Approximate solutions of vector-valued control-approximation problems

Christiane Tammer

Martin-Luther-University Halle-Wittenberg, Dept. of Mathematics and Informatics, D-06099 Halle, Germany

Abstract: We consider a class of vector-valued control-approximation problems in which each objective function is a sum of two terms, a linear function and a power of a norm of a linear vector function. Necessary conditions for approximate solutions of these problems were derived using a vector-valued variational principle, the subdifferential calculus and the directional derivative, respectively.

1 Introduction

Several authors have studied scalar and vectorial location and approximation problems from the theoretical as well as the computational point of view (e.g. [4], [5], [6], [7], [11], [14], [15], [18], [19], [21], [23], [29]). In [12] and [13] we have derived necessary conditions for approximately efficient elements of a class of abstract approximation problems in which the objective function takes its values in a reflexive Banach space. However, these conditions are not easy to utilize such that for more special problems it would be worth to find necessary conditions which are better to handle. The aim of this paper is to derive necessary conditions for approximately efficient solutions for another class of approximation problems having a finite dimensional image space of the objective function, which is more usefull for control problems in comparison with the approximation problem in [12] and [13]. In our proofs it is possible to use the special structure of the subdifferential of a power of the norm and a vector-valued variational principle given in [10].

The paper is organized as follows: In Chapter 1.1 we introduce the vectorvalued control-approximation problem (P) and summarize some practically important special cases of (P). Notations and preliminary results, especially the subdifferential calculus for convex vector-valued functions and a variational principle for vector optimization problems are presented in Chapter 1.2. Using this variational principle and the subdifferential calculus we derive in Chapter 2 necessary conditions for approximately efficient elements of the vector-valued control-approximation problem (P). Finally, in Chapter 3 we show necessary conditions for approximately efficient elements of the vector-valued controlapproximation problem (P) using the directional derivative of the objective function. For special cases of (P), for instance for the case of a real-valued control-approximation problems, ϵ-variational inequalities will be presented.

1.1 Formulation of the problem

In our paper we suppose that

(A) (X, ||.||_{X}) and (Y_{i}, ||.||_{i}) (i = 1, ..., n) are real Banach spaces, x ∈ X, a^{i} ∈ Y_{i}, α_{i} ≥ 0 , β_{i} ≥ 1, A_{i} ∈ L (X, Y_{i}), (i = 1, ..., n), C ∈ L(X, R^{n}), (L(X, Y_{i}) denotes the space of linear continuous operators from X to Y_{i}),

(B) D_{j} ⊆ X ( j = 1, ..., m) are closed and convex sets, and D = ⋂_{j=1}^{m} D_{j} is

nonempty, (C) K ⊂ R^{n} is a pointed closed convex cone with nonempty interior and

K + R_{+}^{n} ⊆ K.

In the following A_{i}^{∗} denotes the adjoint operator to A_{i} (i = 1, ..., n), and

K^{∗} := {z^{∗} ∈ R^{n} | z^{∗}(k) ≥ 0 for all k ∈ K}

the dual cone to K. Moreover, ||.||_{∗} denotes the dual norm to ||.||_{X} and ||.||_{i^{∗}} the dual norm to ||.||_{i}. Let us recall that the dual norm || . ||_{∗} to ||.||_{X} is defined by

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

Now we consider the following vectorial control-approximation problem

(P) :  Compute the set  Eff(f[D], K),

where

f(x) := C(x) + ( \begin{matrix}α_{1} || A_{1}(x) − a^{1} ||_{1}^{β_{1}} \\ ⋯ \\ α_{n} || A_{n}(x) − a^{n} ||_{n}^{β_{n}} \end{matrix} )  (1)

is the objective vector function and

Eff(F, K) := {z ∈ F | F ∩ (z − (K \ {0})) = ∅}

is the set of efficient elements of a set F with respect to K. In the objective function f in (1) the norms in its components may be different. Additionally, we assume that

(D) f : X ⟶ R^{n} is bounded from below on D, i.e., there exists some z ∈ R^{n} with f[D] ⊂ z + K.

It is well-known, that the set Eff(f[D], K) may be empty in the general noncompact case (see [16]). 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 [22], [26], [28]).

Definition 1. An element f(x_{ϵ}) with x_{ϵ} ∈ D is called an approximately efficient point of f[D] with respect to K, k^{0} ∈ int K and ϵ > 0, if

f[D] ∩ (f(x_{ϵ}) − ϵk^{0} − (K \ {0})) = ∅.

The set of approximately efficient points of f[D] with respect to K, k^{0} and ϵ is denoted by Eff(f[D], K_{ϵk^{0}}), where K_{ϵk^{0}} := ϵk^{0} + K .

Remark. Problem (P) can be considered as an extension of a scalar problem studied by Idrissi, Lefebvre and Michelot [14]:

∑_{i=1}^{n} α_{i} || x − a^{i} ||_{i}⟶ min_{x∈D}

with D_{j} ⊂ R^{p} (j = 1, ..., m), x, a^{i} ∈ R^{p} (i = 1, ..., n), where the minimization in the later problem corresponds to the task to compute the set of efficient points in the former problem (P) if the objective function is real-valued.

Remark. The approximation problem (P) with β_{1} = ... = β_{n} = 1 is a special case of the abstract approximation problem (P 1) in Henkel and Tammer [13]. The case of different values β_{i} ≥ 1 is of interest in control theory, especially linear regulator problems (cf. Benker and Kossert [3] and Barbu [2]). In comparison with [13] we only use the weaker assumption that X and Y_{i} are real Banach spaces.

Remark. The class of problems (P) contains many practically important special cases:

1. Linear vector optimization problems (α_{i} = 0 for all i = 1, ..., n and D is a convex polyhedron).

2. Surrogate problems

Eff(f_{1}[D], K)

with

f_{1}(x) := C(x) + ( \begin{matrix}α_{1} || A_{1}(x) − a^{1} ||_{1} \\ ⋯ \\ α_{n} || A_{n}(x) − a^{n} ||_{n} \end{matrix} )

for vector optimization problems

Eff(φ[D], K),

where

φ(x) := C(x)

and

D = {x ∈ D | A_{i}(x) = a^{i}, i = 1, ..., n},

for which the feasible set D is empty (see Tammer, K. and Tammer, Chr. [23]). Here the set D represents the ”hard” restrictions (which must be fulfilled by all means), whereas the additional terms ||A_{i}(x) − a^{i}||_{i} in the objective function should ensure that the remaining restrictions A_{i}(x) = a^{i}, i = 1, ..., n, will be fulfilled as good as possible.

3. Perturbed respectively regularized linear vector optimization problems (compare Jahn [16]).

4. Vector-valued optimal control problems of the form (cf. Benker and Kossert

[3]):

Compute the set Eff(ϕ[U], R_{+}^{2}),

with

ϕ(u) := ( \begin{matrix}|| Au − a ||^{p} \\ || u ||^{p} \end{matrix} ) , u ∈ U ⊂ X,

where C = 0, n = 2, β_{i} = p ≥ 1 (i = 1, 2), 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 usual ordering cone in R^{2}. Here u denotes the so-called control variable, the image z := Au denotes the state variable.

5. Vectorial approximation and location problems (C = 0, cf. Jahn [16], Gerth and Pöhler [8], Wanka [29], [30], Oettli [20]). Vectorial location problems and corresponding algorithms are important especially in town planning (cf. Kuhpfahl, Patz and Tammer [17]).

1.2 Notations and preliminary results

We will derive necessary conditions for approximately efficient elements using a vector-valued variational principle and the subdifferential calculus. In order to apply the subdifferential calculus we have to show that the objective function in (P) is K-convex. At first, we introduce the following definition of K − convexity (cf. Jahn [15]).

Definition 2. A function f : X ⟶ R^{n} is called K-convex on X if for any x_{1}, x_{2} ∈ X, t ∈ [0, 1] it holds

f(tx_{1} + (1 − t)x_{2}) ∈ tf(x_{1}) + (1 − t)f(x_{2}) − K.

Lemma 3. Under the assumptions (A), (B) and (C) the objective function f in (1) is K-convex.

Proof: For β_{i} ≥ 1, i = 1, ..., n, each component

f_{i}(x) = c_{i}(x) + α_{i} || A_{i}(x) − a^{i} ||_{i}^{β_{i}}

of the vector-valued objective function f is convex. So we get for any x_{1}, x_{2} ∈ X, t ∈ [0, 1]:

f(tx_{1}+(1−t)x_{2}) ∈ tf(x_{1})+(1−t)f(x_{2})−R_{+}^{n} ⊂ tf(x_{1})+(1−t)f(x_{2})−K.

Lemma 4. Let F be a nonempty subset of a partially ordered reflexive Banach space Z with a pointed nontrivial ordering cone K. If the set F + K is convex and has a nonempty topological interior, then for each efficient element z̄ of F with respect to K there exists a linear functional z^{∗} ∈ K^{∗} \ {0} with the property

z^{∗}(z̄) ≤ z^{∗}(z)  for all  z ∈ F.

In section 2 we will derive necessary conditions for approximately efficient elements of (P) using the subdifferential calculus for vector-valued functions.

Definition 5. The set

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

is called the subdifferential of the K-convex function f : X ⟶ R^{n} at x_{0} ∈ X.

For the special case of a real-valued objective function f : X ⟶ R with K = R_{+} Definition 5 coincides with the well-known definition of the subdifferential of a real-valued function f at x_{0} ∈ X:

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

In order to prove our main result we need the following assertion of Aubin and Ekeland [1] concerning the subdifferential of norm terms:

Lemma 6. If X is a Banach space then we have:

∂ || x ||= \{ \begin{matrix}\{p ∈ L(X, R) | p(x) =|| x || , || p ||_{∗}= 1\}  if & x ≠ 0, \\ \{p ∈ L(X, R) | || p ||_{∗}≤ 1\}  if & x = 0. \end{matrix}

and for β > 1

∂(\frac{1}{β} || . ||^{β})(x) = \{p ∈ L(X, R) | || p ||_{∗}=|| x ||^{β−1}, p(x) =|| x ||^{β}\}.

In order to derive necessary conditions for approximately efficient elements of (P) we will use besides the subdifferential the directional derivative.

Definition 7. The directional derivative of the function f : X ⟶ R^{n} at x ∈ D in the direction v ∈ X is defined by

f′(x)(v) := lim_{t→+0} \frac{f(x + tv) − f(x)}{t}.

Furthermore, we will introduce a vector-valued variational principle due to Göpfert and Tammer [10] (cf. also Tammer [23]), which we will apply in the proof of our main result. The following Theorem 8 is an assertion about the existence of an efficient solution of a slightly perturbed vector optimization problem in a certain neighbourhood of an approximately efficient element of the original vector optimization problem. Without any compactness assumptions we derive an assertion for approximately efficient elements. Approximately efficient elements exist under very weak assumptions (cf. Tammer [25]), whereas the set of efficient points may be empty in the general noncompact case. Moreover, numerical algorithms usually generate only approximately efficient elements if we stop after a finite number of iterations. We say that f is lower semicontinuous with respect to k^{o} ∈ int K and K (for short: f is (k^{o}, K)-lower semicontinuous), if

M_{r}^{X} := {x ∈ X | f(x) ∈ rk^{o} − K}

is a closed set for each r ∈ R.

Theorem 8. X and Z are considered to be real Banach spaces and K ⊂ Z is a pointed, closed and convex cone with nonempty interior, D is a nonempty closed subset of X. Assume that f : X ⟶ Z is a (k^{0}, K) - lower semicontinuous function which is bounded from below on D. Then for any ϵ > 0 and any x^{0} ∈ D with f(x^{0}) ∈ Eff(f[D], K_{ϵk^{0}}) there exists an element x_{ϵ} ∈ D with the following properties:

1. f(x_{ϵ}) ∈ f(x^{0}) − \sqrt{ϵ} || x_{ϵ} − x^{0} ||_{X} k^{0} − K,

2. || x_{ϵ} − x^{0} ||_{X}≤ \sqrt{ϵ},

3. f_{ϵk^{0}}(x_{ϵ}) ∈ Eff(f_{ϵk^{0}}[D], K),

where f_{ϵk^{0}}(x) := f(x) + \sqrt{ϵ} || x − x_{ϵ} ||_{X} k^{0}.

The main result in Theorem 8 is Condition 3, which implies necessary conditions for approximately efficient elements. Such necessary conditions can be used in order to derive numerical algorithms. In the following theorem we have derived necessary conditions using the directional derivative of f.

Theorem 9. Let be satisfied the assumptions of Theorem 8 and suppose, additionally, that the directional derivative f′(x)(v) exists for all x ∈ X and for all directions v ∈ X. Then for any ϵ > 0 and any x^{0} ∈ D with f(x^{0}) ∈ Eff (f[D], K_{ϵk^{0}}) there exists an element x_{ϵ} ∈ D with

f(x_{ϵ}) ∈ f(x^{0}) − \sqrt{ϵ} || x_{ϵ} − x^{0} ||_{X} k^{0} − K, || x_{ϵ} − x^{0} ||_{X}≤ \sqrt{ϵ}

and

f′(x_{ϵ})(v) ∉ − \sqrt{ϵ} k^{0} − int K  (2)

for any feasible direction v at x_{ϵ} with respect to D having || v ||_{X}= 1.

2 Necessary conditions using the subdifferential calculus

In this section under the assumptions (A), (B), (C) and (D) we consider the vector-valued approximation problem (P) introduced in Section 1. In the next theorem we will derive necessary conditions for approximately efficient solutions of Eff(f[X], K_{ϵk^{o}}) using the subdifferential calculus. At first we state necessary conditions for approximately efficient elements of the vector-valued approximation problem (P) with D = X.

Theorem 10. Under the assumptions (A), (C) and (D) for any ϵ > 0 and any x^{0} ∈ X with f(x^{0}) ∈ Eff(f[X], K_{ϵk^{0}}) there exist an element x_{ϵ} ∈ X, a functional z^{∗} ∈ K^{∗} \ {0} and linear continuous mappings M_{iϵ} ∈ L(Y_{i}, R) with

M_{iϵ}(A_{i}(x_{ϵ}) − a^{i}) =|| A_{i}(x_{ϵ}) − a^{i} ||_{i}^{β_{i}}

and

||M_{iϵ}||_{i^{∗}} \{ \begin{matrix}≤ 1 & if β_{i} = 1 and A_{i}(x_{ϵ}) = a^{i}, \\ = || A_{i}(x_{ϵ}) − a^{i} ||_{i}^{β_{i}−1} & otherwise, \end{matrix}

for all i = 1, ..., n, such that

(α) f(x_{ϵ}) ∈ f(x^{0}) − \sqrt{ϵ} || x^{o} − x_{ϵ} ||_{X} k^{0} − K,

(β) || x_{ϵ} − x^{0} ||_{X}≤ \sqrt{ϵ},

(γ) ∥z^{∗}C + ∑_{i=1}^{n} α_{i}β_{i}A_{i}^{∗}z_{i}^{∗}M_{iϵ}∥_{∗} ≤ \sqrt{ϵ}z^{∗}(k^{o}).

Proof: We assume f(x^{0}) ∈ Eff(f[X], K_{ϵk^{0}}). Under the given assumptions Theorem 8 implies the existence of an element x_{ϵ} ∈ X with

α′) f(x_{ϵ}) ∈ f(x^{0}) − \sqrt{ϵ} || x^{0} − x_{ϵ} ||_{X} k^{0} − K,

β′) || x^{0} − x_{ϵ} ||_{X}≤ \sqrt{ϵ},

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

where f_{ϵk^{0}}(x) := f(x) + \sqrt{ϵ} || x − x_{ϵ} ||_{X} k^{0}.

Because of the K-convexity of f (Lemma 3) and by assertion γ′) Lemma 4 implies that there exists a functional z^{∗} ∈ K^{∗} \ {0} with

z^{∗}(f(x_{ϵ})) ≤ inf_{x∈X} (z^{∗}(f(x)) + \sqrt{ϵ} || x − x_{ϵ} ||_{X} z^{∗}(k^{0})).  (3)

This means that x_{ϵ} minimizes the function

x ⟶ z^{∗}(f(x)) + \sqrt{ϵ} || x − x_{ϵ} ||_{X} z^{∗}(k^{0}),

and then from the subdifferential calculus of convex functionals (cf. Aubin and Ekeland [1], Corollary 4.3.6) we get

0 ∈ ∂(z^{∗} ∘ f)(x_{ϵ}) + \sqrt{ϵ}z^{∗}(k^{0})B,

where B is the closed unit ball in X. It follows immediately that there is a linear continuous functional l_{ϵ} : X ⟶ R belonging to the subdifferential of the scalarized function z^{∗} ∘ f at the point x_{ϵ} ∈ X with

∥l_{ϵ}∥_{∗} ≤ \sqrt{ϵ} z^{∗}(k^{0}).  (4)

Furthermore, we have

∂(z^{∗} ∘ f)(x_{ϵ}) = ∂(z^{∗}(C(·)) + ∑_{i=1}^{n} α_{i}z_{i}^{∗} || A_{i}(·) − a^{i} ||_{i}^{β_{i}} )(x_{ϵ}),

because z^{∗} is a linear functional. The rule of sums for subdifferentials yields the relation

\begin{aligned}∂(z^{∗}(C(·))+ & ∑_{i=1}^{n} α_{i}z_{i}^{∗} || A_{i}(·) − a^{i} ||_{i}^{β_{i}} )(x_{ϵ}) = \\ & z^{∗}∂^{≤}(C(·))(x_{ϵ}) + ∑_{i=1}^{n} α_{i}z_{i}^{∗}∂ || A_{i}(·) − a^{i} ||_{i}^{β_{i}} (x_{ϵ}). \end{aligned}  (5)

Applying Lemma 6 relation (5) implies

∂(z^{∗} ∘ f)(x_{ϵ}) = z^{∗}C + ∑_{i=1}^{n} α_{i}A_{i}^{∗}z_{i}^{∗}∂(|| u ||_{i}^{β_{i}}) |_{u=A_{i}(x_{ϵ})−a^{i}}=

{z^{∗}C + ∑_{i=1}^{n} α_{i}β_{i}A_{i}^{∗}z_{i}^{∗}M_{iϵ} | M_{iϵ} ∈ L(Y_{i}, R), M_{iϵ}(A_{i}(x_{ϵ}) − a^{i}) =|| A_{i}(x_{ϵ}) − a^{i} ||_{i}^{β_{i}},

||M_{iϵ}||_{i^{∗}} \{ \begin{matrix}≤ 1 & if β_{i} = 1 and A_{i}(x_{ϵ}) = a^{i}, \\ = || A_{i}(x_{ϵ}) − a^{i} ||_{i}^{β_{i}−1} & otherwise, \end{matrix} i = 1, ..., n\}.

(6)

From (4), (5) and (6) we get the desired inequality

∥z^{∗}C+∑_{i=1}^{n} α_{i}β_{i}A_{i}^{∗}z_{i}^{∗}M_{iϵ}∥_{∗} ≤ \sqrt{ϵ} z^{∗}(k^{0}).

In the next theorem we derive necessary conditions for approximately efficient elements of the vectorial approximation problem (P) with restrictions. In order to formulate necessary conditions we need the indicator functions χ_{D_{j}} of D_{j} defined by χ_{D_{j}}(x) = 0 if x ∈ D_{j} and χ_{D_{j}}(x) = +∞ otherwise. It is wellknown that the subdifferential of the indicator function χ_{D_{j}} at x^{0} ∈ X is the normal cone N_{D_{j}}(x^{0}) to D_{j} at x^{0} ∈ X, defined by

N_{D_{j}}(x^{0}) = \{ \begin{matrix}\{p ∈ L(X, R) : p(x′ − x^{0}) ≤ 0 ∀x′ ∈ D_{j}\} & if x^{0} ∈ D_{j} \\ ∅ & otherwise. \end{matrix}

Theorem 11. Under the assumptions (A), (B), (C) and (D) for any ϵ > 0 and any x^{0} ∈ D with f(x^{0}) ∈ Eff(f[D], K_{ϵk^{0}}) there exist an element x_{ϵ} ∈ D, a functional z^{∗} ∈ K^{∗} \ {0}, linear continuous mappings M_{iϵ} ∈ L(Y_{i}, R) with

M_{iϵ}(A_{i}(x_{ϵ}) − a^{i}) =|| A_{i}(x_{ϵ}) − a^{i} ||_{i}^{β_{i}},

||M_{iϵ}||_{i^{∗}} \{ \begin{matrix}≤ 1 & if β_{i} = 1 and A_{i}(x_{ϵ}) = a^{i}, \\ = || A_{i}(x_{ϵ}) − a^{i} ||_{i}^{β_{i}−1} & otherwise, \end{matrix}

for all i = 1, ..., n, and elements r_{j} ∈ N_{D_{j}}(x_{ϵ}) for all j = 1, ..., m, such that

(α) f(x_{ϵ}) ∈ f(x^{0}) − \sqrt{ϵ} || x^{o} − x_{ϵ} ||_{X} k^{0} − K,

(β) || x_{ϵ} − x^{0} ||_{X}≤ \sqrt{ϵ},

(γ) ∥z^{∗}C + ∑_{i=1}^{n} α_{i}β_{i}A_{i}^{∗}z_{i}^{∗}M_{iϵ} + ∑_{j=1}^{m} r_{j}∥_{∗} ≤ \sqrt{ϵ}z^{∗}(k^{o}).

Proof: We can follow the line of the proof of Theorem 10 taking into consideration that our problem has the feasible set D instead of X. Then we get instead of inequality (3) the inequality

z^{∗}(f(x_{ϵ})) ≤ inf_{x∈D} (z^{∗}(f(x)) + \sqrt{ϵ} || x − x_{ϵ} ||_{X} z^{∗}(k^{0})).

This problem is equivalent to the following unconstrained minimization problem

F (x) := (z^{∗}(f(x)) + \sqrt{ϵ} || x − x_{ϵ} ||_{X} z^{∗}(k^{0}) + ∑_{j=1}^{m} χ_{D_{j}}(x) ⟶ min_{x∈X} .

Taking into account the fact that the subdifferential of the indicator function χ_{D_{j}} at x_{ϵ} is the normal cone to D_{j} at x_{ϵ} we can conclude again in the same way as in the proof of Theorem 10.

3 Necessary conditions using the directional derivative

Under the assumptions (A), (B), (C) and (D) the vector-valued approximation problem (P) introduced in Section 1 with α_{i} = 1, β_{i} = 1 for all i = 1, ...n and K = R_{+}^{n}, namely

(P 1) :  Compute the set  Eff(f_{1}[D], K)

with

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

is a special case of problem (P).

The following lemma is a direct consequence of Theorem 9 if we use the directional derivative of the norms || A_{i}(x) − a^{i} ||_{i}, i = 1, ..., n (see Jahn [16], Theorem 2.27). So we derive necessary conditions for approximately efficient solutions of the problem (P 1). In order to formulate the next lemma we introduce a set of linear continuous mappings

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

Lemma 12. Under the assumptions given above, for any ϵ > 0 and any x^{0} ∈ D with

f_{1}(x^{0}) ∈ Eff(f_{1}[D], K_{ϵk^{o}}),

there exists some x_{ϵ} ∈ D with

f_{1}(x_{ϵ}) ∈ f_{1}(x^{o}) − \sqrt{ϵ} || x_{ϵ} − x^{o} ||_{X} k^{o} − K, || x_{ϵ} − x^{0} ||_{X}≤ \sqrt{ϵ},

such that for any feasible direction v at x_{ϵ} with respect to D having || v ||_{X}= 1 there is a linear continuous mapping l_{ϵ} ∈ L_{1}(x_{ϵ}), with

( \begin{matrix}l_{ϵ1}A_{1}(v) \\ ⋯ \\ l_{ϵn}A_{n}(v) \end{matrix} ) ∈ \{( \begin{matrix}l_{1}A_{1}(v) \\ ⋯ \\ l_{n}A_{n}(v) \end{matrix} )\} + K  for all  l = (l_{1}, ..., l_{n}) ∈ L_{1} (7)

and

C(v) + ( \begin{matrix}(A_{1}^{∗} l_{ϵ1})(v) \\ (A_{2}^{∗} l_{ϵ2})(v) \\ ⋯ \\ (A_{n}^{∗} l_{ϵn})(v) \end{matrix} ) ∉ −\sqrt{ϵ} k^{o} − int K.  (8)

In comparison with Theorem 11 the linear continuous mappings l ∈ L_{1}(x_{ϵ}) depend on the direction v ∈ V. Moreover, in Lemma 12 we have additionally the ”maximality property” (7). Finally, we will study the special case of a real-valued approximation problem. We suppose that X and Y are real Banach spaces, α_{i} = 1 for all i = 1, ..., n, k^{0} = 1, A ∈ L(X, Y), and a ∈ Y. In this case we study the real-valued objective function

f_{2}(x) = c(x)+ || A(x) − a ||

which is to minimize over D. According to Lemma 12 for K = R_{+}, k^{0} = 1 we get for K = R_{+}, k^{0} = 1 the following ϵ-Kolmogorov-condition:

max\{c(v)+A^{∗}l(v) | l ∈ L(Y, R), l(A(x_{ϵ})−a) =|| A(x_{ϵ})−a ||, || l ||_{∗}= 1\} ≥ −\sqrt{ϵ}.

(9)

Here the maximality property (7) in Lemma 12 implies that we have to maximize the left hand side of the inequality (9) with respect to l ∈ L(Y, R) satisfying

l(A(x_{ϵ}) − a) =|| A(x_{ϵ}) − a || and || l ||_{∗}= 1

and the ϵ-variational inequality (8) implies the right hand side of the inequality (9). If we suppose that A is the identical operator and c = 0, then we derive from (7) and (8) in Lemma 12 the following ϵ-variational inequality:

max\{ l(v) | l ∈ L(Y, R), l(x_{ϵ} − a) =|| x_{ϵ} − a ||, || l ||_{∗}= 1\} ≥ −\sqrt{ϵ}.

Furthermore, by putting ϵ = 0 in the last inequality we get a well-known necessary condition for solutions of the real-valued approximation problem (see Jahn [16]):

max{l(v) | l ∈ L(Y, R), l(x_{ϵ} − a) =|| x_{ϵ} − a ||, || l ||_{∗}= 1} ≥ 0.

But we recall that the assertion of Lemma 12 is only true for ϵ > 0, since for ϵ = 0 the existence of efficient solutions is not guaranteed.

References

1. Aubin, J. P. and Ekeland, I. (1984): Applied Nonlinear Analysis. John Wiley, New York.

2. Barbu, V. (1994): Mathematical methods in optimization of differential systems. Kluwer Academic Publisher, Dordrecht.

3. Benker, H. and Kossert, S. (1982): Remarks on quadratic optimal control problems in Hilbert spaces. Zeitschrift für Analysis und ihre Anwendungen 1 (3), 13-21.

4. Benker, H.; Hamel, A.; Tammer, Chr. (1994): A Proximal Point Algorithm for Control Approximation Problems. To appear: ZOR.

5. Durier, R.: On Pareto Optima, the Fermat - Weber Problem, and Polyhedral Gauges. Mathematical Programming 47 (1990), 65 - 79.

6. Durier, R. ; Michelot, C. (1986): Efficient points in a normed space. Journal of Mathematical Analysis and Applications 117, 506-528.

7. Francis, R.L. ; White, J.A. (1974): Facility Layout and Location: An Analytical Approach. Prentice-Hall, Englewood Cliffs, NJ.

8. Gerth (Tammer), Chr.; Pöhler, K. (1988): Dualität und algorithmische Anwendung beim vektoriellen Standortproblem. Optimization 19, 491-512.

9. Göpfert, A. (1973): Mathematische Optimierung in Allgemeinen Vektorräumen. BSB B.G. Teubner Verlagsgesellschaft, Leipzig.

10. Göpfert, A. and Tammer, Chr. (1995): A new maximal point theorem. Zeitschr. f. Analysis und ihre Anwendungen 14, 379-390.

11. Hansen, P.; Peeters, D.; Thisse (1983): Public facility location models: a selective survey. In: Thisse, J.F.; Zoller, H.G. (Eds.): Locational Analysis of Public Facilities. North-Holland, Amsterdam.

12. Henkel, E.-C., Tammer, Chr. (1996): ϵ-variational inequalities in partially ordered spaces. Optimization 36, 105-118.

13. Henkel, E.-C., Tammer, Chr. (1995): ϵ- inequalities for vector approximation problems. To appear: Optimization.

14. Idrissi, H.; Lefebvre, O.; Michelot, C.(1988): A Primal-Dual Algorithm for a constrained Fermat-Weber Problem Involving Mixed Norms. Recherche operationnelle / Operations Research 22 (4), 313-330.

15. Jahn, J. (1983): Zur vektoriellen linearen Tschebyscheff - Approximation. Math. Operationsforsch. Statist., Ser. Optim. 14, 577-591.

16. Jahn, J. (1986): Mathematical Vector Optimization in Partially Ordered Spaces. Lang Verlag Frankfurt, Bern, New York .

17. Kuhpfahl, I., Patz, R., Tammer, Chr. (1995): Location problems in town planning. In: D. Schweigert: Methods of Multicriteria Decision Theory. University Kaiserslautern, 101-112.

18. Love, R.F.; Morris, J.G.; Wesolowsky, G.O. (1988): Facilities location: models and methods. North-Holland, New York.

19. Michelot, C.; Lefebvre, O. (1987): A Primal-dual Algorithm for the Fermat-Weber Problem Involving Mixed Gauges. Mathematical Programming 39, 319-335.

20. Oettli, W. (1995): Kolmogorov conditions for vectorial optimization problems. OR Spektrum 17, 227-229.

21. Plastria, F. (1995): Continuous location problems. In: Drezner, Z. (Ed.): Facility location: a survey of applications and methods. Springer, New York.

22. Staib, T. (1988): On generalizations of Pareto minimality. Journ. Optimization Theory Applications 59, 289–306.

23. Tammer, Ch.; Tammer, K. (1991): Generalization and sharpening of some duality relations for a class of vector optimization problems. ZOR-Methods and Models of Operations Research (35), 249-265.

24. Tammer, Ch. (1993): Necessary conditions for approximately efficient solutions of vector approximation problems. Proceedings of the 2nd Conference on Approximation and Optimization, Havana 1993, (to appear).

25. Tammer, Chr. (1993): Existence results and necessary conditions for ϵ-efficient elements. In: Multicriteria Decision. (Eds.: B. Brosowski, J. Ester, S. Helbig and R. Nehse). Frankfurt am Main: Verlag P. Lang 1993, 97 - 109.

26. Tammer, Chr. (1992): A generalization of Ekeland’s variational principle. Optimization 25, 129 - 141.

27. Tammer, Chr. (1994): Stability results for approximately efficient solutions. OR Spektrum 16, 47-52.

28. Valyi, (1986): On approximate vector optimization. Working Paper 86-7 IIASA Laxenburg.

29. Wanka, Gert (1991): Duality in Vectorial Control Approximation Problems with Inequality Restrictions. Optimization 22, 755-764.

30. Wanka, G. (1994): Kolmogorov Conditions for Vectorial Approximation Problems. OR Spektrum 16, 53-58.

Acknowledgement The author is thankful to the referees for their helpful comments and corrections.

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