Come contributions to the theory of generalized polyhedral optimization problems

with respect tolcHtvs locally convex Hausdorff topological vector space gpcs generalized polyhedral convex set gpcf generalized polyhedral convex function PLVOP piecewise linear vector o

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

NGUYEN NGOC LUAN

SOME CONTRIBUTIONS

TO THE THEORY OF GENERALIZED

POLYHEDRAL OPTIMIZATION PROBLEMS

DISSERTATION

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

HANOI - 2019

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

INSTITUTE OF MATHEMATICS

NGUYEN NGOC LUAN

SOME CONTRIBUTIONS

TO THE THEORY OF GENERALIZED

POLYHEDRAL OPTIMIZATION PROBLEMS

DISSERTATION

SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN MATHEMATICS

Supervisor: Prof Dr.Sc NGUYEN DONG YEN

HANOI - 2019

This dissertation was written on the basis of my research works carried out

at Institute of Mathematics, Vietnam Academy of Science and Technologyunder the supervision of Prof Dr.Sc Nguyen Dong Yen All the presentedresults have never been published by others

August 06, 2019The author

Nguyen Ngoc Luan

Viet-my special appreciation to Prof Hoang Xuan Phu, Assoc Prof Ta DuyPhuong, Assoc Prof Phan Thanh An, and other members of the weeklyseminar at Department of Numerical Analysis and Scientific Computing, In-stitute of Mathematics, as well as all the members of Prof Nguyen DongYen’s research group for their valuable comments and suggestions on my re-search results In particular, I would like to express my sincere thanks to

Dr Thai Doan Chuong for his significant comments and suggestions cerning the research related to Chapters 1 and 5 of this dissertation

con-I would like to thank the Assoc Prof Truong Xuan Duc Ha, Prof LeDung Muu, Assoc Prof Pham Ngoc Anh, Assoc Prof Tran Dinh Ke,Assoc Prof Nguyen Thi Thu Thuy, and Dr Le Hai Yen, and the twoanonymous referees, for their careful readings of this dissertation and valuablecomments

I am sincerely grateful to Prof Jen-Chih Yao from China Medical versity and National Sun Yat-sen University, Taiwan, for granting severalshort-termed scholarships for my PhD studies

Uni-Furthermore, I would like to thank my colleagues at Department of ematics and Informatics, Hanoi National University of Education for theirefficient help during the years of my Master and PhD studies

Math-Finally, I would like to thank my family for their endless love and ditional support

uncon-The research related to this dissertation was supported by Vietnam tional Foundation for Science and Technology Development (NAFOSTED)and Hanoi National University of Education.

1.1 Preliminaries 1

1.2 Representation Formulas for Generalized Convex Polyhedra 2

1.3 Characterizations via the Finiteness of the Faces 12

1.4 Images via Linear Mappings and Sums of Generalized Polyhe-dral Convex Sets 17

1.5 Convex Hulls and Conic Hulls 23

1.6 Relative Interiors of Polyhedral Convex Cones 27

1.7 Solution Existence in Linear Optimization 31

1.8 Conclusions 34

Chapter 2 Generalized Polyhedral Convex Functions 35 2.1 Generalized Polyhedral Convex Function as a Maximum of Finitely Many Affine Functions 35

2.2 Piecewise Linearity of Generalized Polyhedral Convex Func-tions and an Application 39

2.3 Directional Derivatives 41

2.4 Infimal Convolutions 43

2.5 Conclusions 45

3.1 Normal Cones 46

3.2 Polars 51

3.3 Conjugate Functions 52

3.4 Subdifferentials 53

3.5 Conclusions 57

Chapter 4 Generalized Polyhedral Convex Optimization Prob-lems 58 4.1 Motivations 58

4.2 Solution Existence Theorems 61

4.3 Optimality Conditions 69

4.4 Duality 77

4.5 Conclusions 84

Chapter 5 Linear and Piecewise Linear Vector Optimization Problems 85 5.1 Preliminaries 85

5.2 The Weakly Efficient Solution Set in Linear Vector Optimization 86 5.3 The Efficient Solution Set in Linear Vector Optimization 89

5.4 Structure of the Solution Sets in the Convex Case 94

5.5 Structure of the Solution Sets in the Nonconvex Case 102

5.6 Conclusions 111

Table of Notations

¯

A ⊂ B A is a subset of B (the case A = B is not

ex-cluded)

int A the topological interior of A

¯

C[a, b] the linear space of continuous real-valued

functions on the interval [a, b]

dom f the effective domain of a function f

sup

x∈D

f (x) the supremum of the set {f (x) | x ∈ D}

inf

x∈Df (x) the infimum of the set {f (x) | x ∈ D}

TC(x) the tangent cone of C at x

∂f (x) the subdifferential of f at x

f0(x; h) the directional derivative of f at x

with respect to a direction h

M : X → Y an operator from X to Y

M∗ : Y∗ → X∗ the adjoint operator of M

span {xj | j = 1, , m} the linear subspace generated by

vectors xj, j = 1, , m

w.r.t with respect to

lcHtvs locally convex Hausdorff topological vector space

gpcs generalized polyhedral convex set

gpcf generalized polyhedral convex function

PLVOP piecewise linear vector optimization problem

Vector optimization has a rich history and diverse applications Vectoroptimization (sometimes called multiobjective optimization) is a natural de-velopment of scalar optimization F.Y Edgeworth (1881) and V Pareto(1906) defined a notion, which later was called Pareto solution This so-lution concept remains the most important in vector optimization Otherbasic solution concepts of this theory are weak Pareto solution and propersolution The latter has been defined in different ways by A.M Geoffrion,J.M Borwein, H.P Benson, M.I Henig, and other authors

Vector optimization has numerous applications in economics, managementscience, and engineering; see, e.g., [2, 22, 41, 67]

One calls a vector optimization problem (VOP) linear if the objectivefunctions are linear (affine) functions and the constraint set is polyhedralconvex (i.e., it is a intersection of a finite number of closed half-spaces) If

at least one of the objective functions is nonlinear (non-affine, to be moreprecise) or the constraint set is not a polyhedral convex set (for example, it

is merely a closed convex set or, more general, a solution set of a system ofnonlinear inequalities), then the VOP is said to be nonlinear

Linear VOPs have been considered in many books (see, e.g., [50, 51]) and

in numerous papers (see, e.g., [3, 34, 38, 39]) The classical Blackwell Theorem (the ABB Theorem; see, e.g., [3, 50]) asserts that, for

Arrow-Barankin-a lineArrow-Barankin-ar vector optimizArrow-Barankin-ation problem, the PArrow-Barankin-areto solution set Arrow-Barankin-and the weArrow-Barankin-akPareto solution set are connected by line segments and are the unions offinitely many faces of the constraint set This is an example of qualitativeproperties of vector optimization problems Quantitative aspects (i.e., solu-tion methods) are also very important in vector optimization Observe that,the second part of the recent book [51] of D.T Luc on linear vector optimiza-tion discusses qualitative properties, while the entire third part is devoted to

quantitative aspects of the problems in question.

Nonlinear VOPs have been considered in many books (see, e.g., [2, 41, 50])and research papers (see, e.g., [37, 75, 76, 79, 82])

This dissertation focuses on linear VOPs and several related nonlinearscalar optimization problems, as well as nonlinear vector optimization prob-lems Namely, apart from linear VOPs in locally convex Hausdorff topologicalvector spaces, which are the main subjects of our research, we will study poly-hedral convex optimization problems and piecewise linear vector optimizationproblems

The dissertation is put on the framework of functional analysis, convexanalysis, and convex optimization The book by Rudin [65] is main source

of the facts from functional analysis used herein Observe that sive results on convex analysis and convex optimization in locally convexHausdorff topological vector spaces can be found in the books by Ioffe andTihomirov [40], Z˘alinescu [78]

comprehen-The fundamental concepts used in this dissertation are polyhedral convexset and polyhedral convex function on locally convex Hausdorff topologicalvector spaces About one half of the dissertation is devoted to these con-cepts Another half of the dissertation shows how our new results on poly-hedral convex sets and polyhedral convex functions can be applied to scalaroptimization problems and VOPs

The notions of polyhedral convex set – also called a convex polyhedron, andgeneralized polyhedral convex set – also called a generalized convex polyhedron,stand in the crossroad of several mathematical theories

First, let us briefly review some basic facts about polyhedral convex set

in a finite-dimensional setting By definition, a polyhedral convex set in afinite-dimensional Euclidean space is the intersection of a finite family ofclosed half-spaces (By convention, the intersection of an empty family ofclosed half-spaces is the whole space Therefore, emptyset and the wholespace are two special polyhedra.) So, a polyhedral convex set is the solutionset of a system of finitely many inhomogenous linear inequalities This is theanalytical definition of a polyhedral convex set

According to Klee [46, Theorem 2.12] and Rockafellar [63, Theorem 19.1],for every given convex polyhedron one can find a finite number of points and

a finite number of directions such that the polyhedron can be represented asthe sum of the convex hull of those points and the convex cone generated bythose directions The converse is also true This celebrated theorem, which

is a very deep geometrical characterization of polyhedral convex set, is tributed [63, p 427] primarily to Minkowski [55] and Weyl [73, 74] By usingthe result, it is easy to derive fundamental solution existence theorems inlinear programming It is worthy to stress that the above cited represen-tation formula for finite-dimensional polyhedral convex set has many otherapplications in mathematics As an example, we refer to the elegant proofs

at-of the necessary and sufficient second-oder conditions for a local solution andfor a locally unique solution in quadratic programming, which were given byContesse [18] in 1980; see [49, pp 50–63] for details

For polyhedral convex sets, there is another important characterization: Aclosed convex set is a polyhedral convex set if and only if it has finitely manyfaces; see [46, Theorem 2.12] and [63, Theorem 19.1] for details

A bounded polyhedral convex set is called a polytope Leonhard Euler’sTheorem stating a relation between the numbers of faces of different dimen-sions of a polytope is a profound classical result The reader is referred

to [33, pp 130–142b] for a comprehensive exposition of that theorem andsome related results

Functions can be identified with their epigraphs, while sets can be tified with their indicator functions As explained by Rockafellar [63, p xi],

iden-“These identifications make it easy to pass back and forth between a geometricapproach and an analytic approach” In that spirit, it seems reasonable to call

a function generalized polyhedral convex when its epigraph is a generalizedpolyhedral convex set

Now, let us discuss the existing facts about polyhedral convex sets andgeneralized polyhedral convex sets in an infinite-dimensional setting Ac-cording to Bonnans and Shapiro [14, Definition 2.195], a subset of a locallyconvex Hausdorff topological vector space (lcHtvs) is said to be a generalizedpolyhedral convex set (gpcs), or a generalized convex polyhedron, if it is theintersection of finitely many closed half-spaces and a closed affine subspace

of that topological vector space When the affine subspace can be chosen asthe whole space, the generalized polyhedral convex set is called a polyhedralconvex set (pcs), or a convex polyhedron The theories of generalized lin-

ear programming in locally convex Hausdorff topological vector spaces andquadratic programming in Banach spaces (see [14, Sections 2.5.7 and 3.4.3])are based on the concept of generalized convex polyhedron It is worthy tostress that this concept allows one to obtain such beautiful and importantresults as Hoffman’s lemma for systems of equalities and inequalities in Ba-nach spaces [14, Theorem 2.200], the generalized Farkas lemma [14, Propo-sition 2.201], an analogue of the Walkup-Wets theorem in a Banach spacesetting (see [72] and [14, Theorem 2.207]), Robinson’s theorem on the localupper Lipschitzian property for polyhedral multifunctions in a Banach spacesetting (see [62] and [14, Theorem 2.207]), an extension of Frank-Wolfe’s andEaves’ solution existence theorems for quadratic programming in a Hilbertspace setting (see [14, Theorem 3.128] and [49]) Theorem 3.128 of [14] re-quires that the quadratic form must be a Legendre form Recently, by con-structing an elegant example, Dong and Tam [19, Example 3.3] have shownthat the requirement cannot be dropped.

Many applications of polyhedral convex sets and piecewise linear functions

in normed spaces to vector optimization can be found in the papers by Yangand Yen [75], Zheng [80], Zheng and Ng [81], Zheng and Yang [82]

Numerous applications of generalized polyhedral convex sets and ized polyhedral multifunctions in Banach spaces to variational analysis, op-timization problems, and variational inequalities can be found in the works

general-by Henrion, Mordukhovich, and Nam [36], Ban, Mordukhovich, and Song [7],Gfrerer [29, 30], Ban and Song [8]

In 2009, using a result related to the Banach open mapping theorem (see,e.g., [65, Theorem 5.20]), Zheng [80, Corollary 2.1] has clarified the relation-ships between convex polyhedra in Banach spaces and the finite-dimensionalconvex polyhedra

It is well known that any infinite-dimensional normed space equipped withthe weak topology is not metrizable, but it is a locally convex Hausdorff topo-logical vector space Similarly, the dual space of any infinite-dimensionalnormed space equipped with the weak∗ topology is not metrizable, but it

is a locally convex Hausdorff topological vector space Actually, the justmentioned two models provide us with the most typical examples of locallyconvex Hausdorff topological vector spaces, whose topologies cannot be given

by norms It is clear that Zheng’s results in [80] cannot be used neither for a

infinite-dimensional normed space equipped with the weak topology, nor forthe dual space of any infinite-dimensional normed space equipped with theweak∗ topology.

The introduction of these concepts poses an interesting problem Namely,since the entire Section 19 of [63] is devoted to establishing a variety of basicproperties of polyhedral convex sets and polyhedral convex functions whichhave numerous applications afterwards, one may ask whether a similar studycan be done for generalized polyhedral convex sets and generalized polyhedralconvex functions, or not

The systematic study of generalized polyhedral convex sets and generalizedpolyhedral convex function in this dissertation can serve as a basis for furtherinvestigations on minimization of a generalized polyhedral convex function

on a generalized polyhedral convex set – a generalized polyhedral convex mization problem, which is a special infinite-dimensional convex programmingproblem If the objective function is linear, then the just mentioned prob-lem collapse to the generalized linear programming problem introduced andtreated in detail by Bonnans and Shapiro [14, Chapter 2 and p 571] Theconcepts of polyhedral convex optimization problem have attracted much at-tention from researchers (see Rockafellar and Wets [64], Bertsekas, Ned´ıc,and Ozdaglar [12], Boyd and Vandenberghe [15], Bertsekas [10, 11], and thereferences therein) As observed by Bonnans and Shapiro [14, p 133], suchproblems can be viewed as particular cases of conic linear problems whenthe ordering cones in the primal and image spaces are generalized polyhe-dral convex It is worthy to stress that semi-infinite linear programs, themass-transfer problem, maximal flow in a dynamic network, continuous lin-ear programs, and other infinite linear programs can be viewed as conic linearproblems (see Anderson and Nash [1])

opti-Piecewise linear vector optimization problem (PLVOP) is a natural opment of polyhedral convex optimization The study of the structures andcharacteristic properties of these solution sets of PLVOPs is useful in the de-sign of efficient algorithms for solving these PLVOPs Zheng and Yang [82]have proved that for a PLVOP, where the spaces are normed and the con-strain set is a polyhedral convex set, the weak efficient solutions set is theunion of finitely many polyhedral convex sets Moreover, if the objectivefunction is convex w.r.p cone, then the weak efficient solutions set is con-

devel-nected by line segments In order to describe the structure of the efficientsolutions set of PLVOP and obtain sufficient conditions for its connected-ness, Yang and Yen [75] have applied the image space approach [31, 32] tooptimization problems and variational systems and proposed the notion ofsemi-closed polyhedral convex set On account of [75, Theorem 2.1], if thespaces are normed, the image space is of finite dimension, the ordering cone

is a pointed cone, and the constrain set is a polyhedral convex set, then theefficient solution set is the union of finitely many semi-closed polyhedra Inthis setting, if the objective function is convex with respect to a cone, thenthe efficient solutions set is the union of finitely many polyhedra and it isconnected by line segments; see [75, Theorem 2.2] Observe that the maintool for proving the latter results is the representation formula for convexpolyhedra in Rn via a finite number of points and a finite number of direc-tions Theorem 2.3 of [75] is an infinite-dimensional version of the classicalArrow-Barankin-Blackwell Theorem

Fang, Meng, and Yang [24] have studied multiobjective optimization lems with either continuous or discontinuous piecewise linear objective func-tions and polyhedral convex constraint sets They obtained an algebraicrepresentation of a semi-closed polyhedron and apply it to show that theimage of a semi-closed polyhedron under a continuous linear function is al-ways a semi-closed polyhedron They proposed an algorithm for finding thePareto point set of a continuous piecewise linear bi-criteria program and gen-eralized it to the discontinuous case The authors applied that algorithm tosolve discontinuous bi-criteria portfolio selection problems with an `∞ riskmeasure and transaction costs Some examples with the historical data ofthe Hong Kong Stock Exchange are discussed Other results in this directionwere given in [23] and [25] Later, Zheng and Ng [81] have investigated themetric subregularity of piecewise polyhedral multifunctions and applied thisproperty to piecewise linear multiobjective optimization

prob-The dissertation has five chapters, a list of the related papers of the author,

a section of general conclusions, and a list of references

Chapter 1 gives a series of fundamental properties of generalized polyhedralconvex sets

In Chapter 2, we discuss some basic properties of generalized polyhedralconvex functions

Chapter 3 is devoted to several dual constructions including the concepts

of conjugate function and subdifferential of a generalized polyhedral convexfunction

Generalized polyhedral convex optimization problems in locally convexHausdorff topological vector spaces are studied systematically in Chapter 4

We establish solution existence theorems, necessary and sufficient optimalityconditions, weak and strong duality theorems In particular, we show thatthe dual problem has the same structure as the primal problem, and thestrong duality relation holds under three different sets of conditions

Chapter 5 discusses structure of efficient solutions sets of linear vectoroptimization problems and piecewise linear vector optimization problems.The dissertation is written on the basis of 5 papers in the List of Au-thor’s Related Papers on page 113: Paper [A1] published in Optimization,paper [A2] published in Applicable Analysis, paper [A3] published in Numer-ical Functional Analysis and Optimization, paper [A4] published in Journal

of Global Optimization, and paper [A5] published in Acta Mathematica namica

Viet-The results of this dissertation have been presented at

- The weekly seminar of the Department of Numerical Analysis and tific Computing, Institute of Mathematics, Vietnam Academy of Science andTechnology;

Scien The 14th Workshop on “Optimization and Scientific Computing” (April21–23, 2016, Ba Vi, Hanoi);

- The 15th Workshop on “Optimization and Scientific Computing” (April20–22, 2017, Ba Vi, Hanoi);

- The 16th Workshop on “Optimization and Scientific Computing” (April19–21, 2018, Ba Vi, Hanoi);

- The 9th Vietnam Mathematical Congress (August 14–18, 2018, NhaTrang, Khanh Hoa)

Chapter 1

Generalized Polyhedral Convex Sets

In this chapter, we first establish a representation formula for generalizedconvex polyhedra A series of fundamental properties of generalized polyhe-dral convex sets will be obtained in Sections 2-5 In Section 6, by using therepresentation formulas for generalized polyhedral convex sets we will provesolution existence theorems in generalized linear programming

The main theorems of Section 1 below (see Theorems 1.2 and 1.5), whichcan be considered as geometrical descriptions of generalized convex polyhedraand convex polyhedra, are not formal extensions of Theorem 19.1 from [63]and Corollary 2.1 of [80] Recently, Yen and Yang [77] have used Theorem 1.2

to study infinite-dimensional affine variational inequalities (AVIs) on normedspaces It is shown that infinite-dimensional quadratic programming prob-lems and infinite-dimensional linear fractional vector optimization problemscan be studied by using AVIs They have obtained two basic facts aboutinfinite-dimensional AVIs: the Lagrange multiplier rule and the solution setdecomposition

The present chapter is written on the basis of the papers [A1], [A2],and [A3] in the List of Author’s Related Papers on page 113

From now on, if not otherwise stated, X is a locally convex Hausdorfftopological vector space over the reals This means (see, e.g., [65, Defini-tions 1.6, 1.8, and Theorem 1.12]) that

(a) X is a vector space over the field R of reals number;

(b) X is equipped with a topology τ ;

(c) The vector space operations are continuous with respect to τ ;

(d) For any distinct points u, v in X, there exist a neighborhood U of u and

a neighborhood V of v such that U ∩ V = ∅;

(e) There is a base B of neighborhoods of 0 such that every neighborhood

U ∈ B is a convex set

We denote by X∗ the dual space of X and by hx∗, xi the value of x∗ ∈ X∗

at x ∈ X If X is a Hilbert space with the scalar product (x, y), then by theRiesz theorem one can identify X∗ with X Namely, for each x∗ ∈ X∗ thereexists a unique vector y ∈ X such that, for all x ∈ X, (y, x) = hx∗, xi Takingaccount of the last identity, one would prefer to replace (y, x) by hy, xi Thisway of writing the scalar product in a Hilbert space or in an Euclidean space

is used in the whole dissertation

For a subset Ω ⊂ X of a locally convex Hausdorff topological vector space,

we denote its interior by int Ω, and its topological closure by Ω The convexhull of a subset Ω is denoted by conv Ω

One says that a nonempty subset K ⊂ X is a cone if tK ⊂ K for every

t > 0 A cone K ⊂ X is said to be a pointed cone if `(K) = {0}, where

`(K) := K ∩ (−K) For a subset Ω ⊂ X, by cone Ω we denote the smallestconvex cone containing Ω, that is, cone Ω = {tx | t > 0, x ∈ conv Ω}

Any normed space is a locally convex Hausdorff topological vector space.Its is also well known (see, e.g., [65, Sections 3.12, 3.14]) that if X is a normedspace, then X (resp., X∗) equipped with the week topology (resp the weak∗topology) is a locally convex Hausdorff topological vector space

Polyhedra

We begin this section with the definition of generalized polyhedral convexset due to Bonnans and Shapiro [14]

Definition 1.1 (See [14, p 133]) A subset D ⊂ X is said to be a generalizedpolyhedral convex set, or a generalized convex polyhedron, if there exist some

x∗i ∈ X∗, αi ∈ R, i = 1, 2, , p, and a closed affine subspace L ⊂ X, suchthat

D = 

x ∈ X | x ∈ L, hx∗i, xi ≤ αi, i = 1, , p

If D can be represented in the form (1.1) with L = X, then we say that

it is a polyhedral convex set, or a convex polyhedron (Hence, the notion ofpolyhedral convex set is more specific than that of generalized polyhedralconvex set.)

Let D be given as in (1.1) According to [14, Remark 2.196], there exists acontinuous surjective linear mapping A from X to a locally convex Hausdorfftopological vector space Y and a vector y ∈ Y such that

L =

x ∈ X | A(x) = y

;then

D = 

x ∈ X | A(x) = y, hx∗i, xi ≤ αi, i = 1, , p

Set I = {1, , p} and I(x) = {i ∈ I | hx∗i, xi = αi} for x ∈ D

From Definition 1.1 it follows that every generalized polyhedral convex set

is a closed set If X is finite-dimensional, a subset D ⊂ X is a generalizedpolyhedral convex set if and only if it is a polyhedral convex set In thatcase, we can represent a given affine subspace L ⊂ X as the solution set of asystem of finitely many linear inequalities

Our further investigations are motivated by the following fundamental sult [63, Theorem 19.1] about polyhedral convex sets in finite-dimensionaltopological vector spaces, which has origin in the works of Minkowski [55]and Weyl [73, 74] (see also Klee [46, Theorem 2.12])

re-Theorem 1.1 (See [63, re-Theorem 19.1]) For any nonempty convex set C

in Rn, the following properties are equivalent:

(a) C is a convex polyhedron;

(b) C is finitely generated, i.e., C can be represented as

for some ui ∈ Rn, i = 1, , k, and vj ∈ Rn, j = 1, , `;

(c) C is closed and it has only a finite number of faces

From (1.3) it follows that ui ∈ C for i = 1, , k

A natural question arises: Is there any analogue of the representation (1.3)for convex polyhedra in locally convex Hausdorff topological vector spaces, ornot? In order to give an answer in the affirmative to this question, we willneed several results from functional analysis

Lemma 1.1 (Closedness of the sum two linear subspaces; see [65, rem 1.42]) Suppose X0 and X1 are linear subspaces of X, X0 is closed, and

Theo-X1 has finite dimension Then X0+ X1 is closed

Lemma 1.2 (The Hahn-Banach extension theorem; see [65, Theorem 3.6])

If x∗ is a continuous linear functional on a linear subspace M of X, thenthere exists xe∗ ∈ X∗ such that hxe∗, xi = hx∗, xi for all x ∈ M

The forthcoming lemma follows from a theorem in [65] A proof is providedhere for the sake of clarity of our presentation

Lemma 1.3 If Y and Z are Hausdorff finite-dimensional topological vectorspaces of dimension n and if g : Y → Z is a linear bijective mapping, then g

is a homeomorphism

Proof Let {e1, e2, , en} be a basis of the Euclidean space Rn, which isequipped with the natural topology Let {v1, v2, , vn} be a basis of Y Setting wi = g(vi) for i = 1, , n, we see that {w1, w2, , wn} is a basis

of Z Clearly, there is an unique linear bijection Φ : Rn → Y satisfying theconditions Φ(ei) = vi for all i Similarly, there is an unique linear bijection

Ψ : Rn → Z with Ψ(ei) = wi for all i By [65, Theorem 1.21(a)], Φ and Ψare homeomorphisms (Note that the quoted result was obtained for Cn and

topological vector spaces over the complex field C Nevertheless, the method

of proof is valid for the case of Rn and topological vector spaces over R.)Since g = Ψ ◦ Φ−1 and g−1 = Φ ◦ Ψ−1 by our construction, it follows that both

We are now in a position to extend Corollary 2.1 from [80], which wasgiven in a normed spaces setting, to the case of convex polyhedra in locallyconvex Hausdorff topological vector spaces

Proposition 1.1 A nonempty subset D ⊂ X is a convex polyhedron if only

if there exist closed linear subspaces X0, X1 of X and a convex polyhedron

D1 ⊂ X1 such that

X = X0 + X1, X0 ∩ X1 = {0}, dim X1 < +∞, (1.4)and

Proof Necessity: If D is a convex polyhedron, then there exist x∗i ∈ X∗,

αi ∈ R, i = 1, , p, such that

D = {x ∈ X | hx∗i, xi ≤ αi, i = 1, , p} Let

X0 := {x ∈ X | hx∗i, xi = 0, i = 1, , p} Because X0 is a closed linear subspace of finite codimension, one can find

a finite-dimensional linear subspace X1 of X, such that X = X0 + X1 and

X0∩ X1 = {0} By [65, Theorem 1.21(b)], X1 is closed Clearly,

Sufficiency: Let X0, X1 be closed subspaces of X satisfying the conditions

in (1.4) Let D1 ⊂ X1 be a convex polyhedron in X1 and let D be defined

Let π0 : X → X/X0, x 7→ x + X0 for all x ∈ X, be the canonical projectionfrom X on the quotient space X/X0 It is clear that the operator

Φ0 : X/X0 → X1, x1+ X0 7→ x1for all x1 ∈ X1, is a linear bijective mapping On one hand, by [65, Theo-rem 1.41(a)], π0 is a linear continuous mapping On the other hand, Φ0 is ahomeomorphism by Lemma 1.3 So, the operator π := Φ0 ◦ π0 : X → X1 islinear and continuous Put x∗j = u∗j ◦ π, j = 1, , m Take any x = x1+ x0with x1 ∈ D1 and x0 ∈ X0 It is clear that

hx∗

j, xi = hu∗j ◦ π, xi = hu∗

j, π(x)i = hu∗j, x1i ≤ βjfor all j = 1, , m Conversely, take any x ∈ X satisfying hx∗j, xi ≤ βj forevery j = 1, , m Let x0 ∈ X0 and x1 ∈ X1 be such that x = x0+ x1 Since

βj ≥ hx∗j, x0+ x1i = hu∗j ◦ Φ0 ◦ π0, x0 + x1i = hu∗j, x1ifor all j = 1, , m, we see that x1 ∈ D1 Hence x ∈ D1+ X0 It follows that

D1+ X0 = 

x ∈ X | hx∗j, xi ≤ βj, j = 1, , m

.Therefore D = D1+ X0 is a convex polyhedron in X 2The main result of this section is formulated as follows

Theorem 1.2 A nonempty subset D ⊂ X is a generalized convex polyhedron

if and only if there exist u1, , uk ∈ X, v1, , v` ∈ X, and a closed linearsubspace X0 ⊂ X such that

L = {x ∈ X | A(x) = y}

Fix an element x0 ∈ D and set D0 = D − x0 It is easy to verify that

D0 = {u ∈ X | A(u) = 0, hx∗i, ui ≤ αi− hx∗i, x0i, i = 1, , p}

As D0 is a convex polyhedron in ker A := {u ∈ X | A(u) = 0}, by sition 1.1 we can find closed linear subspaces X0,A and X1,A of ker A and aconvex polyhedron D1,A ⊂ X1,A such that

Propo-ker A = X0,A + X1,A, X1,A ∩ X0,A = {0}, dimX1,A < +∞,

and

D0 = D1,A + X0,A.Because X1,A ⊂ ker A is closed and ker A is a closed linear subspace of X,

X1,A is a closed linear subspace of X Since D1,A is a convex polyhedron of thefinite-dimensional space X1,A, invoking Theorem 1.1 we can represent D1,A as

We have thus found a representation of the form (1.6) for D

Sufficiency: Suppose that D is of the form (1.6) Let

linear subspace W1 ⊂ W , such that W = X0 + W1 and X0 ∩ W1 = {0}.Consider the linear mapping π : W → W1 be defined by π(x) = w1, where

x1,0 := x1− π(x1) ∈ X0

So x = π(x1) + x1,0+ x0 belongs to the set π(D1) + X0 Conversely, for any

x = π(z1) + x0 with z1 ∈ D1 and x0 ∈ X0, we have

x = z1+ π(z1) − z1 + x0 = z1+ (x0 − (z1 − π(z1))) ∈ D1+ X0

Since D = D1 + X0 = π(D1) + X0, D is a convex polyhedron in W byProposition 1.1 Hence there exist w∗1, , wm∗ ∈ W∗ and α1, , αm ∈ R suchthat

D = {x ∈ W | hwi∗, xi ≤ αi, i = 1, , m} According to Lemma 1.2, there exist x∗i ∈ X∗, i = 1, , m, such that

Theorem 1.3 A nonempty subset D ⊂ X is a convex polyhedron if and only

if there exist u1, , uk ∈ X, v1, , v` ∈ X, and a closed linear subspace

X0 ⊂ X of finite codimension such that (1.6) is valid

The next example is an illustration for Theorem 1.3

Example 1.1 Let X = C[a, b] be the linear space of continuous real valuedfunctions on the interval [a, b] with the norm defined by

||x|| = max

t∈[a,b]|x(t)|

The Riesz representation theorem (see, e.g., [47, Theorem 6, p 374] and [53,Theorem 1, p 113]) asserts that the dual space of X is X∗ = N BV [a, b], thenormalized space of functions of bounded variation on [a, b], i.e., functions

y : [a, b] → R of bounded variation, y(a) = 0, and y(·) is continuous from theright at every point of (a, b) Let x∗1, x∗2 ∈ X∗ be defined by

D := {x ∈ X | hx∗1, xi ≤ α1, hx∗2, xi ≤ α2} (1.8)

It is clear that X0 := {x ∈ X | hx∗1, xi = 0, hx∗2, xi = 0} is a closed linearsubspace of finite codimension of X For x = η1ω1+ η2ω2 with η1, η2 ∈ R, we

hx∗1, xi = hx∗1, ui + µ1hx∗1, v1i + µ2hx∗1, v2i + hx∗1, x0i = α1− µ2δ ≤ α1and

hx∗2, xi = hx∗2, ui + µ1hx∗2, v1i + µ2hx∗2, v2i + hx∗2, x0i = α2− µ1δ ≤ α2,

we have x ∈ D Now, take any x ∈ D Put

µ1 = δ−1(α2− hx∗2, xi) , µ2 = δ−1(α1− hx∗1, xi) ,and

x0 = x − (u + µ1v1+ µ2v2) Note that µ1 ≥ 0, µ2 ≥ 0 and x = u + µ1v1+ µ2v2+ x0 Since hx∗i, x0i = 0 for

i = 1, 2, we see that x0 ∈ X0 The formula (1.9) has been proved

Based on the preceding example, we can easily construct an illustrativeexample for polyhedral convex sets in locally convex Hausdorff topologicalvector spaces

Example 1.2 Keeping all the notations of Example 1.1, let us consider

X = C[a, b] with the weak topology Then X is a locally convex Hausdorfftopological vector space whose topology is not a norm topology The analysisgiven above shows that the set D in (1.8) admits the representation (1.6).From Theorem 1.2 we can obtain a representation formula for generalizedpolyhedral convex cones

Theorem 1.4 A nonempty set K ⊂ X is a generalized polyhedral convexcone if and only if there exist vj ∈ K, j = 1, , `, and a closed linear sub-space X0 such that

tiui ∈ K for all i = 1, , k, and ti ≥ 0, by choosing v`+i = ui for i = 1, , k,

by (1.11) we see that K admits the representation (1.10) where ` is replaced

by ` + k

Sufficiency: If K has the form (1.10) then it is a cone In addition, K is ageneralized polyhedral convex set by Theorem 1.2 2Combining Theorem 1.3 with Theorem 1.4, we obtain a representationformula for polyhedral convex cones

Theorem 1.5 A nonempty set K ⊂ X is a polyhedral convex cone if andonly if there exist vj ∈ K, j = 1, , `, and a closed linear subspace X0 ⊂ X

of finite codimension such that (1.10) is valid

1.3 Characterizations via the Finiteness of the Faces

In this section, we show how a generalized polyhedral convex set can becharacterized via the finiteness of the number of its faces In order to obtainthe desired results, we first recall some definitions

Definition 1.2 (See [14, p 20]) The relative interior ri C of a convex subset

C ⊂ X is the interior of C in the induced topology of the closed affine hullaff C of C

By [63, Theorem 6.4] and Lemma 1.3, if X is a finite-dimensional Hausdorfftopological vector space, and C ⊂ X is a nonempty convex set, then u ∈ ri C

if and only if, for every x ∈ C, there exists ε > 0 such that u − ε(x − u)belongs to C

Remark 1.1 If X is finite-dimensional and C ⊂ X is a nonempty convexsubset, ri C is nonempty by [63, Theorem 6.2] If X is infinite-dimensional,

it may happen that ri C = ∅ for certain nonempty convex subsets C ⊂ X

To justify the claim, it suffices to choose X = `2 – the Hilbert space of allreal sequences x = (xk)∞k=1 such that

If C ⊂ X is a nonempty generalized polyhedral convex set, then by [14,Proposition 2.197] we know that ri C 6= ∅ The latter fact shows that gener-alized polyhedral convex sets have a nice topological structure

Definition 1.3 (See [63, p 162]) A convex subset F of a convex set C ⊂ X

is said to be a face of C if for every x1, x2 in C satisfying (1 − λ)x1+ λx2 ∈ Fwith λ ∈ (0, 1) one has x1 ∈ F and x2 ∈ F

Definition 1.4 (See [63, p 162]) A convex subset F of a convex set C ⊂ X

is said to be an exposed face of C if there exists x∗ ∈ X∗ such that

F = 

u ∈ C | hx∗, ui = inf

x∈Chx∗, xi

From the above definitions it is immediate that if F is an exposed face of

a convex set C, then F is a face of C To see that the converse may not true

in general, it suffices to choose



and F = {(1, 0)}

Clearly, a convex set C ⊂ X itself is not only a face, but also an exposedface of it The emptyset is a face of C, but it is not necessarily an exposedface of C For example, a nonempty compact convex C does not have theemptyset as an exposed face of it

In the spirit of Theorem 1.1, for a nonempty convex subset D ⊂ X, we areinterested in establishment of relations between the following properties:(a) D is a generalized polyhedral convex set ;

(b) There exist u1, , uk ∈ X, v1, , v` ∈ X, and a closed linear subspace

As shown in Theorem 1.2, (a) and (b) are equivalent Now, let us provethat (a) implies (c)

Theorem 1.6 Every generalized polyhedral convex set has a finite number offaces and all the nonempty faces are exposed

Proof Let D be a generalized polyhedral convex set given by (1.1) For anysubset J ⊂ I, using the definition of face and formula (1.1), it is not difficult

Indeed, put xt := x−t(x0−x) where t > 0 and observe that xt ∈ L, because

xt = (1 + t)x + (−t)x0 and x, x0 belong to the closed affine subspace L Foreach i ∈ I(x) ⊂ I(x0), we have

hx∗

i, xti = hx∗

i, xi − thx∗i, x0− xi = αi.Since hx∗j, xi < αj for all j ∈ I \ I(x), we can find t > 0 such that hx∗j, xti < αjfor every j ∈ I \ I(x) Hence, for the chosen t, we have xt ∈ D As x ∈ Fand x = 1+t1 xt+ 1+tt x0, we must have x0 ∈ F

Claim 2 If F is a nonempty face of D, then there exists J ⊂ I such that

F = FJ Hence, the number of faces of D is finite Moreover, F is an exposedface

Indeed, given a nonempty face F of D, we define J = T

x∈F

I(x) It is clearthat F ⊂ FJ To have the inclusion FJ ⊂ F , we select a point x0 ∈ F suchthat the number of elements of I(x0) is the minimal one among the numbers

of elements of I(x), x ∈ F Let us show that I(x0) = J Suppose, on thecontrary, that I(x0) 6= J Then there must exist a point x1 ∈ F and anindex i0 ∈ I(x0) \ I(x1) By the convexity of F , ¯x := 12x0+ 12x1 belongs to F Since hx∗i0, x1i < αi0, we have hx∗i0, ¯xi < αi0, that is, i0 ∈ I(¯/ x) If j /∈ I(x0),i.e., hx∗j, x0i < αj, then hx∗j, ¯xi < αj; so j /∈ I(¯x) Thus, I(¯x) ⊂ I(x0) andI(¯x) 6= I(x0) This contradicts the minimality of I(x0) For any x ∈ FJ, it isclear that J ⊂ I(x) Since x0 ∈ F and I(x0) = J ⊂ I(x), by Claim 1 we canassert that x ∈ F The inclusion FJ ⊂ F has been proved Thus F = FJ

As J ⊂ I and I is finite, the above obtained result shows that the number

of faces of D is finite

If J = ∅, then FJ = D For x∗ := 0, one has D = argmin

hx∗, xi | x ∈ D

;hence D is an exposed face of it It follows that F∅ is an exposed face Now,suppose that J 6= ∅ Let k denote the number of elements of J Setting

j0 ∈ J with h−x∗

j 0, xi > −αj0, while h−x∗j, xi ≥ −αj for all j ∈ J ) Hence,

Remark 1.2 The point x0, constructed in the proof of Theorem 1.6, belongs

to ri F Conversely, for any ¯x ∈ ri F , I(¯x) has the minimality property

of I(x0) The proof of these claims is omitted

Theorem 1.7 Let D ⊂ X be a closed convex set with nonempty relativeinterior If D has finitely many faces, then D is a generalized polyhedralconvex set

Proof By our assumption ri D 6= ∅ We, first, consider the case, whereint D 6= ∅ We have D = int D ∪ ∂D, where ∂D = D \ int D is the boundary

of D If ∂D = ∅, then D = X because D is both open and closed in X,which is a connected topological space So D is a convex polyhedron If

∂D 6= ∅, we pick a point ¯x ∈ ∂D As {¯x} ∩ int D = ∅ and since {¯x} andint D are convex sets, by the separation theorem [65, Theorem 3.4(a)], thereexists ϕx¯ ∈ X∗\ {0} such that hϕx¯, ¯xi ≥ hϕx¯, xi for all x ∈ int D Since D isconvex and int D 6= ∅, it follows that

hϕx¯, ¯xi ≥ hϕx ¯, xi, ∀x ∈ D (1.12)Let αx¯ := hϕx¯, ¯xi and Fx,ϕ¯ x¯ := {x ∈ D | hϕx¯, xi = αx¯} It is easy to showthat Fx,ϕ¯ x¯ is a face of D and ¯x ∈ Fx,ϕ¯ x¯ As D has finitely many faces, we canfind a finite sequence of points x1, , xk in ∂D such that, for every u ∈ ∂D,there exists i ∈ {1, , k} with Fu,ϕu = Fxi,ϕxi Let

T := {t ∈ [0, 1] | ut := (1 − t)u0+ tu1 ∈ D}

is a closed convex subset of [0, 1] Note that 0 ∈ T , but 1 /∈ T Hence,

T = [0, ¯t] for some ¯t ∈ [0, 1) As u0 ∈ intD, we must have ¯t > 0 It is easy

to show that ¯u := (1 − ¯t) u0 + ¯tu1 belongs to ∂D Hence, Fu,ϕ¯ u¯ = Fxi,ϕxi forsome i ∈ {1, , k} Since u0 ∈ intD and ϕxi 6= 0, from (1.12) it follows that

hϕxi, u0i < αxi (1.14)

As ¯u ∈ Fxi,ϕxi, one has

hϕxi, ¯ui = αxi (1.15)From the equality ¯u = (1 − ¯t) u0+¯tu1 we can deduce that u1 = 1¯tu+ 1 −¯ 1¯t
u0.Since 1 − 1¯t < 0, by (1.14) and (1.15) we have



αxi = αxi.Then we obtain hϕxi, u1i > αxi, contradicting the assumption u1 ∈ D0 Wehave thus proved that D0 = D Therefore, by (1.13) we can conclude that D

is a polyhedral convex set

Now, let us consider the case intD = ∅ As riD 6= ∅, the interior of D inthe induced topology of affD is nonempty Take any x0 ∈ D Applying theabove result for the closed convex subset D0 := D − x0 of the locally convexHausdorff topological vector space X0 := affD − x0, we find x∗i ∈ X∗

Remark 1.3 Note that Maserick [54] introduced the concept of convex tope, which is very different from the notion of generalized polyhedral convexset in [14, Definition 2.195] On one hand, any convex polytope in the sense ofMaserick must have nonempty interior, while a generalized polyhedral convexset in the sense of Bonnans and Shapiro may have empty interior (so it is not

a convex polytope in general ) On the other hand, there exist convex topes in the sense of Maserick which cannot be represented as intersections

poly-of finitely many closed half-spaces and a closed affine subspace poly-of that logical vector space For example, the closed unit ball ¯B of c0 – the Banach

topo-space of the real sequences x = (x1, x2, ), xi ∈ R for all i, lim

i→∞xi = 0, withthe norm kxk = sup{|xi| | i = 1, 2, } – is a convex polytope in the sense

of Maserick (see Theorem 4.1 on page 632 in [54]) However, since ¯B has

an infinite number of faces, it cannot be a generalized polyhedral convex set

in the sense of Bonnans and Shapiro (see Theorem 1.6) Subsequently, theconcept of convex polytope of [54] has been studied by Maserick and otherauthors (see, e.g., Durier and Papini [20], Fonf and Vesely [27]) However,after consulting many relevant research works which are available to us, we

do hope that the results obtained herein are new

Gener-alized Polyhedral Convex Sets

Let us consider the following question: Given locally convex Hausdorff logical vector spaces X and Y , whether the image of a generalized polyhedralconvex set via a linear mapping from X to Y is a generalized polyhedral convexset, or not? The answers in the affirmative are given in [63, Theorem 19.3]for the case where X and Y are finite-dimensional, in [82, Lemma 3.2] for thecase where X is a Banach space and Y is finite-dimensional

topo-We are now in a position to extend Lemma 3.2 from the paper of Zhengand Yang [82], which was given in a normed space setting, to the case ofconvex polyhedra in locally convex Hausdorff topological vector spaces.Proposition 1.2 If T : X → Y is a linear mapping between locally convexHausdorff topological vector spaces with Y being a space of finite dimensionand if D ⊂ X is a generalized polyhedral convex set, then T (D) is a convexpolyhedron of Y

Proof Suppose that D is of the form (1.6) We have

closed linear subspace Hence, by Theorem 1.2, T (D) is a polyhedral convex

One may wonder: Whether the assumption on the finite dimensionality

of Y can be removed from Proposition 1.2, or not? Let us solve this question

by an example

Example 1.3 Let X = C[0, 1] be the linear space of continuous real valuedfunctions on the interval [0, 1] with the norm defined by ||x|| = max

t∈[0,1]|x(t)|.Let

Y = C0[0, 1] := y ∈ C[0, 1] | y(0) = 0 and let T : X → Y be the bounded linear operator given by

It is easily seen that {qk} converges uniformly to y in Y and {qk} ⊂ T (X)

As T (X) 6= Y , we see that T (X) is a non-closed linear subspace set of Y Hence, T (X) cannot be a generalized polyhedral convex set

A careful analysis of Example 1.3 leads us to the following question: ther the image of a generalized polyhedral convex set via a surjective linearoperator from a Banach space to another Banach space is a generalized poly-hedral convex set, or not?

Whe-Example 1.4 Let X = C[0, 1] × C[0, 1] with the norm defined by

where integral is Riemannian Clearly, T is a surjective continuous linearmapping from X to Y Note that D := C[0, 1] × {0} is a generalized polyhe-dral convex set of X, but

T (D) = ny ∈ C0[0, 1] | y is continuously differentiable on (0, 1)o

is not a generalized polyhedral convex set of Y

In the above mentioned example, one sees that the image of a closed linearsubspace of X via a continuous surjective linear operator may be not closed;hence it can be not a generalized polyhedral convex set

The above results motivate the following proposition

Proposition 1.3 Suppose that T : X → Y is a linear mapping between cally convex Hausdorff topological vector spaces and D ⊂ X, Q ⊂ Y arenonempty generalized polyhedral convex sets Then, T (D) is a generalizedpolyhedral convex set If T is continuous, then T−1(Q) is a generalized poly-hedral convex set

lo-Proof Suppose that D is of the form (1.6) Then T (D) = D0 + T (X0),where

D0 := convT (ui) | i = 1, , k + coneT (vj) | i = 1, , `

Since T (X0) ⊂ Y is a linear subspace, T (X0) is a closed linear subspace of Y

by [65, Theorem 1.13(c)]; so D0 + T (X0) is a generalized polyhedral convexset by Theorem 1.2 In particular, D0+ T (X0) is closed Hence, the inclusion

T (D) ⊂ D0+ T (X0) yields

According to [65, Theorem 1.13(b)], we have

Combining (1.17) with (1.18) implies that T (D) = D0 + T (X0) Therefore

T (D) is a generalized polyhedral convex set

Now, suppose that Q ⊂ Y is a generalized polyhedral convex set given by

Q = y ∈ Y | B(y) = z, hyj∗, yi ≤ βj, j = 1, , q ,where B : Y → Z is a continuous linear mapping between two locally convexHausdorff topological vector spaces, z ∈ Z and yj∗ ∈ Y∗, βj ∈ R, j = 1, , q

Then we have

T−1(Q) =

x ∈ X | B(T (x)) = z, hyj∗, T (x)i ≤ βj, j = 1, , q

=x ∈ X | (B ◦ T )(x) = z, hT∗(y∗j), xi ≤ βj, j = 1, , q ,where T∗ : Y∗ → X∗ is the adjoint operator of T Since T : X → Y and

B : Y → Z are linear continuous mappings, B ◦ T : X → Z is a continuouslinear mapping Hence, the above expression for T−1(Q) shows that the set

Proposition 1.4 If D1, , Dm are nonempty generalized polyhedral convexsets in X, so is D1+ · · · + Dm

Proof Consider the linear mapping T : Xm → X given by

T (x1, , xm) = x1+ · · · + xm ∀(x1, , xm) ∈ Xm,and observe that T (D1× · · · × Dm) = D1+ · · · + Dm Since Dk is a generalizedpolyhedral convex set in X for k = 1, , m, using Definition 1.1, one canshow that D1× · · · × Dm is a generalized polyhedral convex set in Xm Then,

T (D1 × · · · × Dm) is a generalized polyhedral convex set by Proposition 1.3.Hence, D1 + · · · + Dm is a generalized polyhedral convex set in X 2Remark 1.4 One may ask: Whether the statement of Corollary 1.4 is validalso for the sum of the sets Di, i = 1, , m, without the closure operation.When X is a finite-dimensional space, the sum of finitely many polyhedralconvex sets in X is a polyhedral convex set (see, e.g., [46, Corollary 2.16], [63,Corollary 19.3.2]) However, when X is an infinite-dimensional space, thesum of a finite number of generalized polyhedral convex sets may be not ageneralized polyhedral convex set To see this, one can choose a suitablespace X and closed linear subspaces X1, X2 of X satisfying X1 + X2 = Xand X1+ X2 6= X (see [9, Example 3.34] for an example of subspaces in anyinfinite-dimensional Hilbert space, [16, Exercise 1.14] for an example in `1,and [65, Exercise 20, p 40] for an example in L2(−π, π)) Clearly, X1, X2are generalized polyhedral convex sets in X Since X1+ X2 is non-closed, itcannot be a generalized polyhedral convex set

Concerning the question stated in Remark 1.4, in the two following tions we shall describe some situations where the closure sign can be dropped

proposi-Proposition 1.5 If D1, D2 are generalized polyhedral convex sets of X andaffD1 is finite-dimensional, then D1 + D2 is a generalized polyhedral convexset.

Proof According to Theorem 1.2, for each m ∈ {1, 2}, we can represent Dm

as Dm = Dm0 + Xm,0 with Xm,0 being a closed linear subspace of X,

Dm0 = conv {um,1, , um,km} + cone {vm,1, , vm,`m}for some um,1, , um,km, vm,1, , vm,`m in X Since affD1 is finite-dimensional,

we must have dimX1,0 < ∞ By Lemma 1.1, X1,0 + X2,0 is a closed linearsubspace of X Let W be the finite-dimensional linear subspace generated

by the vectors um,1, , um,km, vm,1, , vm,`m, for m = 1, 2 Since D01 and D02are polyhedral convex sets in W due to Theorem 1.1, D10 + D20 is a polyhedralconvex set in W by [63, Corollary 19.3.2] On account of Theorem 1.1, onecan choose u1, , uk in W , v1, , v` in W such that

D01+ D02 = conv {ui | i = 1, , k} + cone {vj | j = 1, , `}

It follows that

D1+ D2 = conv {ui | i = 1, , k} + cone {vj | j = 1, , `} + X1,0+ X2,0.Recalling that the linear subspace X1,0+X2,0 is closed, we can use Theorem 1.2

to assert that D1 + D2 is a generalized polyhedral convex set 2Before going further, let us present a useful lemma

Lemma 1.4 If X1 and X2 are linear subspaces of X with X1 being closed andfinite-codimensional, then X1+ X2 is closed and codim(X1 + X2) < ∞.Proof Since X1 ⊂ X is finite-codimensional, there exists a finite-dimensionallinear subspace X10 ⊂ X such that X = X1 ∪ X0

1, is a linear bijective mapping

On one hand, by [65, Theorem 1.41(a)], π1 is a linear continuous mapping

On the other hand, Φ1 is a homeomorphism by Lemma 1.3 So, the operator

π := Φ1 ◦ π1 : X → X10 is linear and continuous Note that π(X2) is closed,because it is a linear subspace of X10, which is finite-dimensional Since π

is continuous and X1 + X2 = π−1(π(X2)), we see that X1 + X2 is closed.The codimX1 < ∞ clearly forces codim(X1+ X2) < ∞ 2

Proposition 1.6 If D1 ⊂ X is a polyhedral convex set and D2 ⊂ X is ageneralized polyhedral convex set, then D1+ D2 is a polyhedral convex set.

Proof By Theorem 1.3, there exist u1,1, , u1,k1 in X, v1,1, , v1,`1 in Xand a closed finite-codimensional linear subspace X1,0 ⊂ X such that

D1 = D10 + X1,0with D01 = conv {u1,1, , u1,k1} + cone {v1,1, , v1,`1} According to Theo-rem 1.2, there exist u2,1, , u2,k2 in X, v2,1, , v2,`2 in X and a closed linearsubspace X2,0 of X satisfying D2 = D20 + X2,0 with

D20 = conv {u2,1, , u2,k2} + cone {v2,1, , v2,`2} Let W be the finite-dimensional linear subspace generated by the vectors

u1,1, , u1,k1, v1,1, , v1,`1, u2,1, , u2,k2, v2,1, , v2,`2 Since D01 and D02 arepolyhedral convex sets in W by Theorem 1.1, Corollary 19.3.2 of [63] impliesthat D10 + D02 is a polyhedral convex set Applying Theorem 1.1 for thepolyhedral convex set D01 + D02 of W , one can find u1, , uk and v1, , v`

The next result is an extension of [63, Corollary 19.3.2] to an dimensional setting

infinite-Corollary 1.1 Suppose that D1 ⊂ X is a polyhedral convex set and D2 ⊂ X

is a generalized polyhedral convex set If D1 ∩ D2 = ∅, then there exists

x∗ ∈ X∗ such that

sup{hx∗, ui | u ∈ D1} < inf{hx∗, vi | v ∈ D2} (1.20)Proof By Proposition 1.6, D2− D1 = D2+ (−D1) is a polyhedral convex set

in X; hence it is closed Since D2− D1 is a closed convex set and 0 /∈ D2− D1,

by the strongly separation theorem [65, Theorem 3.4(b)] there exist x∗ ∈ X∗

and γ ∈ R such that

hx∗, 0i < γ ≤ hx∗, xi, ∀x ∈ D2− D1.This implies that

sup{hx∗, ui | u ∈ D1} + γ ≤ inf{hx∗, vi | v ∈ D2};

The assertion of Corollary 1.1 would be false if D1 is only assumed to be

a generalized polyhedral convex set Indeed, an answer in the negative forthe question in [16, Exercise 1.14] assures us that there exist closed affinesubspaces D1 and D2 in X = `1 such that one cannot find any x∗ ∈ X∗\ {0}satisfying

sup{hx∗, ui | u ∈ D1} ≤ inf{hx∗, vi | v ∈ D2}

So, with the chosen generalized polyhedral convex sets D1 and D2, one cannothave (1.20) for any x∗ ∈ X∗ = `∞

As in [63, p 61], the recession cone 0+C of a convex set C ⊂ X is given by

Clearly, if D is represented in the form (1.6), then

0+D = cone{v1, , v`} + X0

We are now in a position to extend Theorem 19.6 from the book of afellar [63], which was given in Rn, to the case of generalized polyhedralconvex in locally convex Hausdorff topological vector spaces

Rock-Theorem 1.8 Suppose that D1, , Dm are generalized polyhedral convex sets

in X Let D be the smallest closed convex subset of X that contains Di forall i = 1, , m Then D is a generalized polyhedral convex set If at leastone of the sets D1, , Dm is polyhedral convex, then D is a polyhedral convexset

Proof By removing all the empty sets from the system D1, , Dm, we mayassume that Di 6= ∅ for all i ∈ I := {1, , m} Due to Theorem 1.2, foreach i ∈ I, one can find ui,1, , ui,ki and vi,1, , vi,`i in X and a closed linearsubspace Xi,0 ⊂ X such that

Di = conv{ui,1, , ui,ki} + cone{vi,1, , vi,`i} + Xi,0 (1.21)Since X1,0+ · · · + Xm,0 ⊂ X is a linear subspace, X0 := X1,0+ · · · + Xm,0 is aclosed linear subspace of X by [65, Theorem 1.13(c)] Let

D0 : = conv {ui,j | i ∈ I, j = 1, , ki}

+ cone {vi,j | i ∈ I, j = 1, , `i} + X0 (1.22)

On account of Theorem 1.2, D0 is a generalized polyhedral convex set Inparticular, D0 is convex and closed From (1.21) and (1.22) it follows that

Di ⊂ D0 for every i ∈ I Hence, by the definition of D, we must have

D ⊂ D0 Let us show that D0 ⊂ D Since ui,j belongs to Di ⊂ D for i ∈ Iand j ∈ {1, , ki}, and since D is convex,

cone {vi,j | i ∈ I, j = 1, , `i} + X0 ⊂ 0+D (1.24)Combining (1.22), (1.23) with (1.24) yields D0 ⊂ D Thus we have provedthat D0 = D Since D0 is a generalized polyhedral convex set, D is also ageneralized polyhedral convex set

Now, suppose that at least one of the set D1, , Dm is polyhedral convex.Then, by Theorem 1.3, in the representation (1.21) for D1, , Dm we mayassume that at least one of the sets X1,0, , Xm,0 is finite-codimensional