Some contributions to the theory of generalized polyhedral optimization problems (tt)

Namely, apart from linear VOPs in locally convex Hausdorff topological vector spaces, which are the main subjects of ourresearch, we will study polyhedral convex optimization problems an

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY

HANOI - 2019

The dissertation was written on the basis of the author’s research works carried at Institute

of Mathematics, Vietnam Academy of Science and Technology

Supervisor: Prof Dr.Sc Nguyen Dong Yen

First referee:

Second referee:

Third referee:

To be defended at the Jury of Institute of Mathematics, Vietnam Academy of Science and Technology:

on , at o’clock

The dissertation is publicly available at:

• The National Library of Vietnam

• The Library of Institute of Mathematics

Vector optimization has a rich history and diverse applications Vector optimization times called multiobjective optimization) is a natural development of scalar optimization F.Y.Edgeworth (1881) and V Pareto (1906) defined a notion, which later was called Pareto solu-tion This solution concept remains the most important in vector optimization Other basicsolution concepts of this theory are weak Pareto solution and proper solution The latter hasbeen defined in different ways by A.M Geoffrion, J.M Borwein, H.P Benson, M.I Henig, andother authors

(some-One calls a vector optimization problem (VOP) linear if the objective functions are linear(affine) functions and the constraint set is polyhedral convex (i.e., it is a intersection of a finitenumber of closed half-spaces) If at least one of the objective functions is nonlinear (non-affine,

to be more precise) 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 of nonlinear inequalities), thenthe VOP is said to be nonlinear

Linear VOPs have been considered in many books and in numerous papers The classicalArrow-Barankin-Blackwell Theorem asserts that, for a linear vector optimization problem,the Pareto solution set and the weak Pareto solution set are connected by line segments andare the unions of finitely many faces of the constraint set This is an example of qualitativeproperties of vector optimization problems

This dissertation focuses on linear VOPs and several related nonlinear scalar optimizationproblems, as well as nonlinear vector optimization problems Namely, apart from linear VOPs

in locally convex Hausdorff topological vector spaces, which are the main subjects of ourresearch, we will study polyhedral convex optimization problems and piecewise linear vectoroptimization problems The fundamental concepts used in this dissertation are polyhedralconvex set and polyhedral convex function on locally convex Hausdorff topological vectorspaces About one half of the dissertation is devoted to these concepts Another half ofthe dissertation shows how our new results on polyhedral convex sets and polyhedral convexfunctions can be applied to scalar optimization problems and VOPs

According to Bonnans and Shapiro (2000), a subset of a locally convex Hausdorff topologicalvector space is said to be a generalized polyhedral convex set, if it is the intersection of finitelymany closed half-spaces and a closed affine subspace of that topological vector space Whenthe affine subspace can be chosen as the whole space, the generalized polyhedral convex set

is called a polyhedral convex set

Many applications of polyhedral convex sets and piecewise linear functions in normed spaces

to vector optimization can be found in the papers of Yang and Yen (2010), Zheng (2009), Zhengand Ng (2014), Zheng and Yang (2008)

Numerous applications of generalized polyhedral convex sets and generalized polyhedralmultifunctions in Banach spaces to variational analysis, optimization problems, and variationalinequalities can be found in the works by Henrion, Mordukhovich, and Nam (2010), Ban,Mordukhovich, and Song (2011), Gfrerer (2013, 2014), Ban and Song (2016)

The introduction of these concepts poses an interesting problem Namely, since the entireSection 19 of the book “Convex Analysis” of Rockafellar (1970) is devoted to establishing avariety of basic properties of polyhedral convex sets and polyhedral convex functions whichhave numerous applications afterwards, one may ask whether a similar study can be done forgeneralized polyhedral convex sets and generalized polyhedral convex functions, or not.The systematic study of generalized polyhedral convex sets and generalized polyhedral con-vex function in this dissertation can serve as a basis for further investigations on minimization

of a generalized polyhedral convex function on a generalized polyhedral convex set – a alized polyhedral convex optimization problem, which is a special infinite-dimensional convexprogramming problem

gener-Piecewise linear vector optimization problem (PLVOP) is a natural development of dral convex optimization The study of the structures and characteristic properties of thesesolution sets of PLVOPs is can be found in the papers of Zheng and Yang (2008), Yang andYen (2010), Fang, Meng, and Yang (2012), Fang, Huang and Yang (2012), Fang, Meng andYang (2015) Zheng and Ng (2014)

polyhe-The dissertation has five chapters, a list of the related papers of the author, a section ofgeneral conclusions, and a list of references

Chapter 1 gives a series of fundamental properties of generalized polyhedral convex sets

In Chapter 2, we discuss some basic properties of generalized polyhedral convex functions.Chapter 3 is devoted to several dual constructions including the concepts of conjugatefunction and subdifferential of a generalized polyhedral convex function

Generalized polyhedral convex optimization problems in locally convex Hausdorff ical vector spaces are studied systematically in Chapter 4 We establish solution existencetheorems, necessary and sufficient optimality conditions, weak and strong duality theorems

topolog-In particular, we show that the dual problem has the same structure as the primal problem,and the strong duality relation holds under three different sets of conditions

Chapter 5 discusses structure of efficient solutions sets of linear vector optimization lems and piecewise linear vector optimization problems

prob-Chapter 1

Generalized Polyhedral Convex Sets

In this chapter, we first establish a representation formula for generalized convex dra A series of fundamental properties of generalized polyhedral convex sets are obtained inSections 2-5 In Section 6, by using the representation formulas for generalized polyhedralconvex sets we prove solution existence theorems in generalized linear programming

polyhe-The main theorems of Section 1 below (see polyhe-Theorems 1.2 and 1.5), which can be considered

as geometrical descriptions of generalized convex polyhedra and convex polyhedra, are notformal extensions of Theorem 19.1 from a book of Rockafellar (1970) and Corollary 2.1 of a

paper of Zheng (2009) Recently, Yen and Yang (2018) have used Theorem 1.2 to study dimensional affine variational inequalities (AVIs) on normed spaces It is shown that infinite-dimensional quadratic programming problems and infinite-dimensional linear fractional vectoroptimization problems can be studied by using AVIs They have obtained two basic facts aboutinfinite-dimensional AVIs: the Lagrange multiplier rule and the solution set decomposition.

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 smallest convex cone containing Ω

of Minkowski (1910) and Weyl (1934) (see also Klee (1959) and Rockafellar (1970))

Theorem 1.1 (see Rockafellar (1970)) For any nonempty convex set C in Rn, the followingproperties 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

A natural question arises: Is there any analogue of the representation (1.3) for convexpolyhedra in locally convex Hausdorff topological vector spaces, or not?

The following proposition extends a result of Zheng (2009), which was given in a normedspaces setting, to the case of convex polyhedra in locally convex Hausdorff topological vectorspaces

Proposition 1.1 A nonempty subset D ⊂ X is a convex polyhedron if only if there exist closedlinear subspaces X0, X1 of X and a convex polyhedron D1 ⊂ X1 such that X = X0+ X1,

X0∩ X1 = {0}, dim X1< +∞, and D = D1+ X0

The 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 ifthere exist u1, , uk ∈ X, v1, , v`∈ X, and a closed linear subspace X0⊂ X 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 codimensionsuch that (1.4) is valid

Some illustrative examples for Theorem 1.3 are given in the dissertation

From Theorem 1.2 we can obtain a representation formula for generalized polyhedral convexcones

Theorem 1.4 A nonempty set K ⊂ X is a generalized polyhedral convex cone if and only ifthere exist vj ∈ K, j = 1, , `, and a closed linear subspace X0 such that

polyhe-Theorem 1.5 A nonempty set K ⊂ X is a polyhedral convex cone if and only if there exist

vj ∈ K, j = 1, , `, and a closed linear subspace X0 ⊂ X of finite codimension such that (1.5)

is valid

Definition 1.2 (see Bonnans and Shapiro (2000)) The relative interior ri C of a convex subset

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

If C ⊂ X is a nonempty generalized polyhedral convex set, then ri C 6= ∅ (see Bonnans andShapiro (2000)) The latter fact shows that generalized polyhedral convex sets have a nicetopological structure

Definition 1.3 (see Rockafellar (1970)) 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 ∈ F with λ ∈ (0, 1) one has

x1 ∈ F and x2∈ F

Definition 1.4 (see Rockafellar (1970)) 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

In the spirit of Theorem 1.1, for a nonempty convex subset D ⊂ X, we are interested inestablishment of relations between the following properties:

(a) D is a generalized polyhedral convex set ;

(b) D is closed and has only a finite number of faces

The next theorems shows that a generalized polyhedral convex set can be characterized viathe finiteness of the number of its faces

Theorem 1.6 Every generalized polyhedral convex set has a finite number of faces and all thenonempty faces are exposed

Theorem 1.7 Let D ⊂ X be a closed convex set with nonempty relative interior If D hasfinitely many faces, then D is a generalized polyhedral convex set

1.4 Images via Linear Mappings and Sums of

General-ized Polyhedral Convex Sets

Let us consider the following question: Given locally convex Hausdorff topological vectorspaces X and Y , whether the image of a generalized polyhedral convex set via a linear mappingfrom X to Y is a generalized polyhedral convex set, or not? The answers in the affirmativefor the case where X and Y are finite-dimensional (see Rockafellar (1970)), for the case where

X is a Banach space and Y is finite-dimensional (see Zheng and Yang (2008))

The following proposition extends a lemma from the paper of Zheng and Yang (2008),which was given in a normed space setting, to the case of convex polyhedra in locally convexHausdorff topological vector spaces

Proposition 1.2 If T : X → Y is a linear mapping between locally convex Hausdorff logical vector spaces with Y being a space of finite dimension and if D ⊂ X is a generalizedpolyhedral convex set, then T (D) is a convex polyhedron of Y

topo-One may wonder: Whether the assumption on the finite dimensionality of Y can be removedfrom Proposition 1.2, or not? In the dissertation, some examples have been given to showthat if Y is a infinite-dimensional space then T (D) may not be a generalized polyhedral convexset

Proposition 1.3 Suppose that T : X → Y is a linear mapping between locally convex dorff topological vector spaces and D ⊂ X, Q ⊂ Y are nonempty generalized polyhedral convexsets Then, T (D) is a generalized polyhedral convex set If T is continuous, then T−1(Q) is ageneralized polyhedral convex set

Haus-Proposition 1.4 If D1, , Dm are nonempty generalized polyhedral convex sets in X, so is

D1+ · · · + Dm

One may ask: Whether the statement of Corollary 1.4 is valid also for the sum of the sets

Di, i = 1, , m, without the closure operation When X is a finite-dimensional space, thesum of finitely many polyhedral convex sets in X is a polyhedral convex set (see Klee (1959)).However, when X is an infinite-dimensional space, the sum of a finite number of generalizedpolyhedral convex sets may be not a generalized polyhedral convex set Concerning thisquestion, in the two following propositions we shall describe some situations where the closuresign can be dropped

Proposition 1.5 If D1, D2 are generalized polyhedral convex sets of X and affD1 is dimensional, then D1+ D2 is a generalized polyhedral convex set

finite-Proposition 1.6 If D1 ⊂ X is a polyhedral convex set and D2⊂ X is a generalized polyhedralconvex set, then D1+ D2 is a polyhedral convex set

The next result is an extension of a result from Rockafellar (1970) to an infinite-dimensionalsetting

Corollary 1.1 Suppose that D1⊂ X is a polyhedral convex set and D2⊂ X is a generalizedpolyhedral convex set If D1∩ D2 = ∅, then there exists x∗∈ X∗ such that

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

1.5 Convex Hulls and Conic Hulls

As in Rockafellar (1970), the recession cone 0+C of a convex set C ⊂ X is given by

0+C =v ∈ X | x + tv ∈ C, ∀x ∈ C, ∀t ≥ 0

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 for all i = 1, , m Then D is ageneralized polyhedral convex set If at least one of the sets D1, , Dm is polyhedral convex,then D is a polyhedral convex set

From Theorem 1.8 we obtain the following corollary

Corollary 1.2 If a convex subset D ⊂ X is the union of a finite number of generalizedpolyhedral convex sets (resp., of polyhedral convex sets) in X, then D is a generalized polyhedralconvex (resp., polyhedral convex) set

It turns out that the closure of the cone generated by a generalized polyhedral convex set

is a generalized polyhedral convex cone Hence, next proposition extends a theorem fromRockafellar (1970) to a locally convex Hausdorff topological vector spaces setting

Proposition 1.7 If a nonempty subset D ⊂ X is generalized polyhedral convex, then cone D

is a generalized polyhedral convex cone In addition, if 0 ∈ D then cone D is a generalizedpolyhedral convex cone; hence cone D is closed

An analogue of Proposition 1.7 for polyhedral convex sets can be formulated as follows

Proposition 1.8 If a nonempty subset D ⊂ X is polyhedral convex, then cone D is a dral convex cone In addition, if 0 ∈ D then cone D is a polyhedral convex cone; hence cone D

polyhe-is closed

In convex analysis, to every convex set and a point belonging to it, one associates a gent cone The forthcoming proposition shows that the tangent cone to a generalized poly-hedral convex set at a given point is a generalized polyhedral convex cone By definition, the(Bouligand-Severi) tangent cone TD(x) to a closed subset D ⊂ X at x ∈ D is the set of all

tan-v ∈ X such that there exist sequences tk → 0+ and vk → v such that x + tkvk ∈ D for every

k If D is convex, then TD(x) = cone (D − x)

Proposition 1.9 If D ⊂ X is a generalized polyhedral convex set (resp., a polyhedral convexset) and if x ∈ D, then TD(x) is a generalized polyhedral convex cone (resp., a polyhedralconvex cone) and one has TD(x) = cone (D − x)

In this section, we obtain a formula for the relative interiors of a generalized polyhedralconvex cone and the dual cone of a polyhedral convex cone

Theorem 1.9 Suppose that C ⊂ X is a generalized polyhedral convex cone in a locally convexHausdorff topological vector space If C =

K∗ := y∗ ∈ Y∗ | hy∗, yi ≥ 0 ∀y ∈ K By using the set K \ `(K), we can be describe therelative interior of the dual cone K∗ as follows

Theorem 1.10 If K is not a linear subspace of Y , then a vector y∗ ∈ Y∗ belongs to ri K∗ ifand only if hy∗, yi > 0 for all y ∈ K \ `(K)

Our aim in this section is to apply the representation formula for generalized polyhedralconvex sets to proving solution existence theorems for generalized linear programming prob-lems

Consider a generalized linear programming problem

is nonempty, then (LP) has a solution if and only if hx∗, vi ≥ 0 for every v ∈ 0+D

Theorem 1.12 (The Frank–Wolfe-type existence theorem; see Bonnans and Shapiro (2000))

If D is nonempty, then (LP) has a solution if and only if there is a real number γ such that

hx∗, xi ≥ γ for every x ∈ D

We are interested in studying the region G of all x∗ for which (LP) has a nonempty solutionset, assuming that the constraint set D is nonempty and fixed

Proposition 1.10 If D has the form (1.4), then G is a generalized polyhedral convex cone

of X∗ which has the representation G = X0⊥∩ {x∗ ∈ X∗| hx∗, vji ≥ 0, j = 1, , `}

Next, for each x∗ ∈ G, we want to describe the solution set of (LP), which is denoted

by S(x∗) For doing so, let us suppose that D is given by (1.4) and consider the index sets

I(x∗) := {i0 ∈ {1, , k} | hx∗, ui0i ≤ hx∗, uii ∀i = 1, , k} ,and J (x∗) := {j0 ∈ {1, , `} | hx∗, vj0i = 0}

Proposition 1.11 If x∗ ∈ G and D is given by (1.4), then

S(x∗) =

 X

Max-imum of Finitely Many Affine Functions

Let X be a locally convex Hausdorff topological vector space and f a function from X to

¯

R := R∪ {±∞} The effective domain and the epigraph of f are defined, respectively, bysetting dom f = {x ∈ X | f (x) < +∞} and epi f =(x, α) ∈ X ×R| x ∈ dom f, f (x) ≤ α

If dom f is nonempty and f (x) > −∞ for all x ∈ X, then f is said to be proper We say that

f is convex if epi f is a convex set in X ×R

According to Rockafellar (1970), a real-valued function defined on Rn is called polyhedralconvex if its epigraph is a polyhedral convex set in Rn+1 The following notion of generalizedpolyhedral convex function appears naturally in that spirit.

Definition 2.1 Let X be a locally convex Hausdorff topological vector space A function

f : X → ¯R is called generalized polyhedral convex (resp., polyhedral convex ) if its epigraph

is a generalized polyhedral convex set (resp., a polyhedral convex set) in X ×R If −f is ageneralized polyhedral convex function (resp., a polyhedral convex function), then f is said

to be a generalized polyhedral concave function (resp., a polyhedral concave function)

Complete characterizations of a generalized polyhedral convex function (resp., a polyhedralconvex function) in the form of the maximum of a finite family of continuous affine functionsover a certain generalized polyhedral convex set (resp., a polyhedral convex set) are given innext theorem

Theorem 2.1 Suppose that f : X → ¯Ris a proper function Then f is generalized polyhedralconvex (resp., polyhedral convex) if and only if dom f is a generalized polyhedral convex set(resp., a polyhedral convex set) in X and there exist vk∗∈ X∗, βk ∈R, for k = 1, , m, suchthat

f (x) =

(

maxhvk∗, xi + βk | k = 1, , m if x ∈ dom f,

Con-vex Functions and an Application

We will need the following infinite-dimensional generalization of the concept of piecewiselinear function on Rn of Rockafellar and Wets (1998)

Definition 2.2 A proper function f : X → ¯R, which is defined on a locally convex Hausdorfftopological vector space, is said to be generalized piecewise linear (resp., piecewise linear ) ifthere exist generalized polyhedral convex sets (resp., polyhedral convex sets) D1, , Dm in

X, v1∗, , vm∗ ∈ X∗, and β1, , βm ∈ R such that dom f =

m

S

k=1

Dk and f (x) = hv∗k, xi + βkfor all x ∈ Dk, k = 1, , m

Theorem 2.1 provides us with a general formula for any generalized polyhedral convex tion on a locally convex Hausdorff topological vector space For polyhedral convex functions

func-on Rn, there is another important characterization: A proper convex function f is polyhedralconvex if and only if f is piecewise linear (see Rockafellar and Wets (1998)) It is of interest

to obtain analogous results for generalized polyhedral convex functions and polyhedral convexfunctions on a locally convex Hausdorff topological vector space

The forthcoming theorem clarifies the relationships between generalized polyhedral convexfunctions and generalized piecewise linear functions

Theorem 2.2 Suppose that f : X → ¯R is a proper convex function Then the function f isgeneralized polyhedral convex (resp., polyhedral convex) if and only if f is generalized piecewiselinear (resp., piecewise linear)

Based on Theorem 2.2, we can prove that the class of generalized polyhedral convex tions (resp., the class of polyhedral convex functions) is invariant with respect to the addition

func-of functions

Theorem 2.3 Let f1, f2 be two proper functions on X If f1, f2 are generalized polyhedralconvex (resp., polyhedral convex) and (dom f1)∩(dom f2) is nonempty, then f1+f2is a propergeneralized polyhedral convex function (resp., a polyhedral convex function)

In convex analysis, it is well known that the concept of directional derivative has an portant role We are going to discuss a property of the directional derivative mapping of ageneralized polyhedral convex function (resp., a polyhedral convex function) at a given point

im-If f : X → ¯R is a proper convex function and x ∈ X is such that f (x) is finite, thedirectional derivative f0(x; h) := lim

t→0 +

f (x + th) − f (x)

t of f at x with respect to a direction

h ∈ X, always exists (it can take values −∞ or +∞) Moreover, the closure of the epigraph

of f0(x; ·) coincides with the tangent cone to epi f at (x, f (x)), i.e.,

epi f0(x; ·) = Tepi f(x, f (x)) (2.2)

We know that if f : Rn → ¯R is proper polyhedral convex, then the closure sign in (2.2)can be omitted and f0(x; ·) is a proper polyhedral convex function The last two facts can beextended to polyhedral convex functions on locally convex Hausdorff topological vector spacesand generalized polyhedral convex functions as follows

Theorem 2.4 Let f be a proper generalized polyhedral convex function (resp., a proper hedral convex function) on a locally convex Hausdorff topological vector space X For any

poly-x ∈ dom f , f0(x; ·) is a proper generalized polyhedral convex function (resp., a proper dral convex function) In particular, epi f0(x; ·) is closed and, by (2.2) one has

polyhe-epi f0(x; ·) = Tepi f(x, f (x))

In this section, we are interested in the concept of infimal convolution function, whichwas introduced by Fenchel (1953) According to Rockafellar (1970), the infimal convolutionoperation is analogous to the classical formula for integral convolution and, in a sense, is dual

to the operation of addition of convex functions

Although the infimal convolution of a finite family of functions can be defined (see Ioffeand Tihomirov (1979)), for simplicity, we will only consider the infimal convolution of twofunctions By induction, one can easily extends the result obtained in Proposition 2.1 below

to infimal convolutions of finite families of generalized polyhedral convex functions, providedthat one of them is polyhedral convex

Definition 2.3 (see Ioffe and Tihomirov (1979)) Let f1, f2 be two proper functions on alocally convex Hausdorff topological vector space X The infimal convolution of f1, f2 is the