover, even if there were anyone ambitious enough to do so. Instead, the material is organized as far as possible by subject matter; for example, all the pertinent facts about relative interiors of convex sets, whether of major or minor importance, are collected in one place (§6) rather than derived here and there in the course of other developments. This type of organization may make it easier to refer to basic results, at least after one has some acquaintance with the subject, yet it can get in the...

    • SECTION 1 Affine Sets
      (pp. 3-9)

      Throughout this book,Rdenotes the real number system, andRnis the usual vector space of realn-tuplesx= (ξ1, … ,ξn). Everything takes place inRnunless otherwise specified. The inner product of two vectorsxandx*inRnis expressed by

      \[\left\langle x,{{x}^{*}} \right\rangle ={{\xi }_{1}}\xi _{1}^{*}+\cdots +{{\xi }_{n}}\xi _{n}^{*}\]

      The same symbolAis used to denote anm×nreal matrixAand the corresponding linear transformationxAxfromRntoRm. The transpose matrix and the corresponding adjoint linear transformation fromRmtoRnare denoted byA*, so that one has the identity

      \[\left\langle Ax,{{y}^{*}} \right\rangle =\left\langle x,{{A}^{*}}{{y}^{*}} \right\rangle .\]

      {In a...

    • SECTION 2 Convex Sets and Cones
      (pp. 10-15)

      A subsetCofRnis said to beconvexif (1 -λ)x+λyCwheneverxC,YCand 0 <λ< 1. All affine sets (including Ø andRnitself) are convex. What makes convex sets more general than affine sets is that they only have to contain, along with any two distinct pointsxandy, a certain portion of the line throughxandy, namely

      \[\{(1-\lambda )x+\lambda y|0\le \lambda \le 1\}.\]

      This portion is called the (closed)line segment between x and y. Solid ellipsoids and cubes inR3, for instance, are convex but not...

    • SECTION 3 The Algebra of Convex Sets
      (pp. 16-22)

      The class of convex sets is preserved by a rich variety of algebraic operations.

      For instance, ifCis a convex set inRnthen so is every translateC+aand everyscalar multipleλC, where

      \[\lambda C=\{\lambda x|x\in C\}.\]

      In geometric terms, ifλ> 0,λCis the image ofCunder the transformation which expands (or contracts)Rnby the factorλwith the origin fixed.

      Thesymmetric reflectionofCacross the origin is −C= (−1)C. A convex set is said to besymmetricif −C=C. Such a set (if non-empty) must contain the origin, since...

    • SECTION 4 Convex Functions
      (pp. 23-31)

      Letfbe a function whose values are real or ±∞ and whose domain is a subsetSofRn. The set

      \[\{(x,\mu )|x\in S,\ \mu \in R,\ \mu \ge f(x)\}.\]

      is called theepigraphoffand is denoted by epifWe definefto be aconvexfunction onSif epifis convex as a subset ofRn+1. Aconcavefunction onSis a function whose negative is convex. Anaffinefunction onSis a function which is finite, convex and concave.

      Theeffective domainof a convex functionfonS, which we denote by domf, is the projection...

    • SECTION 5 Functional Operations
      (pp. 32-40)

      How can new convex functions be obtained from functions already known to be convex? There are many operations which preserve convexity, as it turns out. Some of the operations, like pointwise addition of functions, are familiar from ordinary analysis. Others, like taking the convex hull of a collection of functions, are geometrically motivated. Often the constructed function is expressed as a constrained infimum, thereby suggesting applications to the theory of extremum problems.

      Familiarity with the operations below is helpful, especially, when one has to prove that a given function with a complicated formula is a convex function.

      Theorem 5.1.Let...

    • SECTION 6 Relative Interiors of Convex Sets
      (pp. 43-50)

      TheEuclidean distancebetween two pointsxandyinRnis by definition

      \[d(x,y)=\left| x-y \right|={{\left\langle x-y,x-y \right\rangle }^{1/2}}\]

      The functiond, theEuclidean metric, is convex as a function onR2n. (This follows from the fact thatdis obtained by composing the Euclidean normf(z)=|z| with the linear transformation (x,y) →xyfromR2ntoRn.) The familiar topological concepts of closed set, open set, closure, and interior inRnare usually introduced in terms of convergence of vectors with respect to the Euclidean metric. But such convergence is, of course, equivalent to the convergence of a sequence of...

    • SECTION 7 Closures of Convex Functions
      (pp. 51-59)

      The continuity of a linear function is a consequence of an algebraic property, linearity. With convex functions, things are not quite so simple, but still a great many topological properties are implied by convexity alone. These can be deduced by applying the theory of closures and relative interiors of convex sets to the epigraphs or level sets of convex functions. One of the principal conclusions which can be reached is that lower semi-continuity is a “constructive” property for convex functions. It will by demonstrated below, namely, that there is a simple closure operation which makes any proper convex function lower...

    • SECTION 8 Recession Cones and Unboundedness
      (pp. 60-71)

      Closed bounded subsets ofRnare usually easier to work with than unbounded ones. When the sets are convex, however, the difficulties with unboundedness are very much less, and that is fortunate, since so many of the sets we need to consider, like epigraphs, are unbounded by their nature.

      Unbounded closed convex sets have a simple behavior “at infinity,” according to one’s intuition. Suppose thatCis such a set andxis a point ofC. It seems thatCmust actually contain some entire half-line starting atx, or the unboundedness would be contradicted. Thedirectionsof such...

    • SECTION 9 Some Closedness Criteria
      (pp. 72-81)

      There are many operations for convex sets which preserve relative interiors but have a more complicated behavior with respect to closures. For example, given a convex setCand a linear transformationA, one has ri (AC) =A(riC), but in general only cl (AC) ⊃A(clC) (Theorem 6.6). When is cl (AC) actually equal toA(clC)? When is the image of a closed convex set closed?

      Such questions are worth careful attention. One reason is that they are connected with the preservation of lower semi-continuity. The epigraph of the imageAhof a proper convex function...

    • SECTION 10 Continuity of Convex Functions
      (pp. 82-92)

      The closure operation for convex functions alters a function “slightly” to make it lower semi-continuous. We shall now describe some common situations where a convex functionfis automatically upper semi-continuous, so that clf(orfitself to the extent that it agrees with clf) is actually continuous. Uniform continuity and equicontinuity will also be considered. In every case, a strong conclusion about continuity follows from an elementary hypothesis, because of convexity.

      A functionfonRnis said to becontinuous relative to a subset SofRnif the restriction offtoSis a...

    • SECTION 11 Separation Theorems
      (pp. 95-101)

      The notion of separation has proved to be one of the most fertile notions in convexity theory and its applications. It is based on the fact that a hyperplane inRndividesRnevenly in two, in the sense that the complement of the hyperplane is the union of two disjoint open convex sets, the open half-spaces associated with the hyperplane.

      LetC1andC2be non-empty sets inRn. A hyperplaneHis said toseparate C1andC2ifC1is contained in one of the closed half-spaces associated withHandC2lies in the opposite closed...

    • SECTION 12 Conjugates of Convex Functions
      (pp. 102-111)

      There are two ways of viewing a classical curve or surface like a conic, either as a locus of points or as an envelope of tangents. This fundamental duality enters the theory of convexity in a slightly different form: a closed convex set inRnis the intersection of the closed half-spaces which contain it (Theorem 11.5). Many intriguing duality correspondences exist as embodiments of this fact, among them conjugacy of convex functions, polarity of convex cones or of other classes of convex sets or functions, and the correspondence between convex sets and their support functions. The basic theory of...

    • SECTION 13 Support Functions
      (pp. 112-120)

      A common sort of extremum problem is that of maximizing a linear function$\left\langle \cdot ,{{x}^{*}} \right\rangle $over a convex setCinRn. One fruitful approach to such a problem is to study what happens asx*varies. This leads to the consideration of the function which expresses the dependence of the supremum onx*, namely thesupport function${{\delta }^{*}}(\cdot |C)$ofC:

      \[{{\delta }^{*}}({{x}^{*}}|C)=\sup \{\left\langle x,{{x}^{*}} \right\rangle |x\in C\}.\]

      The appropriateness of theδ*notation for the support function will be clear below.

      Minimization of linear functions overC, as well as maximization, can be studied in terms of${{\delta }^{*}}(\cdot |C)$, because

      \[\inf \left\{ \left\langle x,{{x}^{*}} \right\rangle |x\in C \right\}=-{{\delta }^{*}}(-{{x}^{*}}|C).\]

      The support function ofC...

    • SECTION 14 Polars of Convex Sets
      (pp. 121-127)

      The correspondence between convex sets and their support functions reflects a certain duality between positive homogeneity and the property of being an indicator function. Namely, supposefis a proper convex function onRn. Iffis an indicator function, its conjugatef*is positively homogeneous (Theorem 13.2). Iffis positively homogeneous,f*is an indicator function (Corollary 13.2.1). It follows that, iffis a positively homogeneous indicator function, thenf*is a positively homogeneous indicator function. Of course, the positively homogeneous indicator functions are simply the indicator functions of cones. Thus, iff(x) =δ(x|K)...

    • SECTION 15 Polars of Convex Functions
      (pp. 128-139)

      A functionkonRnwill be called agaugeifkis a non-negative positively homogeneous convex function such thatk(0) = 0, i.e. if epikis a convex cone inRn+1containing the origin but not containing any vectors (x, μ) such thatμ< 0. Gauges are thus the functionsksuch that

      \[k(x)=\gamma (x|C)=\inf \{\mu \ge 0|x\in \mu C\}\]

      for some non-empty convex setC. Of course,Cis not uniquely determined bykin general, although one always has$\gamma (\cdot |C)=k$for

      \[C=\{x|k(x)\le 1\}.\]

      Ifkis closed, the latterCis theunique closedconvex set containing the origin such that$\gamma (\cdot |C)=k$...

    • SECTION 16 Dual Operations
      (pp. 140-150)

      Suppose we perform some operation on given convex functionsf1, … ,fm, such as adding them. How is the conjugate of the resulting function related to the conjugate functions$f_{1}^{*},\ldots ,f_{m}^{*}$? Similar questions can be asked about the behavior of set or functional operations under the polarity correspondences. In most cases, it turns out that the duality correspondence converts a familiar operation into another familiar operation (modulo some details about closures). The operations thus arrange themselves in dual pairs.

      We begin with some simple cases already covered by Theorem 12.3. Lethbe any convex function onRn. If...

    • SECTION 17 Carathéodory’s Theorem
      (pp. 153-161)

      IfSis a subset ofRn, the convex hull ofScan be obtained by forming all convex combinations of elements ofS. According to the classical theorem of Carathéodory, it is not really necessary to form combinations involving more thann+ 1 elements at a time. One can limit attention to convex combinationsλlxl+ … +λmxmsuch thatmn+ 1 (or even to combinations such thatm=n+ 1, if one does not insist on the vectorsxibeing distinct).

      Carathéodory’s Theorem is the fundamental dimensionality result in convexity theory,...

    • SECTION 18 Extreme Points and Faces of Convex Sets
      (pp. 162-169)

      Given a convex setC, there exist various point setsSsuch thatC= convS. For any suchS, the points ofCcan be expressed as convex combinations of the pointsSas in Carathéodory’s Theorem. One may call this an “internal representation” ofC, in distinction to an “external representation” ofCas the intersection of some collection of half-spaces. Representations of the formC= convSorC= cl (convS) can also be considered in whichScontains both points and directions, as in the preceding section. Of course, the smaller or...

    • SECTION 19 Polyhedral Convex Sets and Functions
      (pp. 170-178)

      Apolyhedralconvex set inRnis by definition a set which can be expressed as the intersection of some finite collection of closed half-spaces, i.e. as the set of solutions to some finite system of inequalities of the form

      \[\left\langle x,{{b}_{i}} \right\rangle \le {{\beta }_{i}},\quad i=1,\ldots ,m.\]

      Actually, of course, the set of solutions to any finite mixed system of linear equations and weak linear inequalities is a polyhedral convex set, since an equation ⟨x, b⟩ =βcan always be expressed as two inequalities: ⟨x, b⟩ ≤βand ⟨x, −b⟩ ≤ −β. Every affine set (including the empty set andRn) is polyhedral (Corollary...

    • SECTION 20 Some Applications of Polyhedral Convexity
      (pp. 179-184)

      In this section, we shall show how certain separation theorems, closure conditions and other results which were proved earlier for general convex sets and functions may be refined when some of the convexity is polyhedral.

      We begin with the general formula for the conjugate of a sum of proper convex functions (Theorem 16.4):

      \[{{\text{(cl }{{f}_{1}}+\cdots +\text{cl }{{f}_{m}})}^{*}}=\text{cl }(f_{1}^{*}\square \cdots \square f_{m}^{*}).\]

      Suppose that everyfi; is polyhedral, and that

      \[\text{dom }{{f}_{1}}\cap \cdots \cap \text{dom }{{f}_{m}}\ne \phi .\]

      Then clfi=fi, andf1+ … +fmis a proper polyhedral convex function (Theorem 19.4). The conjugate off1+ … +fmmust be proper too, so$f_{1}^{*}\square \cdots \square f_{m}^{*}$must be proper. Every...

    • SECTION 21 Helly’s Theorem and Systems of Inequalities
      (pp. 185-197)

      By a system ofconvexinequalities inRn, we shall mean a system which can be expressed in the form

      \[{{f}_{i}}(x)\le {{\alpha }_{i}},\quad \forall i\in {{I}_{1,}}\]

      \[{{f}_{i}}(x)\le {{\alpha }_{i}},\quad \forall i\in {{I}_{2,}}\]

      whereI=I1I2is an arbitrary index set, each,fiis a convex function onRn, and -∞ ≤ αi≤ +∞. The set of solutionsxto such a system is, of course, a certain convex set inRn, the intersection of the convex level sets

      \[(x|{{f}_{i}}(x)\le {{\alpha }_{i}}\},\quad i\in {{I}_{1}},\]

      \[(x|{{f}_{i}}(x)\le {{\alpha }_{i}}\},\quad i\in {{I}_{2}},\]

      If everyfi; is closed and there are no strict inequalities (i.e.I2= ɸ), the set of solutions is closed. The system is said to...

    • SECTION 22 Linear Inequalities
      (pp. 198-210)

      This section treats the theory of finite systems of (weak or strict) linear inequalities. First we shall state various existence results as special cases of relatively difficult theorems that have been established in §21 for more general systems of inequalities. An alternate method of development will then be presented which yields the same results in an elementary way independent of the general theory of convexity.

      Theorem 22.1.Let aiRnand αiR for i= 1, … ,m. Then one and only one of the following alternatives holds:

      (a)There exists a vector xRnsuch...

    • SECTION 23 Directional Derivatives and Subgradients
      (pp. 213-226)

      Convex functions have many useful differential properties, and one of these is the fact that one-sided directional derivatives exist universally. Just as the ordinary two-sided directional derivatives of a differentiable functionfcan be described in terms of gradient vectors, which correspond to tangent hyperplanes to the graph off, the one-sided directional derivatives of any proper convex functionf, not necessarily differentiable, can be described in terms of “subgradient” vectors, which correspond to supporting hyperplanes to the epigraph off

      Letfbe any function fromRnto [-∞, +∞], and letxbe a point wherefis...

    • SECTION 24 Differential Continuity and Monotonicity
      (pp. 227-240)

      Letfbe a closed proper convex function onRn. The subdifferential mapping of∂fdefined in the preceding section assigns to eachxRna certain closed convex subset∂f(x) ofRn. Theeffective domainof∂f, which is the set

      \[\text{dom }\partial f=\{x|\partial f(x)\ne \phi \},\]

      is not necessarily convex, but it differs very little from being convex, in the sense that

      \[\text{ri }(\text{dom }f)\subset \text{dom }\partial f\subset \text{dom }f\]

      (Theorem 23.4). Therangeof∂fas a multivalued mapping is defined by

      \[\text{range }\partial f=\cup \{\partial f(x)|x\in {{R}^{n}}\}.\]

      The range of∂fis the effective domain of∂f*by Corollary 23.5.1, so

      \[\text{ri }(\text{dom }{{f}^{*}})\subset \text{range }\partial f\subset \text{dom }{{f}^{*}}.\]

      Certain continuity and monotonicity properties of∂fand the set...

    • SECTION 25 Differentiability of Convex Functions
      (pp. 241-250)

      Letfbe a function fromRnto [-∞, +∞], and letxbe a point wherefis finite. According to the usual definition,fisdifferentiableatxif and only if there exists a vectorx*(necessarily unique) with the property that

      \[f(z)=f(x)+\left\langle {{x}^{*}},z-x \right\rangle +o(\left| z-x \right|),\]

      or in other words

      \[\underset{z\to x}{\mathop{\lim }}\,\frac{f(z)-f(x)-\left\langle {{x}^{*}},z-x \right\rangle }{\left| z-x \right|}=0.\]

      Such anx*, if it exists, is called thegradientoffatxand is denoted by ∇f(x).

      Suppose thatfis differentiable atx. Then by definition, for anyy≠ 0,

      \[0=\underset{\lambda \downarrow 0}{\mathop{\lim }}\,\frac{f(x+\lambda y)-f(x)-\left\langle \nabla f(x),\lambda y \right\rangle }{\lambda \left| y \right|} =[f^\prime (x;y)-\left\langle \nabla f(x),y \right\rangle ]/\left| y \right|.\]

      Thereforef(x;y) exists and is a linear function ofy:

      \[{{f}^{\prime }}(x;y)=\left\langle \nabla f(x),y \right\rangle ,\quad \quad \forall y.\]

      In particular,...

    • SECTION 26 The Legendre Transformation
      (pp. 251-260)

      The classical Legendre transformation for differentiable functions defines a correspondence which, for convex functions, is intimately connected with the conjugacy correspondence. The Legendre transformation will be investigated here in the light of the general differential theory of convex functions. We shall show that the case where it is well-defined and involutory is essentially the case where the subdifferential mapping of the convex function is single-valued and in fact one-to-one.

      A multivalued mappingρwhich assigns to eachxRna setρ(x) ⊂Rnis said to besingle-valued, of course, ifρ(x) containsat mostone elementx*...

    • SECTION 27 The Minimum of a Convex Function
      (pp. 263-272)

      The great importance of extremum problems and variational principles in applied mathematics leads one to the general study of the minimum or maximum (or of certain minimax extrema) of a functionhover a setC. When a sufficient amount of convexity is present, the study is greatly simplified, and many significant theorems can be established, particularly as regards duality and characterizations of the points where the extrema are attained.

      In this section we shall study the minimum of a convex functionhover a convex setCinRn. There is no real loss of generality in assumingh...

    • SECTION 28 Ordinary Convex Programs and Lagrange Multipliers
      (pp. 273-290)

      The theory of Lagrange multipliers tells how to transform certain constrained extremum problems into extremum problems involving fewer constraints but more variables. Here we shall present the branch of the theory which is applicable to problems of minimizing convex functions subject to “convex” constraints.

      By anordinary convex program(P) (as opposed to a “generalized” convex program, to be defined in §29), we shall mean a problem of the following form: minimizef0(x) overCsubject to constraints

      \[{{f}_{1}}(x)\le 0,\ldots ,{{f}_{r}}(x)\le 0,\quad {{f}_{r+1}}(x)=0,\ldots ,{{f}_{m}}(x)=0,\]

      whereCis a non-empty convex set inRn,fiis a finite convex function onCfori= 0,...

    • SECTION 29 Bifunctions and Generalized Convex Programs
      (pp. 291-306)

      In an ordinary convex program (P), one is interested in minimizing a certain convex function onRn, the objective function for (P), whose effective domain is the set of all feasible solutions to (P). But there is more to (P) than just this abstract minimization problem. Another ordinary convex program can have the same objective function as (P) and yet have a different Lagrangian and different Kuhn–Tucker coefficients. If one is to have a full generalization of the concept of “convex program,” one must somehow take this fact into account.

      The Kuhn–Tucker vectors corresponding to an ordinary convex...

    • SECTION 30 Adjoint Bifunctions and Dual Programs
      (pp. 307-326)

      A fundamental fact about generalized convex programs is that each such “minimization problem with perturbations” has a dual, which is a certain “maximization problem with perturbations,” a generalized concave program. In most cases, two programs dual to each other have the same optimal-value, and the optimal solutions to one are the Kuhn–Tucker vectors for the other.

      The duality theory for convex programs is based on a concept of the “adjoint” of a convex bifunction. The adjoint operation for bifunctions may be regarded as a generalization of the adjoint operation for linear transformations, and a considerable “convex algebra” parallel to...

    • SECTION 31 Fenchel’s Duality Theorem
      (pp. 327-341)

      Fenchel’s duality theorem pertains to the problem of minimizing a differencef(x) −g(x), wherefis a proper convex function onRnandgis a proper concave function onRn. This problem includes, as a special case, the problem of minimizingfover a convex setC(take$g=-\delta (\cdot |C))$. In general,fgis a certain convex function onRn. The duality resides in the connection between minimizingfgand maximizing the concave functiong*f*, wheref*is the (convex) conjugate offandg*is the (concave) conjugate ofg. This...

    • SECTION 32 The Maximum of a Convex Function
      (pp. 342-346)

      The theory of the maximum of a convex function relative to a convex set has an entirely different character from the theory of the minimum. For one thing, it is possible, even likely, in a given case that there are many local maxima besides the global maximum. This phenomenon is rather disastrous as far as computation is concerned, because once a local maximum has been found there is, more or less by definition, no local information to tell one how to proceed to a higher local maximum. In particular, there is no local criterion for deciding whether a given local...

    • SECTION 33 Saddle-Functions
      (pp. 349-358)

      LetCandDbe subsets ofRmandRnrespectively, and letKbe a function fromC×Dto [-∞, +∞]. We say thatKis aconcave-convexfunction ifK(u, v) is a concave function ofuCfor eachvDand a convex function ofvDfor eachuC.Convex-concavefunctions are defined similarly. We speak of both kinds of functions assaddle-functions.

      The theory of saddle-functions, like that of purely convex or concave functions, can be reduced conveniently to the case where the functions are everywhere defined...

    • SECTION 34 Closures and Equivalence Classes
      (pp. 359-369)

      A pairing has been established in §33 between the lower closed saddle-functions$\underline{K}$and the upper closed saddle-functions$\bar{K}$onRm×Rn, where each pair corresponds to a uniquely determined closed convex bifunction and its (closed concave) adjoint. This pairing will be extended below to an equivalence relation amongclosedsaddle-functions, “closed” being a slightly weaker notion than “lower closed” or “upper closed.” The structure of closed saddle-functions will be analyzed in detail. We shall show that each “proper” equivalence class of closed saddle-functions is uniquely determined by its “kernel,” which is a finite saddle-function on a product of...

    • SECTION 35 Continuity and Differentiability of Saddle-Functions
      (pp. 370-378)

      The purpose of this section is to show how the main results about regularity properties of convex functions, such as continuity and differentiability, can be extended to saddle-functions. The continuity and convergence theorems in §10 will be dealt with first.

      Theorem 35.1.Let C and D be relatively open convex sets in Rmand Rn, respectively, and let K be a finite concave-convex function on C×D. Then K is continuous relative to C×D. In fact, K is Lipschitzian on every closed bounded subset of C×D.

      Proof. It suffices to show thatKis Lipschitzian...

    • SECTION 36 Minimax Problems
      (pp. 379-387)

      Minimax theory treats a class of extremum problems which involve, not simply minimization or maximization, but a combination of both. LetCandDbe arbitrary non-empty sets, and letKbe a function fromC×Dto [-∞, +∞]. For eachuC, one can take the infimum ofK(u, v) overvDand then take the supremum of this infimum as a function onC. The quantity so obtained is

      \[\underset{u\in C}{\mathop{\sup }}\,\underset{v\in D}{\mathop{\inf }}\,K(u,v).\]

      On the other hand, for eachvDone can take the supremum ofK(u, v) overuCand then...

    • SECTION 37 Conjugate Saddle-Functions and Minimax Theorems
      (pp. 388-398)

      Questions about saddle-values and saddle-points of concave-convex functions can be reduced essentially to questions about (generalized) convex programs and their associated Lagrangian problems, as has been shown in §36. The main existence theorems will be presented here in terms of a conjugacy correspondence among concave-convex functions, much as the main theorems concerning the minimum of a convex function were presented in §27 in terms of the conjugacy correspondence for convex functions.

      The notion of the conjugate of a saddle-function is derived from properties of the inverse operation for convex bifunctions, which was introduced in the preceding section. Thus, as it...

    • SECTION 38 The Algebra of Bifunctions
      (pp. 401-412)

      The adjoint and inverse operations for convex and concave bifunctions generalize the adjoint and inverse operations for linear transformations in the following sense, as has already been pointed out. LetAbe a linear transformation fromRmtoRn, and letFbe the convex indicator bifunction ofA, i.e. the closed proper convex bifunction fromRmtoRndefined by

      \[(Fu)(x)=\delta (x|Au)=\left\{ \begin{matrix} 0 \quad \quad \ \text{if}\quad \text{ }x=Au, \\ +\infty \quad \text{if }\quad x\ne Au. \\ \end{matrix} \right.\]

      The adjointF*ofFis then the concave indicator bifunction of the adjoint linear transformationA*,

      \[({{F}^{*}}{{x}^{*}})({{u}^{*}})=-\delta ({{u}^{*}}|{{A}^{*}}{{u}^{*}})=\left\{ \begin{matrix} 0 \quad \quad \ \text{if}\quad \text{ }{{u}^{*}}={{A}^{*}}{{u}^{*}}, \\ -\infty \quad \text{if }\quad {{u}^{*}}\ne {{A}^{*}}{{u}^{*}}, \\ \end{matrix} \right.\]

      and we have

      \[\left\langle Fu,{{x}^{*}} \right\rangle =\left\langle Au,{{x}^{*}} \right\rangle =\left\langle u,{{A}^{*}}{{x}^{*}} \right\rangle =\left\langle u,{{F}^{*}}{{x}^{*}} \right\rangle .\]

      IfAis nonsingular, the inverseF*ofFis the concave indicator bifunction ofA-1,...

    • SECTION 39 Convex Processes
      (pp. 413-424)

      The notion of a convex process is intermediate between that of a linear transformation and that of a convex bifunction. Convex processes form an algebra of multivalued mappings with many interesting duality properties. These properties can be deduced from theorems already established for bifunctions, which they help to illuminate.

      Aconvex processfromRmtoRnis a multivalued mappingA:uAusuch that

      (a)A(u1+u2) ⊃Au1+Au2, ∀u1, ∀u2,

      (b)A(λU) =λAu, ∀u, ∀λ> 0,

      (c) 0 ∈A0.

      Condition (c) means that the set

      \[\text{graph }A=\{(u,x)|u\in {{R}^{m}},x\in {{R}^{m}},x\in {{R}^{n}},x\in Au\}\]

      contains the origin ofRm+n. Condition (a)...

  13. Comments and References
    (pp. 425-432)
  14. Bibliography
    (pp. 433-446)
  15. Index
    (pp. 447-451)