A Variational Principle and Applications for Vectorial Control Approximation Problems

Christiane Tammer

Abstract

An Ekeland-type variational principle for vector optimization problems is shown using a nonconvex separation theorem. This variational principle is a sharper assertion as in Tammer [36]. Furthermore, some applications for vectorial control approximation problems will be studied.

1 Introduction

Ekeland’s variational principle [14] is a very deep assertion about the existence of an exact solution of a slightly perturbed optimization problem in a neighbourhood of an approximate solution of the original problem. Many authors have published extensions and applications of Ekeland’s variational principle as well as equivalent statements (cf. Brezis and Browder [6], Brondsted [7], Rockafellar [32], Penot [30], Daneš [9], Borwein and Preiss [5], Rolewicz [33], Georgiev [15], Figuereido [10], Takahashi [35], Phelps [31]). The results of Bishop and Phelps [3] from 1962 concerning supporting points of (convex) sets can also be seen as a first contribution to the topic of Ekeland’s variational principle, given in the form of a maximal point theorem.

Extensions of Ekeland’s variational principle to vector optimization were given by several authors. Loridan (1984) [27] has presented a vector-valued variational principle for the finite-dimensional case using a scalarization and Ekeland’s original result. Further, Nemeth (1986) [29], Khanh (1986) [25], Tammer (1992) [36], Dentscheva and Helbig (1994) [11] and Isac (1994) [22] have derived vector-valued variational principles for an objective function, which take its values in general spaces.

In [19] we have given a direct proof of a maximal point theorem in a product space X × Y of two Banach spaces X and Y, which can be considered as an extension of Phelps result [31]. Moreover, from this general maximal point theorem we have derived in [19] a variational principle for vector optimization problems without any scalarization.

The aim of this paper is to show a vector-valued variational principle for a Banach space X and (more general as in [19]) for a topological space Y using a suitable separation theorem for a nonconvex set and a convex cone and a result of Phelps [31]. In the first part of this paper an Ekeland - type variational principle is shown, which is sharper than the variational principle in Tammer [36]. Moreover, some applications for approximation problems will be studied. Let us assume:

(V1): (X, || . ||_{X}) is a Banach space, Y is a linear topological space, K ⊂ Y is a cone with k^{0} ∈ int K , B ⊂ Y is a convex cone with int B ≠ ∅ such that cl B + (K \ {0}) ⊂ int B.

(V2): f : X ⟶ Y is lower semicontinuous with respect to k^{0} and B in the sense that M_{r} = { x ∈ X | f(x) ∈ rk^{0} − cl B } is closed for each r ∈ R and bounded from below, i. e. f[X] ⊂ y + B for a certain y ∈ Y.

In the following we denote the topological interior of a set C by int C, the topological boundary of C by bd C and the topological closure of C by cl C. If K ⊂ Y is a cone then K^{∗} := {y^{∗} ∈ Y^{∗} | y^{∗}(x) ≥ 0 ∀y ∈ K} is called the dual cone for K. Moreover, the set q − int K^{∗} := {y^{∗} ∈ Y^{∗} | y^{∗}(y) > 0 ∀y ∈ K \ {0}} is called the quasi-interior of the dual cone for K, where Y^{∗} is the dual space of linear continuous functionals defined on Y.

Definition 1 Let D be a nonempty subset of the linear topological space Y and y^{1}, y^{2} ∈ Y. A functional z : Y ⟶ R is called D-monotone, if y^{1} ∈ y^{2} + D implies z(y^{1}) ≥ z(y^{2}). A D- monotone functional z z is called strictly D-monotone, if y^{1} ∈ y^{2} + (D \ {0}) implies z(y^{1}) > z(y^{2}).

Now, we consider the following vector optimization problem to determine the efficient point set of f[X] with respect to B :

(P) : Evaluate the set Eff(f[X], B),

where

Eff(f[X], B) = { f(x̄) | x̄ ∈ X and f[X] ∩ (f(x̄) − (B \ {0})) = ∅ }.

It is well known, that the efficient point set may be empty in the general noncompact case (see [18], [23], [28]). In order to derive also in this general case certain existence results we introduce in the following definition the set of approximately efficient elements (compare [34], [37], [42]).

Definition 2 An element f(x_{ϵ}) ∈ f[X] is called an approximately efficient point of f[X] with respect to B, k^{0} ∈ int K and ϵ > 0, if

f[X] ∩ (f(x_{ϵ}) − ϵk^{0} − (B \ {0})) = ∅.  (1)

The approximately efficient point set of f[X] with respect to B, k^{0} and ϵ is denoted by Eff(f[X], B_{ϵk^{0}}), where B_{ϵk^{0}} := ϵk^{0} + B .

In order to prove the announced vectorial variational principle we will apply the following separation theorem (compare Gerth (Tammer) and Weidner [17]):

Theorem 1 [17] Assume that:

(i) Y is a linear topological space; (ii) C ⊂ Y is a convex cone with nonempty interior; (iii) A ⊂ Y is a nonempty subset of Y. Then, A ∩ −int C = ∅ if and only if there exists a continuous sublinear functional z : Y ⟶ R which is strictly (int C) - monotone with the range (−∞, +∞) and

z(A) ≥ 0, z(int A) > 0, z(−C) ≤ 0, z(−bd C) = 0,
z(−int C) < 0, z(C) ≥ 0, z(int C) > 0.

The separating functional z := Y ⟶ R constructed in the proof of Theorem 1 by

z(y) := inf{t ∈ R | y ∈ tk^{0} − cl C}  (2)

has the following properties (cf. [17]):

z(y) < r ⟺ y ∈ −int C + rk^{0},  (3)
z(y) ≤ r ⟺ y ∈ −cl C + rk^{0},  (4)
z(y) = r ⟺ y ∈ −bd C + rk^{0},  (5)
z(y) ≥ r ⟺ y ∉ −int C + rk^{0},  (6)
z(y) > r ⟺ y ∉ −cl C + rk^{0}.  (7)

Furthermore, we need some properties in connection with applications of the functional z in (2), which were shown in [36]:

Lemma 1 [36] Assume (V1). If f : X ⟶ Y is lower semicontinuous with respect to k^{0} and B and z : Y ⟶ R is a functional according to (2) with C = B, then z ∘ f := z(f(x)) is lower semicontinuous for each x ∈ X.

Lemma 2 [36] Assume (V1) and α > 0. Then the functional z : Y ⟶ R according to (2) with C = B has the following properties:

z(0) = 0,  z(k^{0}) = 1,  z(αk^{0}) = α,  z(−αk^{0}) = −z(αk^{0}) = −α.

In order to prove a vector-valued variational principle we will use the following variational principle of Ekeland [14] which is a direct consequence of Phelps’ maximal point theorem [31] :

Theorem 2 ([14], [31]) Suppose that X is a Banach space and f : X ⟶ R̄ is a proper lower semicontinuous real-valued function, which is bounded from below, ϵ > 0 and ξ ∈ X with f(ξ) ≤ inf f +ϵ. Then for any λ > 0 there exists an x^{o} ∈ dom(f) such that

f(x^{o}) ≤ f(ξ) − λ || x^{o} − ξ ||_{X},  ,  || x^{o} − ξ ||_{X}≤ ϵ/λ,
f(x) + λ || x − x^{o} ||_{X}> f(x^{o}) ∀x ≠ x^{o}.

2 An Ekeland-type variational princple for vector optimization problems

Beginning with the paper of Loridan [27] various authors proved variational principles of Ekeland’s type for ϵ-efficient solutions of multicriteria optimization problems by scalarization. Tammer [36] has proved a vector-valued variational principle by using a separation theorem for nonconvex sets and Ekeland’s original result. Here we will show a sharper result as in [36].

Theorem 3 Assume (V1) and (V2). Then for any ϵ > 0, λ > 0 and any f(x_{0}) ∈ Eff(f[X], B_{ϵk^{0}}) there exists an element x_{ϵ} ∈ X such that

α) f(x_{ϵ}) ∈ f(x_{0}) − λ || x_{0} − x_{ϵ} ||_{X} k^{0} − cl B , f(x_{ϵ}) ∈ Eff(f[X], D_{ϵk^{0}}), where D is an open subset of Y with K \ {0} ⊂ D, 0 ∈ bd D and cl D + (K \ {0}) ⊂ D,

β) || x_{0} − x_{ϵ} ||_{X}≤ ϵ/λ,

γ) f_{λk^{0}}(x_{ϵ}) ∈ Eff(f_{λk^{0}}[X], K) ,

are fulfilled, where f_{λk^{0}}(x) := f(x) + λ || x − x_{ϵ} ||_{X} k^{0}.

Proof: Consider an element x_{0} ∈ X with

f[X] ∩ (f(x_{0}) − ϵk^{0} − (B \ {0})) = ∅.

If we put

f̃(x) := f(x) − f(x_{0})

we get

f̃[X] ∩ (f̃(x_{0}) − ϵk^{0} − (B \ {0})) = ∅

and

(f̃[X] + ϵk^{0}) ∩ (f̃(x_{0}) − (B \ {0})) = ∅.

So the assumptions of Theorem 1 are fulfilled for

A = f̃[X] + ϵk^{0} and C = f̃(x_{0}) − (B \ {0}).

Therefore, we get from Theorem 1 the existence of a continuous sublinear strictly Kmonotone functional z : Y ⟶ R according to (2) with C = B and

z(f̃(x_{0})) ≤ z(f̃(x) + ϵk^{0}) ≤ z(f̃(x)) + z(ϵk^{0}) for all x ∈ X.

From Lemma 2 we obtain

z(ϵk^{0}) = ϵ

and so

z(f̃(x_{0})) ≤ inf_{x∈X}z(f̃(x)) + ϵ, ϵ > 0.

Under the given assumptions z ∘ f̃ is a lower semicontinuous functional (see Lemma 1) which is bounded from below. Applying Theorem 2 we get for any ϵ > 0 and λ > 0 that there exists an element x_{ϵ} ∈ X such that

(α’) z(f̃(x_{ϵ})) ≤ z(f̃(x_{0})) − λ || x_{0} − x_{ϵ} ||_{X},

(β’) || x_{0} − x_{ϵ} ||_{X}≤ ϵ/λ,

(γ’)  z(f̃(x_{ϵ})) < z(f̃(x)) + λ || x − x_{ϵ} ||_{X}  ∀x ≠ x_{ϵ} ∈ X,

From (α′) we get

z(f̃(x_{ϵ})) ≤ z(f̃(x_{0})) − λ || x_{0} − x_{ϵ} ||_{X} .  (8)

Because of Theorem 1 we have for f̃(x_{0}) = 0 that z(f̃(x_{0})) = 0 and so (8) implies

z(f̃(x_{ϵ})) ≤ − λ || x_{0} − x_{ϵ} ||_{X} .

This yields together with (4)

f̃(x_{ϵ}) ∈ − cl B − λ || x_{0} − x_{ϵ} ||_{X} k^{0}

and

f(x_{ϵ}) ∈ f(x_{0}) − λ || x_{0} − x_{ϵ} ||_{X} k^{0} − cl B.

On the other hand, from (α′) we get

z(f̃(x_{ϵ})) ≤ z(f̃(x_{0})) − λ || x_{0} − x_{ϵ} ||_{X}≤ z(f̃(x_{0}))
≤ inf_{x∈X} z(f̃(x)) + ϵ ≤ z(f̃(x)) + z(ϵk^{0}) ∀x ∈ X.  (9)

Now, let us define a set

D := {y ∈ Y | z(−y + f̃(x_{ϵ}) − ϵk^{0}) < z(f̃(x_{ϵ}) − ϵk^{0})}  (10)

and a functional z̄: Y ⟶ R with

z̄(y) := z(y + f̃(x_{ϵ}) − ϵk^{0}).  (11)

Then, because of (11), (9), Lemma 2 and the sublinearity of z we get

z̄(f̃[X] + ϵk^{0} − f̃(x_{ϵ})) = z(f̃[X]) ≥ z(f̃(x_{ϵ})) + z(−ϵk^{0}) ≥ z(f̃(x_{ϵ}) − ϵk^{0}) = z̄(0) (12)

Moreover, from (10) and (11) we can conclude that

z̄(−D) = z(−D + f̃(x_{ϵ}) − ϵk^{0}) < z(f̃(x_{ϵ}) − ϵk^{0}) = z̄(0).

Together with (12) this yields

(f̃[X] + ϵk^{0} − f̃(x_{ϵ})) ∩ (−D) = ∅

and

f[X] ∩ (f(x_{ϵ}) − ϵk^{0} − D) = ∅.

The strict K-monotonicity of the functional z implies K \ {0} ⊂ D. Since z is continuous, the functional z̄ is also continuous and so D is open. Furthermore, in order to show cl D + (K \ {0}) ⊂ D we choose y^{1} ∈ cl D and y^{2} ∈ y^{1} + (K \ {0}). Then, because of the strict K-monotonicity of z̄ we have

z̄(−y^{2}) < z̄(−y^{1}) ≤ z̄(0)

and

y^{1} ∈ cl D ⊂ {y | z̄(−y) ≤ z̄(0)}.

This means that y^{2} ∈ cl D + (K \ {0}) implies y^{2} ∈ D. From k^{0} ∈ int K and cl D + (K \ {0}) ⊂ D it follows ϵk^{0} + D ⊂ D. 0 ∈ cl (K \ {0}), K \ {0} ⊂ D and 0 ∉ D result in 0 ∈ cl D \ D. So we obtain assertion α).

Further, β) coincides with (β′).

In order to prove (γ) we suppose that there exists an element x ∈ X such that

f(x) + λ || x − x_{ϵ} ||_{X} k^{0} ∈ f(x_{ϵ}) − (K \ {0}).

Setting f̃(x) = f(x) − f(x_{0}) we get f̃(x) + λ || x − x_{ϵ} ||_{X} k^{0} ∈ f̃(x_{ϵ}) − (K \ {0}). Then we have f̃(x) ∈ f̃(x_{ϵ}) − λ || x − x_{ϵ} ||_{X} k^{0} − (K \ {0}). For the strictly K - monotone and sublinear functional z according to (2) with C = B it follows

z(f̃(x)) < z(f̃(x_{ϵ}) − λ || x − x_{ϵ} ||_{X} k^{0}) ≤ z(f̃(x_{ϵ})) + z(−λ || x − x_{ϵ} ||_{X} k^{0}).

Regarding Lemma 2 we get with z(−λ || x − x_{ϵ} ||_{X} k^{0}) = −λ || x − x_{ϵ} ||_{X} that

z(f̃(x)) + λ || x − x_{ϵ} ||_{X}≤ z(f̃(x_{ϵ})),

which is a contradiction to (γ′). So assertion (γ) is proved.

Remark:

1. From α) in Theorem 3 it follows that f(x_{ϵ}) is contained in Eff(f[X], K_{ϵk^{0}}).

2. For the special case Y = R, K is the set of nonegative real numbers, B is the set of positive real numbers and k^{0} = 1 Theorem 3 coincides with Ekeland’s original result (Theorem 2).

3. Assertion α) in Theorem 3 is sharper than the corresponding assertion in the variational principle for vector optimization problems in Tammer [36].

3 Applications for Vector Approximation Problems

Location and approximation problems have been studied by many authors from the theoretical as well as the computational point of view (Kuhn [26], Durier and Michelot [13], Idrissi and Lefebvre and Michelot [20], Idrissi and Loridan and Michelot [21] and many others). Such problems play an important role in optimization theory and many practical problems can be described as location or approximation problem. Beside problems with one objective function several authors investigated even vector-valued location and approximation problems. In some papers they presented theoretical results (duality assertions- Gerth and Pöhler [16], Tammer and Tammer [41], Wanka [43]; necessary optimality conditions- Durier [12], [44]) and developed algorithms (Wendell and Hurter and Lowe [45], Chalmet and Francis and Kolen [8], Gerth and Pöhler [16]).

3.1 A general vectorial control approximation problem

In this section we will consider a general vectorial control approximation problem and derive necessary conditions for approximate solutions of this problem. In the whole section we assume additionally to (V1) that (X, || . ||_{X}), (Y, || . ||_{Y}) and (Z, || . ||_{Z}) are real reflexive Banach spaces. Moreover, we suppose that K ⊂ Y is a pointed closed convex cone with int K ≠ ∅ having the Daniell - property, which means that every decreasing net (i.e. i ≤ j implies x_{j} ≤ x_{i}) having a lower bound, converges to its infimum (see Jahn [23]). Further, we suppose that K has a weakly compact base. In order to formulate our vectorial control approximation problem, we will introduce a vector-valued norm (compare Jahn [23]): ||| . |||: Z ⟶ K is called a vector-valued norm if ∀ z, z_{1}, z_{2} ∈ Z, ∀ λ ∈ R it holds:

1.  ||| z |||= 0 ⟺ z = 0,
2.  ||| λ z ||| = | λ | ||| z |||,
3. ||| z_{1} + z_{2} ||| ∈ ||| z_{1} ||| + ||| z_{2} ||| − K.

We assume in the following that ||| . ||| is continuous. The set of linear continuous mappings from X to Y is denoted by L(X, Y). Suppose that A ∈ L(X, Z). A^{∗} denotes the adjoint operator to A. Then we consider for x ∈ X and a ∈ Z the vector-valued function

f_{1}(x) :=||| A(x) − a ||| .

Now, we will introduce the following vectorial control approximation problem:

(P 1) : Evaluate the set Eff(f_{1}[X], K).

Remark: The problem (P1) contains the following practically important special cases:

1. Vector-valued optimal control problems of the form (see Benker and Kossert [2]):

(P_{1}) Eff(F_{1}[U], R_{+}^{2}),

with

F_{1}(u) := ( \begin{matrix}|| Au − a ||_{1} \\ || u ||_{2} \end{matrix} ) , u ∈ U ⊂ X,

where H_{1} and H_{2} are Hilbert spaces, A ∈ L(H_{1}, H_{2}), a ∈ H_{2}, U ⊂ H_{1} is a nonempty closed convex set and R_{+}^{2} denotes the usually ordering cone in R^{2}. Here u denotes the so called control variable, the image z = Au denotes the state variable.

2. Vectorial approximation and location problems (cf. Jahn [23], Gerth and Pöhler [16], Wanka [43] ).

Now, we will derive necessary conditions for approximately efficient elements of the problem (P1) by using the directional derivative of f_{1} at x ∈ X in the direction v ∈ X:

f′_{1}(x)(v) := lim_{t⟶+0}(f_{1}(x + tv) − f_{1}(x))/t.

Theorem 4 Under the assumptions of this section for any ϵ > 0, λ > 0 and any approximately efficient element

f_{1}(x^{o}) ∈ Eff(f_{1}[X], B_{ϵk^{o}})

there exists an element x_{ϵ} ∈ X with

f_{1}(x_{ϵ}) ∈ f_{1}(x^{o}) − λ || x^{o} − x_{ϵ} ||_{X} k^{o} − cl B, f_{1}(x_{ϵ}) ∈ Eff(f_{1}[X], K_{ϵk^{o}}),
|| x_{ϵ} − x^{o} ||_{X}≤ ϵ/λ,

such that for any v ∈ X with || v ||_{X}= 1 there is a linear continuous mapping M_{ϵ} ∈ L, where

L := {M ∈ L(Z, Y) | M(A(x_{ϵ})−a) = ||| A(x_{ϵ})−a ||| and ||| z ||| − M(z) ∈ K ∀ z ∈ Z},

with

M_{ϵ}A(v) ∈ {MA(v)} + K  for all  M ∈ L

and

M_{ϵ}A(v) ∉ −λk^{o} − int K.  (13)

Proof: The assertions in Theorem 3 imply the existence of an element x_{ϵ} ∈ X with

f_{1}(x_{ϵ}) ∈ f_{1}(x^{o}) − λ || x^{o} − x_{ϵ} ||_{X} k^{o} − cl B, f_{1}(x_{ϵ}) ∈ Eff(f_{1}[X], K_{ϵk^{o}}),
|| x_{ϵ} − x^{o} ||_{X}≤ ϵ/λ

and

f_{1_{λk^{o}}}(x_{ϵ}) ∈ Eff(f_{1_{λk^{o}}}[X], K).  (14)

Now, let be v ∈ X with || v ||_{X}= 1. If we put u = x_{ϵ} + tv, t > 0, then (14) implies

||| A(x_{ϵ} + tv) − a ||| +λ || (x_{ϵ} + tv) − x_{ϵ} ||_{X} k^{o} ∉ ||| A(x_{ϵ}) − a ||| −(K \ {0}).

This means that

||| A(x_{ϵ} + tv) − a ||| +λ t || v ||_{X} k^{o} ∉||| A(x_{ϵ}) − a ||| −(K \ {0}).

Using || v ||_{X}= 1 and the assumption that K is a cone we get

(||| A(x_{ϵ} + tv) − a ||| − ||| A(x_{ϵ}) − a |||)/t ∉ −λk^{o} − (K \ {0})

and

((||| A(x_{ϵ} + tv) − a ||| − ||| A(x_{ϵ}) − a |||)/t) ∉ −λk^{o} − (K \ {0}).

Under the given assumptions the directional derivative (||| . |||)′(x_{ϵ})(v) of the vectorvalued norm at x_{ϵ} exists (see Lemma 2.24 of Jahn [23] ). So we can conclude from the definition of the directional derivative and Corollary 5.4 (a) in Borwein [4] that

(||| A(x) − a |||)′(x_{ϵ})(v) ∉ −λk^{o} − int K  (15)

for all v ∈ X with || v ||_{X}= 1.

Further, under the assumption, that K is a closed convex cone in Y which has the Daniell property and a weakly compact base, it follows from Theorem 2.27 of Jahn [23] that for each v ∈ X there exists a linear continuous mapping M_{ϵ} ∈ L(Z, Y) with

M_{ϵ}(A(x_{ϵ}) − a) = ||| A(x_{ϵ}) − a |||,
||| z ||| −M_{ϵ}(z) ∈ K ∀z ∈ Z

and

M_{ϵ}A(v) ∈ {MA(v)} + K for all M ∈ L

such that, using a chain rule for operators (similar as given by Kantorowitsch / Akilow [24]),

(||| A(x) − a |||)′(x_{ϵ})(v) = M_{ϵ}A(v).  (16)

From (15) and (16) it follows that

M_{ϵ}A(v) ∉ −λk^{o} − int K.

Remarks:

1. Relation (13) in Theorem 4 can be considered as an ϵ - variational inequality.

2. Putting ϵ = 0 in condition (13) we get

M_{ϵ}A(v) ∉ −int K

for all v ∈ X with || v ||_{X}= 1. For the special case A = I this is a well-known necessary condition for efficient elements of (P1) (cf. Jahn [23] and Wanka [44]).

In the following theorem we will derive necessary conditions for approximately efficient solutions of Eff(f[X], B_{ϵk^{o}}) using the subdifferential calculus. Here we use the dual norm || . ||_{∗} defined by

|| x^{∗} ||_{∗} := sup_{||x||_{X}=1} | x^{∗}(x) | .

Applying the subdifferential calculus we derive a sharper result as in Theorem 4.

Definition 3 The set

∂^{≤}f(x_{0}) := { M ∈ L(X, Y) | M (x − x_{0}) ∈ f(x) − f(x_{0}) − K ∀x ∈ X }

is called the subdifferential of the convex function f : X ⟶ Y at x_{0} ∈ X.

Jahn [23] has shown the stucture of the subdifferential of the vector-valued norm ||| . |||: Z ⟶ Y:

∂^{≤} ||| z_{0} |||= {M ∈ L(Z, Y) | M(z_{0}) =||| z_{0} ||| and ||| z ||| −M(z) ∈ K for all z ∈ Z}.

Theorem 5 Under the assumptions of this section for any ϵ > 0, λ > 0 and any approximately efficient element f_{1}(x^{0}) ∈ Eff(f_{1}[X], B_{ϵk^{0}}) there exist an element x_{ϵ}, a functional z^{∗} ∈ q − int K^{∗} and a linear continuous mapping M_{ϵ} ∈ L(X, Y) with

M_{ϵ}(A(x_{ϵ}) − a) = ||| A(x_{ϵ}) − a |||,
||| z ||| −M_{ϵ}(z) ∈ K ∀z ∈ Z,

such that

(α)  f_{1}(x_{ϵ}) ∈ f_{1}(x^{o}) − λ || x^{o} − x_{ϵ} ||_{X} k^{0} − cl B,  f_{1}(x_{ϵ}) ∈ Eff(f_{1}[S], K_{ϵk^{o}}),

(β) || x_{ϵ} − x^{o} ||_{X}≤ ϵ/λ,

(γ) ∥A^{∗}(z^{∗}M_{ϵ})∥_{∗} ≤ λz^{∗}(k^{o}).

Proof: We assume f(x^{0}) ∈ Eff(f_{1}[X], B_{ϵk^{0}}). Under the given assumptions Theorem 3 implies the existence of an element x_{ϵ} with α′) f_{1}(x_{ϵ}) ∈ f_{1}(x^{0}) − λ || x^{0} − x_{ϵ} ||_{X} k^{0} − cl B , f_{1}(x_{ϵ}) ∈ Eff(f_{1}[X], D_{ϵk^{0}}),

where D is an open subset of Y with K \ {0} ⊂ D, 0 ∈ bd D and cl D + (K \ {0}) ⊂ D,

β′) || x^{0} − x_{ϵ} ||_{X}≤ ϵ/λ,

γ′) f_{1_{λk^{0}}}(x_{ϵ}) ∈ Eff(f_{1_{λk^{0}}}[X], K) ,

are fulfilled, where f_{1_{λk^{0}}}(x) := f_{1}(x) + λ || x − x_{ϵ} ||_{X} k^{0}.

From assertion γ′) it is possible to conclude that there exists a functional z^{∗} ∈ K^{∗}\{0} with

z^{∗}(f_{1}(x_{ϵ})) ≤ inf z^{∗}(f_{1}(x)) + λ || x − x_{ϵ} ||_{X} z^{∗}(k^{0}).

According to the subdifferential calculus of convex functionals (cf. Aubin and Ekeland [1] , Corollary 4.3.6) there is a linear continuous functional l_{ϵ} : X ⟶ R belonging to the subdifferential of the scalarized function z^{∗} ∘ f_{1} at the point x_{ϵ} ∈ X with

∥l_{ϵ}∥_{∗} ≤ λ z^{∗}(k^{0}).

Using the subdifferential calculus (see Theorem 4.3.5 of Aubin and Ekeland [1]) and Corollary 6.4 (a) of Borwein [4] we get the existence of a linear continuous operator M_{ϵ} with

M_{ϵ}(A(x_{ϵ}) − a) = ||| A(x_{ϵ}) − a |||,
||| z ||| −M_{ϵ}(z) ∈ K ∀z ∈ Z

and ∥A^{∗}(z^{∗}M_{ϵ})∥_{∗} ≤ λ z^{∗}(k^{0}).

3.2 A special vector approximation problem

In section 3.1 we have introduced the vector optimization problem (P1) with

f_{1}(x) :=||| A(x) − a ||| .

Let us assume now that x ∈ R^{n}, a^{i} ∈ R^{m_{i}} ∀i = 1, ..., p, A_{i} are matrices of the type (m_{i}, n) and || . ||_{i} are norms in R^{m_{i}} ∀i = 1, ..., p. Suppose that K = R_{+}^{p}, k^{0} ∈ int R_{+}^{p} and B ⊂ R^{p} is an open convex cone with cl B + (K \ {0}) ⊂ int B and k^{0} ∈ int R_{+}^{p}.

We will study a special case of a vector-valued norm given by

||| A(x) − a |||:= ( \begin{matrix}|| A_{1}(x) − a^{1} ||_{1} \\ || A_{2}(x) − a^{2} ||_{2} \\ ⋯ \\ || A_{p}(x) − a^{p} ||_{p} \end{matrix} ) .

Under the assumptions given above we formulate the following special case of problem (P1):

(P 2) : Eff(f_{2}[X], K)

with

f_{2}(x) := ( \begin{matrix}|| A_{1}(x) − a^{1} ||_{1} \\ || A_{2}(x) − a^{2} ||_{2} \\ ⋯ \\ || A_{p}(x) − a^{p} ||_{p} \end{matrix} )

The following corollary is a direct consequence of Theorems 4 and 5 if we use the directional derivative and the subdifferential of the p norms || A_{i}x − a^{i} ||_{i}. So we derive necessary conditions for approximately efficient solutions of the problem (P2):

Corollary 1 Under the assumptions given above for any ϵ > 0, λ > 0 and any approximately efficient element

f_{2}(x^{o}) ∈ Eff(f_{2}[X], B_{ϵk^{o}})

there exists an element x_{ϵ} ∈ R^{n} with

f_{2}(x_{ϵ}) ∈ f_{2}(x^{o}) − λ || x^{o} − x_{ϵ} ||_{R^{n}} k^{0} − cl B, f_{2}(x_{ϵ}) ∈ Eff(f_{2}[X], K_{ϵk^{o}}),
|| x_{ϵ} − x^{o} ||_{R^{n}}≤ ϵ/λ,

such that for any v ∈ X with || v ||_{X}= 1 there is a linear continuous mapping l_{ϵ} ∈ L_{1}, where

L_{1}  :=  {  l = (l_{1}, ..., l_{p}), l_{i} ∈ L(R^{m_{i}}, R) | || l_{i} ||_{i^{∗}}= 1,
l_{i}(A_{i}(x_{ϵ}) − a^{i}) = || A_{i}(x_{ϵ}) − a^{i} ||_{i} ∀i = 1, ..., p },

with

( \begin{matrix}l_{ϵ1}A_{1}(v) \\ ⋯ \\ l_{ϵp}A_{p}(v) \end{matrix} ) ∈ \{( \begin{matrix}l_{1}A_{1}(v) \\ ⋯ \\ l_{p}A_{p}(v) \end{matrix} )\} + K  for all  l = (l_{1}, ..., l_{p}) ∈ L_{1}

and

( \begin{matrix}(A_{1}^{∗} l_{ϵ1})(v) \\ (A_{2}^{∗} l_{ϵ2})(v) \\ ⋯ \\ (A_{p}^{∗} l_{ϵp})(v) \end{matrix} ) ∉ −λk^{o} − int K,

respectively

|| ∑_{i=1}^{p} z_{i}^{∗}(A_{i}^{∗}l_{ϵi}) ||_{∗}≤ λz^{∗}(k^{o})  for a  z^{∗} ∈ K^{∗} \ {0}.  (17)

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