A Representation of Generalized Convex Polyhedra and Applications

TL;DR

This paper extends the representation formulas for convex polyhedra to generalized convex polyhedra in locally convex spaces, enabling new solution existence results for generalized linear programming and vector optimization problems.

Contribution

It develops new representation formulas for generalized convex polyhedra in locally convex spaces, generalizing prior Banach space results.

Findings

Representation formulas for generalized convex polyhedra in locally convex spaces.

Application of formulas to prove solution existence in generalized linear programming.

Application to generalized linear vector optimization problems.

Abstract

It is well known that finite-dimensional polyhedral convex sets can be generated by finitely many points and finitely many directions. Representation formulas in this spirit are obtained for convex polyhedra and generalized convex polyhedra in locally convex Hausdorff topological vector spaces. Our results develop those of X. Y. Zheng (Set-Valued Anal., Vol. 17, 2009, 389-408), which were established in a Banach space setting. Applications of the representation formulas to proving solution existence theorems for generalized linear programming problems and generalized linear vector optimization problems are shown.

Equations

$$C=\left\{x\in X\mid x\in L,⟨x_i^{\ast },x⟩\le {\alpha }_i,i=1,\dots ,p\right\}.$$

$$\displaystyle C=\Bigg{\{}\sum\limits_{i=1}^{k}\lambda_{i}u_{i}+\sum\limits_{j=1}^{\ell}\mu_{j}v_{j}\mid$$

$$\displaystyle\sum\limits_{i=1}^{k}\lambda_{i}=1,\ \,\mu_{j}\geq 0,\ \forall j=1,\dots,\ell\Bigg{\}},$$

$$X=X_0+X_1,X_0\cap X_1=\{0\},\dim X_1<+\infty ,$$

$$D=D_1+X_0.$$

$$D=\left\{x\in X\mid ⟨x_i^{\ast },x⟩\le {\alpha }_i,i=1,\dots ,p\right\}.$$

$$X_0:=\left\{x\in X\mid ⟨x_i^{\ast },x⟩=0,i=1,\dots ,p\right\}.$$

$$D_1:=\left\{x\in X_1\mid ⟨x_i^{\ast },x⟩\le {\alpha }_i,i=1,\dots ,p\right\}$$

$$⟨x_i^{\ast },x_1⟩=⟨x_i^{\ast },x⟩-⟨x_i^{\ast },x_0⟩=⟨x_i^{\ast },x⟩\le {\alpha }_i$$

$$D_1=\left\{u\in X_1\mid ⟨u_j^{\ast },u⟩\le {\beta }_j,j=1,\dots ,m\right\}.$$

$$⟨x_j^{\ast },x⟩=⟨u_j^{\ast }\circ \pi ,x⟩=⟨u_j^{\ast },\pi (x)⟩=⟨u_j^{\ast },x_1⟩\le {\beta }_j$$

$${\beta }_j\ge ⟨x_j^{\ast },x_0+x_1⟩=⟨u_j^{\ast }\circ {\Phi }_0\circ {\pi }_0,x_0+x_1⟩=⟨u_j^{\ast },x_1⟩$$

$$\displaystyle D=\Bigg{\{}\sum\limits_{i=1}^{k}\lambda_{i}u_{i}+\sum\limits_{j=1}^{\ell}\mu_{j}v_{j}\mid$$

$$\displaystyle\sum\limits_{i=1}^{k}\lambda_{i}=1,\ \,\mu_{j}\geq 0,\ \forall j=1,\dots,\ell\Bigg{\}}+X_{0}.$$

$$D=\left\{x\in X\mid x\in L,⟨x_i^{\ast },x⟩\le {\alpha }_i,i=1,2,\dots ,p\right\},$$

$$D_0=\left\{u\in X\mid Au=0,⟨x_i^{\ast },u⟩\le {\alpha }_i-⟨x_i^{\ast },x_0⟩,i=1,\dots ,p\right\}.$$

$$\ker A=X_{0,A}+X_{1,A},X_{1,A}\cap X_{0,A}=\{0\},\dim X_{1,A}<+\infty ,$$

$$D_0=D_{1,A}+X_{0,A}.$$

$$\displaystyle D_{1,A}=\Bigg{\{}\sum\limits_{i=1}^{k}\lambda_{i}u_{i}+\sum\limits_{j=1}^{\ell}\mu_{j}v_{j}\mid$$

$$\displaystyle\sum\limits_{i=1}^{k}\lambda_{i}=1,\ \,\mu_{j}\geq 0,\ \forall j=1,\dots,\ell\Bigg{\}},$$

$$\displaystyle D=\Bigg{\{}\sum\limits_{i=1}^{k}\lambda_{i}\left({u}_{i}+x_{0}\right)+\sum\limits_{j=1}^{\ell}\mu_{j}{v}_{j}\mid$$

$$\displaystyle\sum\limits_{i=1}^{k}\lambda_{i}=1,\ \,\mu_{j}\geq 0,\ \forall j=1,\dots,\ell\Bigg{\}}+{X}_{0,A}.$$

$$X_1=\mathrm{span}\{u_1,\dots ,u_k,v_1,\dots ,v_{\ell }\}$$

$$\displaystyle D_{1}:=\Bigg{\{}\sum\limits_{i=1}^{k}\lambda_{i}u_{i}+\sum\limits_{j=1}^{\ell}\mu_{j}v_{j}\mid$$

$$\displaystyle\sum\limits_{i=1}^{k}\lambda_{i}=1,\mu_{j}\geq 0,\ \forall j=1,\dots,\ell\Bigg{\}}.$$

$$\displaystyle\pi\left(D_{1}\right)=\Bigg{\{}\sum\limits_{i=1}^{k}\lambda_{i}\pi\left(u_{i}\right)+\sum\limits_{j=1}^{\ell}\mu_{j}\pi\left(v_{j}\right)\mid$$

$$\displaystyle\sum\limits_{i=1}^{k}\lambda_{i}=1,\ \,\mu_{j}\geq 0,\ \forall j=1,\dots,\ell\Bigg{\}}.$$

$$x=z_1+\pi (z_1)-z_1+x_0=z_1+\left(x_0-\left(z_1-\pi (z_1)\right)\right)\in D_1+X_0.$$

$$D=\left\{x\in W\mid ⟨w_i^{\ast },x⟩\le {\alpha }_i,i=1,\dots ,m\right\}.$$

$$D=\left\{x\in W\mid ⟨x_i^{\ast },x⟩\le {\alpha }_i,i=1,\dots ,m\right\}.$$

$$⟨x_1^{\ast },x⟩=\underset{a}{\overset{b}{\int }}{\omega }_1(t)x(t)dt,⟨x_2^{\ast },x⟩=\underset{a}{\overset{b}{\int }}{\omega }_2(t)x(t)dt,$$

$${\gamma }_{i,j}:=\underset{a}{\overset{b}{\int }}{\omega }_i(t){\omega }_j(t)dt,$$

$${\left(\underset{a}{\overset{b}{\int }}{x}^2(t)dt\right)}^{1/2}{\left(\underset{a}{\overset{b}{\int }}{y}^2(t)dt\right)}^{1/2}\ge \left|\underset{a}{\overset{b}{\int }}x(t)y(t)dt\right|,$$

Full text

A Representation of Generalized Convex Polyhedra and Applications111This work was supported by National Foundation for Science $\&$ Technology Development (Vietnam) under the grant No. 101.01-2014.37. The second author thanks the Vietnam Institute for Advanced Study in Mathematics for supporting his 6-month stay at the Institute in 2015.

Nguyen Ngoc Luan222Department of Mathematics and Informatics, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam; email: luannn@hnue.edu.vn.   and   Nguyen Dong Yen333Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi 10307, Vietnam; email: ndyen@math.ac.vn.

Abstract. It is well known that finite-dimensional polyhedral convex sets can be generated by finitely many points and finitely many directions. Representation formulas in this spirit are obtained for convex polyhedra and generalized convex polyhedra in locally convex Hausdorff topological vector spaces. Our results develop those of X. Y. Zheng (Set-Valued Anal., Vol. 17, 2009, 389–408), which were established in a Banach space setting. Applications of the representation formulas to proving solution existence theorems for generalized linear programming problems and generalized linear vector optimization problems are shown.

Mathematics Subject Classification (2010). 49N10, 90C05, 90C29, 90C48.

Key Words. Convex polyhedron, generalized convex polyhedron, locally convex Hausdorff topological vector space, representation formula, generalized linear programming problem, generalized linear vector optimization problem, solution existence theorem.

1 Introduction

The intersection of a finite number of closed half-spaces of a finite-dimensional Euclidean space is called a polyhedral convex set (a convex polyhedron in brief). By convention, the intersection of an empty family of closed half-spaces is the whole space. Therefore, emptyset and the whole spaces are two special polyhedra. Due to [11, 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 as the sum of the convex hull of those points and the convex cone generated by those directions. The converse is also true. This celebrated result is attributed [11, p. 427] primarily to Minkowski [10] and Weyl [13, 14]. By using the result, it is easy to derive fundamental solution existence theorems in linear progamming. Note that the just cited representation formula for finite-dimensional polyhedral convex sets has many other applications in mathematics. As an example, one can refer to the elegant proofs of the necessary and sufficient second-oder conditions for a local solution and for a locally unique solution in quadratic programming, which were given by Contesse [2] in 1980; see [7, pp. 50–63] for details.

According to Bonnans and Shapiro [1, Definition 2.195], a subset of locally convex Hausdorff topological vector space is called a generalized polyhedral convex set (or a generalized convex polyhedron) if it is the intersection of finitely many closed half-spaces and a closed affine subspace of that topological vector space. If the affine subspace can be chosen as the whole space, the generalized polyhedral convex set is said to be a polyhedral convex set (or a convex polyhedron). The theories of generalized linear programming and quadratic programming in [1, Sections 2.5.7 and 3.4.3] are based on this concept of generalized convex polyhedron.

In 2009, using a result related to the Banach open mapping theorem (see, e.g., [12, Theorem 5.20]), Zheng [15, Corollary  2.1] has clarified the relationships between convex polyhedra in Banach spaces and the finite-dimensional convex polyhedra.

It is well known that any infinite-dimensional normed space equipped with the weak topology is not metrizable, but it is a locally convex Hausdorff topological vector space. Similarly, the dual space of any infinite-dimensional normed space equipped with the weak∗ topology is not metrizable, but it is a locally convex Hausdorff topological vector space. Actually, the just mentioned two models provide us with the most typical examples of locally convex Hausdorff topological vector spaces, whose topologies cannot be given by norms. It is clear that Zheng’s results in [15] cannot be used neither for a infinite-dimensional normed space equipped with the weak topology, nor for the dual space of any infinite-dimensional normed space equipped with the weak∗ topology.

The aim of our paper is twofold: to find an analogue of the above-mentioned representation of finite-dimensional convex polyhedra via finite families of points and directions for convex polyhedra in locally convex Hausdorff topological vector spaces, and to apply the obtained results to proving solution existence theorems for infinite-dimensional linear programming problems and linear vector optimization problems. Among other things, we will show that the result of Zheng [15, Corollary  2.1] is valid for convex polyhedra in locally convex Hausdorff topological vector spaces.

The organization of the present paper is as follows. In Section 2, we obtain representation formulas for generalized convex polyhedra. Section 3 is devoted to solution existence of generalized linear programs. Solution existence of generalized linear vector optimization problems is studied in Section 4.

2 Representation Formulas for Generalized Convex Polyhedra

Let $X$ be a locally convex Hausdorff topological vector space with the dual space denoted by $X^{*}$. For any $x^{*}\in X^{*}$ and $x\in X$, $\langle x^{*},x\rangle$ indicates the value of $x^{*}$ at $x$.

Definition 2.1

(See [1, p. 133]) A subset $C\subset X$ is said to be a generalized polyhedral convex set (a generalized convex polyhedron for short) if there exist $x^{*}_{i}\in X^{*}$, $\alpha_{i}\in\mathbb{R}$, $i=1,2,\dots,p$, and a closed affine subspace $L\subset X$, such that

[TABLE]

If $C$ admits the last representation for $L=X$ and for some $x^{*}_{i}\in X^{*}$, $\alpha_{i}\in\mathbb{R}$, $i=1,2,\dots,p$, then it is called a polyhedral convex set (or a convex polyhedron).**

The following classical result shows that, for any convex polyhedron in $\mathbb{R}^{n}$, one can find a finite number of points and a finite number of directions such that the polyhedron can be represented as the sum of the convex hull of those points and the convex cone generated by those directions. The converse is also true.

Theorem 2.2

([11, Theorem 19.1])* For any nonempty set $C\subset\mathbb{R}^{n}$, the following two properties are equivalent:

(a) $C$ is a convex polyhedron;

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

[TABLE]

where $u_{i}\in\mathbb{R}^{n}$, $i=1,\dots,k$, and $v_{j}\in\mathbb{R}^{n}$, $j=1,\dots,\ell$.

From (2.1) it follows that $u_{i}\in C$ for $i=1,\dots,k$. A natural question arises: Is there any analogue of the representation (2.1) for convex polyhedra in locally convex Hausdorff topological vector spaces, or not? In order to give an answer in the affirmative to this question, we will need several results from functional analysis. In what follows, $X$ is a locally convex Hausdorff topological vector space.

Lemma 2.3

(Closedness of the sum two linear subspaces; see [12, Theorem 1.42])* Suppose $X_{0}$ and $X_{1}$ are linear subspaces of $X$, $X_{0}$ is closed, and $X_{1}$ has finite dimension. Then $X_{0}+X_{1}$ is closed.*

Lemma 2.4

(The Hahn-Banach extension theorem; see [12, Theorem 3.6])* If $x^{*}$ is a continuous linear functional on a linear subspace $M$ of $X$, then there exists $\widetilde{x}^{*}\in X^{*}$ such that $\langle\widetilde{x}^{*},x\rangle=\langle x^{*},x\rangle$ for all $x\in M$.*

The forthcoming lemma follows from a theorem in [12]. A proof is provided here for the sake of clarity of our presentation.

Lemma 2.5

If $Y$ and $Z$ are Hausdorff finite-dimensional topological vector spaces of dimension $n$ and if $g:Y\rightarrow Z$ is a linear bijective mapping, then $g$ is a homeomorphism.

Proof. Let $\{e_{1},e_{2},\dots,e_{n}\}$ be a basis of the Euclidean space ${\mathbb{R}}^{n}$, which is equipped with the natural topology. Let $\{v_{1},v_{2},\dots,v_{n}\}$ be a basis of $Y$. Setting $w_{i}=g(v_{i})$ for $i=1,\dots,n$, we see that $\{w_{1},w_{2},\dots,w_{n}\}$ is a basis of $Z$. Clearly, there is an unique linear bijection $\Phi:{\mathbb{R}}^{n}\rightarrow Y$ satisfying the conditions $\Phi(e_{i})=v_{i}$ for all $i$. Similarly, there is an unique linear bijection $\Psi:{\mathbb{R}}^{n}\rightarrow Z$ with $\Phi(e_{i})=w_{i}$ for all $i$. By [12, Theorem 1.21(a)], $\Phi$ and $\Psi$ are homeomorphisms. (Note that the quoted result was obtained for ${\mathbb{C}}^{n}$ and topological vector spaces over the complex field $\mathbb{C}$. Nevertheless, the method of proof is valid for the case of ${\mathbb{R}}^{n}$ and topological vector spaces over $\mathbb{R}$.) Since $g=\Psi\circ\Phi^{-1}$ and $g^{-1}=\Phi\circ\Psi^{-1}$ by our construction, it follows that both $g$ and $g^{-1}$ are continuous mappings. $\hfill\Box$

We are now in a position to extend Corollary 2.1 from the paper of Zheng [15], which was given in a normed spaces setting, to the case of convex polyhedra in locally convex Hausdorff topological vector spaces.

Proposition 2.6

A nonempty subset $D\subset X$ is a convex polyhedron if only if there exist closed linear subspaces $X_{0}$, $X_{1}$ of $X$ and a convex polyhedron $D_{1}\subset X_{1}$ such that

[TABLE]

and

[TABLE]

Proof. Necessity: If $D$ is a convex polyhedron, then there exist $x^{*}_{i}\in X^{*}$, $\alpha_{i}\in\mathbb{R}$, $i=1,\dots,p$, such that

[TABLE]

Let

[TABLE]

Because $X_{0}$ is a closed linear subspace of finite codimension, one can find a finite-dimensional linear subspace $X_{1}$ of $X$, such that $X=X_{0}+X_{1}$ and $X_{0}\cap X_{1}=\{0\}$. By [12, Theorem 1.21(b)], $X_{1}$ is closed. Clearly,

[TABLE]

is a convex polyhedron in $X_{1}$. It is easy to verify that $D_{1}+X_{0}\subset D$. The reverse inclusion is also true. Indeed, for each $x\in D$ there exist $x_{0}\in X_{0}$ and $x_{1}\in X_{1}$ satisfying $x=x_{0}+x_{1}$. Since

[TABLE]

for all $i=1,\dots,p$, it follows that $x_{1}\in D_{1}$; hence $x=x_{1}+x_{0}\in D_{1}+X_{0}$. We have thus proved that $D=D_{1}+X_{0}$.

Sufficiency: Let $X_{0}$, $X_{1}$ be closed subspaces of $X$ satisfying the conditions in (2.2). Let $D_{1}\subset X_{1}$ be a convex polyhedron in $X_{1}$ and let $D$ be defined by (2.3). Select $u^{*}_{j}\in X_{1}^{*}$ and $\beta_{j}\in\mathbb{R}$, $j=1,\dots,m$, such that

[TABLE]

Let $\pi_{0}:X\rightarrow X/X_{0}$, $x\mapsto x+X_{0}$ for all $x\in X$, be the canonical projection from $X$ on the quotient space $X/X_{0}$. It is clear that the operator $\Phi_{0}:X/X_{0}\rightarrow X_{1}$, $x_{1}+X_{0}\mapsto x_{1}$ for all $x_{1}\in X_{1}$, is a linear bijective mapping. On one hand, by [12, Theorem 1.41(a)], $\pi_{0}$ is a linear continuous mapping. On the other hand, $\Phi_{0}$ is a homeomorphism by Lemma 2.5. So, the operator $\pi:=\Phi_{0}\circ\pi_{0}:X\rightarrow X_{1}$ is linear and continuous. Put $x^{*}_{j}=u^{*}_{j}\circ\pi$, $j=1,\dots,m$. Take any $x=x_{1}+x_{0}$ with $x_{1}\in D_{1}$ and $x_{0}\in X_{0}$. It clear that

[TABLE]

for all $j=1,\dots,m$. Conversely, take any $x\in X$ satisfying $\langle x^{*}_{j},x\rangle\leq\beta_{j}$ for all $j=1,\dots,m$. Let $x_{0}\in X_{0}$ and $x_{1}\in X_{1}$ be such that $x=x_{0}+x_{1}$. Since

[TABLE]

for all $j=1,\dots,m$, we see that $x_{1}\in D_{1}$. Hence $x\in D_{1}+X_{0}$. It follows that $D_{1}+X_{0}=\left\{x\in X\mid\langle x^{*}_{j},x\rangle\leq\beta_{j},\;j=1,\dots,m\right\}$. Therefore $D=D_{1}+X_{0}$ is a convex polyhedron in $X$. $\hfill\Box$

The main result of this section is formulated as follows.

Theorem 2.7

A nonempty subset $D\subset X$ is a generalized convex polyhedron if and only if there exist $u_{1},\dots,u_{k}\in X$, $v_{1},\dots,v_{\ell}\in X$, and a closed linear subspace $X_{0}\subset X$ such that

[TABLE]

Proof. Necessity: Suppose that $D$ is a generalized convex polyhedron. Then we have

[TABLE]

where $L\subset X$ is a closed affine subspace, $x^{*}_{i}\in X^{*}$ and $\alpha_{i}\in\mathbb{R}$ for $i=1,\dots,p$. Select a locally convex Hausdorff topological vector space $Y$, a continuous linear mapping $A:X\rightarrow Y$, and a point $y\in Y$ such that $L=\{x\in X\mid Ax=y\}$. Fix an element $x_{0}\in D$ and set $D_{0}=D-x_{0}$. It is easy to verify that

[TABLE]

As $D_{0}$ is a convex polyhedron in ${\rm ker}A:=\{u\in X\mid Au=0\}$, by Proposition 2.6 we can find closed linear subspaces $X_{0,A}$ and $X_{1,A}$ of ${\rm{ker}}A$ and a convex polyhedron $D_{1,A}\subset X_{1,A}$ such that

[TABLE]

and

[TABLE]

Because $X_{1,A}\subset{\rm{ker}}A$ is closed and ${\rm{ker}}A$ is a closed linear subspace of $X$, $X_{1,A}$ is a closed linear subspace of $X$. Since $D_{1,A}$ is a convex polyhedron of the finite-dimensional space $X_{1,A}$, invoking Theorem 2.2 we can represent $D_{1,A}$ as

[TABLE]

where $u_{i}\in D_{1,A}$ for $i=1,\dots,k$ and $v_{j}\in X_{1,A}$ for $j=1,\dots,\ell$. Therefore

[TABLE]

We have thus found a representation of the form (2.4) for $D$.

Sufficiency: Suppose that $D$ is of the form (2.4). Let

[TABLE]

be the linear subspace generated by the vectors $u_{1},\dots,u_{k},v_{1},\dots,v_{\ell}$. Put

[TABLE]

By Lemma 2.3, $W:=X_{1}+X_{0}$ is a closed linear subspace of $X$. Because $X_{0}$ is a closed subspace of finite codimension of $W$, one can find a finite-dimensional linear subspace $W_{1}\subset W$, such that $W=X_{0}+W_{1}$ and $X_{0}\cap W_{1}=\{0\}$. Consider the continuous linear mapping $\pi:W\rightarrow W_{1}$ be defined by $\pi(x)=w_{1}$, where $x=x_{0}+w_{1}$ with $\left(w_{1},x_{0}\right)\in W_{1}\times X_{0}$. We have

[TABLE]

By Lemma 2.2, $\pi(D_{1})$ is a convex polyhedron of $W_{1}$. We have $D_{1}+X_{0}=\pi(D_{1})+X_{0}$. Indeed, if $x=x_{1}+x_{0}$ where $x_{1}\in D_{1}$ and $x_{0}\in X_{0}$, then $x_{1,0}:=x_{1}-\pi(x_{1})\in X_{0}$. So $x=\pi(x_{1})+x_{1,0}+x_{0}$ belongs to $\pi(D_{1})+X_{0}$. Conversely, for any $x=\pi(z_{1})+x_{0}$ with $z_{1}\in D_{1}$ and $x_{0}\in X_{0}$, we have

[TABLE]

Since $D=D_{1}+X_{0}=\pi(D_{1})+X_{0}$, $D$ is a convex polyhedron in $W$ by Proposition 2.6. Hence there exist $w^{*}_{1},\dots,w^{*}_{m}\in W^{*}$ and $\alpha_{1},\dots,\alpha_{m}\in\mathbb{R}$ such that

[TABLE]

According to Lemma 2.4, there exist $x^{*}_{i}\in X^{*}$, $i=1,\dots,m$, such that $\langle x^{*}_{i},x\rangle=\langle w^{*}_{i},x\rangle$ for all $x\in W.$ Therefore

[TABLE]

It follows that $D$ is a generalized polyhedral convex set in $X$. $\hfill\Box$

The next example is an illustration for Theorem 2.7.

Example 2.8

Let $X=C[a,b]$ be the linear space of continuous real valued functions on the interval $[a,b]$ with the norm defined by $||x||=\max_{t\in[a,b]}|x(t)|$. By the Riesz representation theorem (see e.g. [6, Theorem 6, p. 374] and [9, Theorem 1, p. 113]), the dual space of $X$ is the normalized space $X^{*}=NBV[a,b]$ of functions of bounded variation, that is functions $y:[a,b]\rightarrow\mathbb{R}$ of bounded variation, $y(a)=0$, and $y(\cdot)$ is continuous from the left at every point of $(a,b)$. Let $x^{*}_{1},x^{*}_{2}\in X^{*}$ be defined by

[TABLE]

where $\omega_{1},\omega_{2}$ in $X\setminus\{0\}$ are chosen such that the vectors $\omega_{1},\omega_{2}$ are linearly independent. The integrals in (2.5) are Riemannian. They equal respectively to the Riemann-Stieltjes integrals (see [6, p. 367]) $\int\limits_{a}^{b}x(t)dy_{1}(t)$ and $\int\limits_{a}^{b}x(t)dy_{2}(t)$, which are given by the $C^{1}$-smooth functions $y_{i}(t)=\int\limits_{a}^{t}\omega_{i}(\tau)d\tau$, $i=1,2$. Set

[TABLE]

for $i,j\in\{1,2\}$. It is clear that $\gamma_{1,2}=\gamma_{2,1},\gamma_{1,1}>0,\gamma_{2,2}>0$. The Cauchy-Schwarz inequality

[TABLE]

which is valid for any functions $x(\cdot),y(\cdot)\in C[a,b]\subset L_{2}[a,b]$, implies that $\delta:=\gamma_{1,1}.\gamma_{2,2}-\gamma_{1,2}^{2}\geq 0$. As the vectors $\omega_{1},\omega_{2}$ are linearly independent, we must have $\delta>0$. Given any $\alpha_{1},\alpha_{2}\in\mathbb{R}$, we want to find a representation of form (2.4) for the convex polyhedron

[TABLE]

Let $X_{0}:=\left\{x\in X\mid\langle x^{*}_{1},x\rangle=0,\;\langle x^{*}_{2},x\rangle=0\right\}.$ For $x=\eta_{1}\omega_{1}+\eta_{2}\omega_{2}$ with $\eta_{1},\eta_{2}\in\mathbb{R}$, we have

[TABLE]

and

[TABLE]

Since $\delta>0$, there exists an unique pair of real numbers $(\eta_{1},\eta_{2})$ satisfying

[TABLE]

Let the point $u$ and the directions $v_{1},v_{2}\in X$ be defined by

[TABLE]

It is easy to verify that $\langle x^{*}_{i},u\rangle=\alpha_{i}$ for $i=1,2$, and

[TABLE]

Let us show that

[TABLE]

Take any $x=u+\mu_{1}v_{1}+\mu_{2}v_{2}+x_{0}$ with $\mu_{1},\mu_{2}\in\mathbb{R}_{+}$ and $x_{0}\in X_{0}$. Because

[TABLE]

and

[TABLE]

we have $x\in D$. Now, take any $x\in D$. Put $\mu_{1}=\delta^{-1}\left(\alpha_{2}-\langle x_{2}^{*},x\rangle\right)$, $\mu_{2}=\delta^{-1}\left(\alpha_{1}-\langle x_{1}^{*},x\rangle\right)$, and $x_{0}=x-\left(u+\mu_{1}v_{1}+\mu_{2}v_{2}\right)$. Note that $\mu_{1}\geq 0$, $\mu_{2}\geq 0$ and $x=u+\mu_{1}v_{1}+\mu_{2}v_{2}+x_{0}$. Since $\langle x^{*}_{i},x_{0}\rangle=0$ for $i=1,2$, we see that $x_{0}\in X_{0}$. The formula (2.7) has been proved.**

Based on the preceding example, we can easily construct an illustrative example for polyhedral convex sets in locally convex Hausdorff topological vector spaces.

Example 2.9

Keeping all the notations of Example 2.8, we consider $X=C[a,b]$ with the weak topology. Then $X$ is a locally convex Hausdorff topological vector space whose topology is not a norm topology. The analysis given above shows that the set $D$ in (2.6) admits the representation (2.4). **

From Theorem 2.7 we can obtain a representation formula for generalized polyhedral convex cones.

Theorem 2.10

A nonempty set $K\subset X$ is a generalized polyhedral convex cone if and only if there exist $v_{j}\in K,j=1,\dots,\ell$, and a closed linear subspace $X_{0}$ such that

[TABLE]

Proof. Necessity: If $K$ is a generalized polyhedral convex cone, then by Theorem 2.7 we can find $u_{i}\in K$, $i=1,\dots,k$, $v_{j}\in X$, $j=1,\dots,\ell$, and a closed linear subspace $X_{0}$ such that

[TABLE]

To show that $v_{j}\in K$ for $j=1,\dots,\ell$, it suffices to observe by (2.9) that $\frac{1}{t}\left(u_{1}+tv_{j}\right)=\frac{1}{t}u_{1}+v_{j}$ belongs to $K$ for all $t>0$, because $K$ is a cone. Letting $t\to\infty$, by the closedness of $K$, we get $v_{j}\in K$. Since $v_{j}\in K$ for $j=1,\dots,\ell$, and since $t_{i}u_{i}\in K$ for all $i=1,\dots,k$, and $t_{i}\geq 0$, by choosing $v_{\ell+i}=u_{i}$ for $i=1,\dots,k,$ by (2.9) we see that $K$ admits the representation (2.8) where $\ell$ is replaced by $\ell+k$.

Sufficiency: If $K$ has the form (2.8) then it is a cone. In addition, $K$ is a generalized polyhedral convex set by Theorem 2.7.$\hfill\Box$

3 Solution Existence in Linear Optimization

Consider a generalized linear programming problem

[TABLE]

where, as before, $X$ is a locally convex Hausdorff topological vector space, $D\subset X$ is a generalized polyhedral convex set, $x^{*}\in X^{*}$. By definition, the recession cone $0^{+}C$ of a convex set $C\subset X$ is given by

[TABLE]

If $D$ is represented in the form (2.4), then $0^{+}D={\rm{cone}}\{v_{1},\dots,v_{\ell}\}+X_{0}$.

The following two existence theorems for (LP) are known results. Actually, in combination, they express the contents of Theorem 2.199 from [1]. The latter, in its turn, is a special case of Theorem 2.198 from [1]. The simple proofs given below show how Theorem 2.7 can be used to study the solution existence of generalized linear programs.

Theorem 3.1

(The Eaves-type Existence Theorem; see [1, Theorem 2.199])* If $D$ is nonempty, then (LP) has a solution if and only if $\langle x^{*},v\rangle\geq 0$ for every $v\in 0^{+}D$.*

Proof. If (LP) has a solution $\overline{x}$, then for each $v\in 0^{+}D$ it holds that

[TABLE]

because $\overline{x}+tv\in D$ for every $t\geq 0$. Hence $\langle x^{*},v\rangle\geq 0$.

Conversely, suppose that $\langle x^{*},v\rangle\geq 0$ for every $v\in 0^{+}D$. Let us represent $D$ in the form (2.4). Select an element $u_{i_{0}}\in\{u_{1},\dots,u_{k}\}$ such that

[TABLE]

By (2.4), for every $x\in D$ there exist $\lambda_{i}\in\mathbb{R}_{+}$, $i=1,\dots,k$, $\sum\limits_{i=1}^{k}\lambda_{i}=1$, and $v\in 0^{+}D$ such that $x=\sum\limits_{i=1}^{k}\lambda_{i}u_{i}+v$. Then we have

[TABLE]

Since $x\in D$ can be chosen arbitrarily, $u_{i_{0}}$ must be a solution of (LP). $\hfill\Box$

Remark 3.2

If $\langle x^{*},v\rangle\geq 0$ for every $v\in 0^{+}D$, then one says that the functional $x^{*}$ is copositive on the recession $0^{+}D$. We called Theorem 3.1 the Eaves-type Existence Theorem in linear optimization to trace back Eaves’ idea [3, Theorem 3 and Corollary 4, p.702] (see also [7, Theorem 2.2]) in using recession cones for existence theorems in quadratic programming.

Theorem 3.3

(The Frank–Wolfe-type Existence Theorem; see [1, Theorem 2.199])* If $D$ is nonempty, then (LP) has a solution if and only if there is a real number $\gamma$ such that $\langle x^{*},x\rangle\geq\gamma$ for every $x\in D$.*

Proof. The necessity is obvious. To prove the sufficiency, suppose that there is a $\gamma\in\mathbb{R}$ such that $\langle x^{*},x\rangle\geq\gamma$ for all $x\in D$. Then, for any $v\in 0^{+}D$ and $x\in D$ we have

[TABLE]

for every $t>0$. It follows that $\langle x^{*},v\rangle\geq 0$ for any $v\in 0^{+}D$. So, by Theorem 3.1, we can assert that (LP) has a solution. $\hfill\Box$

Remark 3.4

Due to the formulation of the existence theorem in quadratic programming of Frank and Wolfe [4, p. 158] (see also [7, Theorem 2.1]), we called Theorem 3.3 the Frank–Wolfe-type Existence Theorem in linear optimization.

We are interested in studying the region $G$ of all $x^{*}$ for which (LP) has a nonempty solution set, assuming that the constraint set $D$ is nonempty and fixed.

Proposition 3.5

If $D$ has the form (2.4), then $G$ is a generalized polyhedral convex cone of $X^{*}$ which has the representation

[TABLE]

where $X_{0}^{\perp}:=\{x^{*}\in X^{*}\mid\langle x^{*},x_{0}\rangle=0,\ \forall x_{0}\in X_{0}\}$ is the annihilator [9, p. 117] of $X_{0}$.

Proof. By Theorem 3.1, $G=\{x^{*}\in X^{*}\mid\langle x^{*},v\rangle\geq 0,\ \forall v\in 0^{+}D\}$. Therefore, given any $x^{*}\in G$, we have $\langle x^{*},v\rangle\geq 0$ for all $v\in 0^{+}D$. Hence, for every $x_{0}\in X_{0}\subset 0^{+}D$ one has $\langle x^{*},x_{0}\rangle=0$ because $-x_{0}\in 0^{+}D$. Thus $x^{*}\in X_{0}^{\perp}$. In addition, for each $j=1,\dots,\ell$, one has $\langle x^{*},v_{j}\rangle\geq 0$ as $v_{j}\in 0^{+}D$. This establishes the inclusion “$\subset$” in (3.11).

Conversely, suppose that $x^{*}\in X_{0}^{\perp}\cap\{x^{*}\in X^{*}\mid\langle x^{*},v_{j}\rangle\geq 0,\ j=1,\dots,\ell\}$. Since $0^{+}D={\rm{cone}}\{v_{1},\dots,v_{\ell}\}+X_{0}$, the last inclusion implies that $\langle x^{*},v\rangle\geq 0$, for all $v\in 0^{+}D$. Hence, by Theorem 3.1 we can conclude that $x^{*}\in G$. The inclusion “$\supset$” in (3.11) has been proved.

From (3.11) it follows that $G$ is a generalized polyhedral convex set. The fact that $G$ is a cone is obvious. $\hfill\Box$

Next, for each $x^{*}\in G$, 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 (2.4) and consider the index sets

[TABLE]

and

[TABLE]

Note that $I(x^{*})$ is nonempty, but it may happen that $J(x^{*})$ is empty.

Proposition 3.6

If $x^{*}\in G$ and $D$ is given by (2.4), then

[TABLE]

In particular, $S(x^{*})$ is a generalized polyhedral convex set.

Proof. First, take an arbitrary element $\bar{x}$ from the set on the right-hand-side of (3.12). Let

[TABLE]

where $\bar{\lambda}_{i}\geq 0$ for all $i\in I(x^{*})$, $\sum\limits_{i\in I(x^{*})}\bar{\lambda}_{i}=1$, $\bar{\mu}_{j}\geq 0$ for all $j\in J(x^{*})$, and $\bar{x}_{0}\in X_{0}$. By (2.4), for each $x\in D$ one can find $\lambda_{i}\geq 0$ for $i=1,\dots,k$, $\sum\limits_{i=1}^{k}\lambda_{i}=1$, $\mu_{j}\geq 0$ for $j=1,\dots,\ell$, and $x_{0}\in X_{0}$ such that

[TABLE]

By Proposition 3.5, $\langle x^{*},x_{0}\rangle=\langle x^{*},\bar{x}_{0}\rangle=0$. If $J(x^{*})\neq\emptyset$, then using Theorem 3.1 and formula $0^{+}D={\rm{cone}}\{v_{1},\dots,v_{\ell}\}+X_{0}$ we get

[TABLE]

Now, selecting an index ${i_{0}}\in I(x^{*})$ and recalling the definition $I(x^{*})$, we get

[TABLE]

It follows that $\langle x^{*},x\rangle\geq\langle x^{*},\bar{x}\rangle$. We have shown that $\bar{x}\in S(x^{*})$.

Second, take any vector $\bar{x}\in S(x^{*})$ and represent it in the form

[TABLE]

where $\bar{\lambda}_{i}\geq 0$ for $i=1,\dots,k$, $\sum\limits_{i=1}^{k}\bar{\lambda}_{i}=1$, $\bar{\mu}_{j}\geq 0$ for $j=1,\dots,\ell$, and $\bar{x}_{0}\in X_{0}$. It is easy to show that $\bar{\lambda}_{i}=0$ for all $i\notin I(x^{*})$ and $\bar{\mu}_{j}=0$ for all $j\notin J(x^{*})$. This implies that $\bar{x}$ belongs to the set on the right-hand-side of (3.12).

The proof is complete. $\hfill\Box$

4 The Weakly Efficient Solution Set in Linear Vector Optimization

Consider a generalized linear vector optimization problem of the form

[TABLE]

with $M:X\rightarrow Y$ being a continuous linear mapping between locally convex Hausdorff topological vector spaces, $D\subset X$ a generalized polyhedron, $K\subset Y$ a polyhedral convex cone.

We say that $u\in D$ is a weakly efficient solution of (VLP) if there does not exist any $x\in D$ such that $Mu-Mx\in{\rm int}K$. The set of all the weakly efficient solutions is denoted by $E^{w}$. We are interested in finding conditions to have $E^{w}\neq\emptyset$.

By a standard scalarization scheme in vector optimization, given any $y^{*}\in Y^{*}$, we define the scalar problem

[TABLE]

To make our presentation easier for reading, we give simple proof for the following known result.

Lemma 4.1

(See [8, Proposition 3.2, p. 95])* If ${\rm int}K$ is nonempty, then $u\in D$ is a weakly efficient solution of (VLP) if and only if there exists $y^{*}\in K^{*}\setminus\{0\}$, where $K^{*}:=\left\{y^{*}\in Y^{*}\mid\langle y^{*},y\rangle\geq 0\ \,\forall y\in K\right\}$, such that*

[TABLE]

Proof. First, suppose that $u\in E^{w}$. Since \big{(}Mu-M(D)\big{)}\cap{\rm int}K=~{}\emptyset and since $Mu-M(D)$ and ${\rm int}K$ are convex sets, by the separation theorem (see, e.g., [1, Theorem 2.13]), there exists $y^{*}\in Y^{*}\setminus\{0\}$ such that

[TABLE]

for all $x\in D$ and $y\in K$. Substituting $x=u$ to the above inequality yields $\langle y^{*},y\rangle\geq 0$ for all $y\in K$. Hence $y^{*}\in K^{*}$. Choosing $y=0$, one has $\langle y^{*},Mu\rangle\leq\langle y^{*},Mx\rangle$ for every $x\in D$. This shows that the inclusion (4.13) is valid.

Now, suppose that $u\in D$ and there is $y^{*}\in K^{*}\setminus\{0\}$ such that (4.13) is satisfied. If $u\not\in E^{w}$, then there exist $x\in D$ and a balanced neighborhood $V$ of 0 satisfying $Mu-Mx+V\subset K$. Hence, for each $v\in V$, one has $\langle y^{*},Mu-Mx+v\rangle\geq 0$. In combination with the inequality $\langle y^{*},Mu-Mx\rangle\leq 0$ which is guaranteed by (4.13), this implies $\langle y^{*},v\rangle\geq 0$. As $V$ is a balanced neighborhood of 0, we can assert that $\langle y^{*},v\rangle=0$ for all $v\in V$. Let $y\in Y$ be such that $\langle y^{*},y\rangle\not=0$. Since there exists $t>0$ with $ty\in V$, we get $\langle y^{*},y\rangle=0$, a contradiction. $\hfill\Box$

Remark 4.2

Looking back to the proof of Lemma 4.1, we can observe that it suffices to assume that $K\subset Y$ is a convex cone. In other words, the polyhedrality of $K$ is superfluous for the assertion of the lemma.**

We have $\langle y^{*},Mx\rangle=\langle M^{*}y^{*},x\rangle$ for all $x\in X$, where $M^{*}:Y^{*}\rightarrow X^{*}$ is the adjoint operator of $M$.

Theorem 4.3

Problem (VLP) has a solution if and only if

[TABLE]

In particular, if

[TABLE]

then (VLP) has a solution.

Proof. By Lemma 4.1, (VLP) has a solution if and only if there exists $y^{*}\in K^{*}\setminus\{0\}$ such that the solution set of ${\rm(LP)}_{y^{*}}$ is non empty. According to Theorem 3.1, this solution set is non-void if and only if $M^{*}y^{*}\in\left(0^{+}D\right)^{*}$. Thus, we have shown that (VLP) has a solution if and only if (4.14) is fulfilled.

Now, suppose that (4.15) is satisfied. Then we can find $y^{*}\in K^{*}$ such that $M^{*}y^{*}\in\left(0^{+}D\right)^{*}$ and $M^{*}y^{*}\neq 0$. Since the later obviously implies that $y^{*}\neq 0$, we have $y^{*}\in M^{*}\left(K^{*}\setminus\{0\}\right)\cap\left(0^{+}D\right)^{*}$. Hence (4.14) holds true, so (VLP) has a solution. $\hfill\Box$

Remark 4.4

The assertions of Theorem 4.3 are valid for the case $K\subset Y$ is an arbitrary convex cone.**

We conclude this section by a statement about the structure of $E^{w}$ which is applicable also the case $K\subset Y$ is an arbitrary convex cone.

Theorem 4.5

The weakly efficient solution set $E^{w}$ of (VLP) is the union of finitely many generalized polyhedral convex sets.

Proof. Using Lemma 4.1, we can represent the weakly efficient solution set of (VLP) as follows

[TABLE]

Setting $x^{*}=M^{*}y^{*}$, we can rewrite (4.16) as

[TABLE]

where $S(x^{*})$ is the solution set of the problem (LP) considered in Section 3. Invoking (3.12) and noting that the number of the index sets $I(x^{*})$ (resp., the number of the index sets $J(x^{*})$) is finite, from (4.17) we obtain the desired conclusion. $\hfill\Box$