# View of Some cone separation results and applications

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Rev. Anal. Num´er. Th´eor. Approx., vol. 37 (2008) no. 1, pp. 37–46 ictp.acad.ro/jnaat

SOME CONE SEPARATION RESULTS AND APPLICATIONS

MARIUS DUREA

Abstract. In this note we present some cone separation results in infinite dimensional spaces. Our approach is mainly based on two different types of cone outer approximation. Then we consider an application to vector optimization.

MSC 2000. Primary: 90C29; Secondary: 90C26, 49J52.

Keywords. Cone separation results, cone enlargements, vector optimiza- tion.

1. INTRODUCTION AND PRELIMINARIES

The separation theorems are among the main tools in optimization theory and are especially used to obtain necessary optimality conditions. Starting with the classical convex separation results, a lot of such results have been used in the last decades in order to obtain new optimality results or in order to generalize the existing ones. In general, the functional separation by means of linear continuous maps is, by far, the most common way to separate two sets.

In this work we consider the method of “cone separation” having the same meaning as in the paper of Henig [6, Definition 2.1, Theorem 2.1]: consider two cones P, Qin a normed vector space withPQ={0},and find another coneK which strictly separate them, i.e.,PK={0}andQ⊂intK (where int denotes the topological interior). Moreover, we want to specify the form of the coneK and we base our analysis on two models of outer approximation of cones: the Henig dilating cones and the plastering cones in the sense of Krasnosel’ski˘ı [8]. Let us note that, in fact, the concept of Henig dilating cones was subsequently extracted by various authors from the proof of the main result in [6].

The cone separation technique we briefly described above has applications in multicriteria optimization because, once we can apply cone separation, we can treat the Pareto minima as weak Pareto minima and it is well-known that in vector optimization, in general, it is much more easier to work with weak minimality. We shall follow this line in the applications section of this paper.

“Al. I. Cuza” University, Faculty of Mathematics, Bd. Carol I, nr. 11, 700506, Ia¸si, Romania, e-mail: [email protected]

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The main results of the paper are done in Section 2. Section 3 is dedicated to some applications of the results in Section 2 to the field of vector optimiza- tion. More exactly, we obtain some necessary optimality conditions for Pareto minima in reflexive Banach spaces.

Let us start with some preliminaries. Throughout Y is a normed vector space, unless otherwise stated. If aY and ρ > 0, we denote by B(a, ρ) (resp. D(a, ρ)) the open (resp. closed) ball centered at awith radius ρ. The symbolS(a, ρ) denotes the sphere centered atawith radius ρ. For simplicity we shall also use the notation SY :=S(0,1).If A, BY are two subsets, the distance from A toB is d(A, B) := inf{ka−bk |aA, bB}.As usual, for a point xY, d(x, A) :=d({x}, A).The distance function will be denoted as dA:Y →R, dA(x) :=d(x, A).

Let KY be a proper cone. The dual cone of the coneKY is defined as

K:={yY |y(y)≥0,∀y ∈K}

and the quasi-interior of K is

K]:={yY |y(y)>0,∀y ∈K\ {0}}.

whereYis the topological dual ofY.It is clear that ifKandQare two cones with K \ {0} ⊂ intQ then QK and Q\ {0} ⊂ K]. A convex set B is said to be a base for the cone K if 0∈/clB and K= coneB, where cl denotes the topological closure and coneB := [0,∞)B is the cone generated by B. A cone which admits a base is called based.

2. CONE SEPARATION RESULTS

In the setting of general vector spaces, the following result is well-known (see [2]) and it puts into light the main assumptions one should assume when deal- ing with cone separation results. We present a proof of it for the completeness and for easy comparison with the further results derived in this paper.

Theorem 1. Let Y be a locally convex space and P, SY be cones s.t.

PS = {0}. If P is closed and S has compact base B, then there exists a convex pointed cone K s.t. S\ {0} ⊂intK andPK ={0}.

Proof. The relationP∩S ={0}implies that 0∈/ P−B.SincePis closed and B is compact, the setPB is closed and, taking into account the topological separation properties of locally convex spaces, there exists a balanced convex neighborhood V of 0 s.t.

(2.1) V ∩(P−B) =∅.

This means that

0∈/P −(B+V).

It is easy to see that the cone

K := cone(B+V)

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satisfies the requirements.

The above proof shows that, in fact, for normed vector spaces the cone K can be taken in the form

K := cone({y∈Y |d(y, B)≤ε}).

The properties of this theoretical construction are presented in the next result.

Proposition 1. ([5, Lemma 3.2.51]) Let KY be a closed convex cone with a base B and take δ = d(0, B) > 0. For ε ∈(0, δ), consider Bε ={y ∈ Y |d(y, B)ε}and Kε= [0,∞)Bε,the cone generated by Bε.Then

(i) Kε is a closed convex cone for everyε∈(0, δ);

(ii) if 0< γ < ε < δ, K\ {0} ⊂Kγ\ {0} ⊂intKε;

(iii) K=∩ε∈(0,δ)Kε=∩n∈NKεn where (εn)⊂(0, δ) converges to 0.

The conesKεconstructed in this way are termed as dilating cones or Henig dilating cones. It is important to note that the cone Kε has a non-empty interior. Moreover, we can specify the exact form of the elements in the dual cone of such a dilating cone with respect to the elements in the dual cone of the original ordering cone. This was done in [3].

Proposition 2. Let KY be a closed convex cone with a base B. For every ε∈(0, d(0, B)),

Kε ={yY | inf

b∈By(b)≥εkyk}.

If we analyze the proof of Theorem 1 we observe that one of the essential facts is thatP−Bis closed. In order to generalize Theorem 1 we use a criterion for the closedness of the difference of two non-convex sets derived by Z˘alinescu in [11]. We present here only the particular case we are interested in and to this end we need the notion of asymptotically compact subset of Y in the general framework whenY is a locally convex space endowed with a topology τ compatible with the duality system (Y, Y). A subset A of Y is called τ- asymptotically compact (τ-a.c. for short) if there exists a neighborhood U of 0 in (Y, τ) s.t. U ∩[0,1]A is τ-relatively compact. In the case when τ is the strong topology, then we shall simply use the term “asymptotically compact”.

If the topology τ in question is the weak topology we call it “asymptotically weakly compact”.

The asymptotic cone of a nonempty setAY (with respect ofτ) is defined by

Aτ:={u∈Y | ∃(tn)→0+,(an)⊂A:tnan τ

u}= \

t>0

clτ([0, t]A) and it is well known that, in general, Aτ is a τ-closed cone. When we deal with the norm topology we omit the superscript.

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Proposition3. LetY be a normed vector space andC, DY be nonempty (weakly) closed sets. If CD = {0} (CwDw = {0}), and C or D is (weakly) a.c., then CD is a (weakly) closed set.

The first cone separation result we would like to mention was mainly ob- tained in [3] and reads as follows.

Theorem 2. Let P, QY be (weakly) closed cones such that Q admits a (weakly) closed base B. Moreover, suppose that P or B is (weakly) a.c. If PQ={0}, then there exists ε∈(0, d(0, B)) s.t. PQε ={0}.

Proof. SincePQ={0},one deduces thatPB =∅,which is equivalent to 0∈/P−B.On the other hand, one has thatPτP, QτQ,becauseP, Q areτ-closed cones (in both cases of weak or strong convergence). Therefore,

{0} ⊂PτBτPQτ=PQ={0},

whence, from Proposition 3, one deduces that PB is a (weakly) closed set, therefore in any case it is norm closed whence α := d(0, PB) > 0. Let us take 0 < ε < min(α/2, d(0, B)) and we prove that 0 ∈/ PBε (in the notations of Proposition 1). This is obvious, because otherwise there would exist bεBεP and bB s.t. kbεbk<2ε.Then

d(0, PB)≤ kbεbk<2ε < α,

a contradiction. Consequently,P∩ −Bε=∅,soP∩ −Qε ={0},and the thesis

is proved.

Since aτ-compact set is automaticallyτ-a.c. one has the following corollary.

Corollary 1. Let P, QY be (weakly) closed cones such that Q admits a (weakly) closed base B. Moreover, suppose that P or B is (weakly) a.c. If PQ={0} then there exists ε∈(0, d(0, B)) s.t. PQε ={0}.

If one compares Theorem 1 with Theorem 2 one can observe (apart from the specific setting) that in the latter result the compactness of the base is replaced by asymptotically compactness. But, if one looks at the proofs, one can observe that Theorem 1 works for any closed set P (not necessarily a cone), while in Theorem 2 one uses thatP is a (τ-closed) cone in the relation PτP.

Next, we use another method to approximate an original cone by a larger one (having nonempty interior). For a given cone K, the following conical ε-enlargement (ε >0) is studied in the literature (see [8], [1]):

Kε={u∈X|d(u, K)εkuk}.

It is clear that the so-defined Kε is a cone (since for every xX and t≥0, d(tx, K) =td(x, K)) which containsK. It is also clear that forε≥1, Kε=X.

It is shown in [1, Proposition 3.2.1] that a closed pointed convex coneKadmits a convexε-enlargement if and only ifK has a bounded base. Other properties of this construction are listed below.

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Proposition 4. Let KY be a cone and Kε be the cone defined above.

Then:

(i) Kε= coneBε, where Bε:={u∈SY |d(u, K)ε};

(ii) for everyyK, D(y,(1 +ε)−1εkyk)⊂Kε;

(iii) ify ∈(Kε) then for everyyK, y(y)≥(1 +ε)−1εkyk kyk. Proof. (i) IfuKε,then d(u, K)εkuk,whenced(kuk−1u, K)ε.It is clear that v:= kuk−1uBε and since u =kukv, one has that u ∈coneBε. Conversely, if u ∈ coneBε then there exists vBε and α ≥ 0 s.t. u = αv.

Therefore,

d(u, K) =d(αv, K) =αd(v, K)αε=εkuk and the first part is proved.

(ii) Let yK and takevD(y,(1 +ε)−1εkyk).It is obvious that kvk ≥ kyk −(1 +ε)−1εkyk

= (1 +ε)−1kyk and, on the other hand,

d(v, K)≤ kv−yk ≤(1 +ε)−1εkyk ≤εkvk. This shows that vKε.

(iii) Let y ∈(Kε) and yK.Following the preceding item,y(v)≥0 for everyvD(y,(1 +ε)−1εkyk).In particular,

y(u)≥(1 +ε)−1εkyky(u)

for everyuSY.This is the conclusion and the proof is complete.

We put into relation the two types of enlargements in a particular situation (see also [1, Propositions 2.1.1, 3.2.1]).

Proposition 5. Let KY be a closed convex cone with a bounded base B. Then for every ε >0 there is a δ >0 s.t. KδKε.

Proof. Since 0 ∈/ clB, there exists µ := d(0, B) > 0 s.t. kbk ≥ µ for every bB. Then, from Proposition 4, (ii) we have that for every bB, D(b,(1 +ε)−1εµ)Kε, whence B(1+ε)−1εµ/2Kε, i.e. the conclusion for

δ = (1 +ε)−1εµ/2.

Definition 1. Let KY be a cone. One says that K has the property (S) if there exist x1, x2, ..., xnY (n∈N\ {0}) s.t.

K ⊂ {u∈Y | kuk ≤ max

i=1,n

{xi(u)}.

Observe that in the casen= 1 in the above definition one gets the definition of the supernormal cone. It is well known that a cone in a normed vector space is supernormal if and only if it has a bounded base (see, e.g., [5, p. 37]).

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However, is it easy to see that there are cones with the property (S) which are not supernormal: take, for example,

K:={(xn)∈l2 |(xn) = (x1,0,0,0...), x1∈R}.

We are interested in considering this property because if a coneK has the property (S) then it is clear that

(yn)⊂K, ynw 0⇒yn→0.

We are now able to present the main result of this note.

Theorem 3. Let Y be a reflexive Banach space, P, QY be cones such that P, Q are weakly closed and Q has the property (S). If PQ={0} then there exists ε >0 s.t. PQε={0}.

Proof. We proceed by contradiction. Suppose that for every n ∈ N\ {0}

there existsunPQ1/n, un6= 0.Since unQ1/n we have d(un, Q)n−1kunk

i.e.

d(kunk−1un, Q)n−1.

By the definition of the distance function, for every n∈N\ {0},there exists vnQ s.t.

kunk−1unvn≤2n−1.

Taking into account that Y is reflexive and the sequence (kunk−1un)n is bounded there exists a subsequence of it (denoted in the same way) weakly convergent towards an element uY. We show that u 6= 0. Indeed, in the contrary case kunk−1un w

→ 0 and since kunk−1unvn → 0 (in norm topol- ogy) we get thatvn w

→0.SinceQhas the property (S),we obtain that (vn) is strongly convergent towards 0,whencekunk−1un→0 and this is not possible becausekunk−1un has the norm 1 for everyn∈N\ {0}.Consequently,u6= 0.

Moreover, since P is weakly closed, uP \ {0}. Finally, vn w

u, so uQ and we arrive at a contradiction. The thesis is proved.

Note that a necessary and sufficient condition to have the conclusion of above theorem (i.e. PQ={0} implies that there is anε >0 s.t. PQε= {0}) are given in [10] and it can be written in our notation as: there exists an ε > 0 s.t. PεQε ={0}. In particular, this holds if P, Q are (norm) closed andP is (norm) locally compact. We are interested here to consider somehow different conditions in view of the applications we envisage and in order to avoid the local compactness condition.

We record the following consequences.

Corollary 2. Let Y be a reflexive Banach space,P, QY be cones such that P, Q are weakly closed andQ has a bounded base. If PQ={0} then:

(i) there existsε >0 s.t. Qε is convex, pointed andPQε={0}.

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(ii) there existsε >0 s.t. PQε={0}.

Proof. (i) Since Q has a bounded base, then it admits a convex ε-enlar- gement. Taking a smaller ε, if necessary, and using Theorem 3 we get the conclusion.

(ii) This part follows easily from Proposition 5. We would like to present as well a proof based on Corollary 1. First, since Q has a base, it is convex.

Moreover, since it is closed and its base B is bounded, then clB is a base as well, so we can consider thatQhas a closed bounded base. Since in a reflexive Banach space a bounded weakly closed set is weakly compact we can apply

Corollary 1 to get the conclusion.

In order to illustrate the application field of this corollary let us consider Y =L2(Ω) where Ω is a non-empty subset of Rn and Q:=L2(Ω) is the well known space of square Lebesgue-integrable functions f : Ω → R. The space L2(Ω) is a Hilbert space and hence (L2(Ω))=L2(Ω) and the natural ordering cone inL2(Ω) is given as

L+2(Ω) ={f ∈L2(Ω)|f(x)≥0 almost everywhere on Ω}.

It is interesting to note that (L+2(Ω)) =L+2(Ω) andL+2(Ω)] is non-empty (see for example [7]). Thus by considering an element xL+2(Ω)] then

B ={x∈L+2(Ω)|x(x) = 1}

is a base for L+2(Ω) and it is is weakly compact and hence bounded.

At the end of this section, we would like to present another possible en- largement, smaller thanKε, but which can be obtained in a similar way. Let KY be a cone. Based on Proposition 4 let us define

Aε:={u∈SY |d(u, SYK)ε}.

and

K1ε:= coneAε.

Sinced(u, K)d(u, SYK) for everyuY,it is clear thatAεBε,whence K1εKε.

Let us prove now that K\ {0} ⊂K1ε.For this, it is enough to prove that KSYK1ε. Take yKSY. Clearly, SYD(y, ε)Aε. Suppose, by way of contradiction that there exists a sequence (yn)→y, yn/K1ε for every n≥1.Therefore,

kynk−1yn→ kyk−1y=y

whence vn :=kynk−1ynSY, vny. Since SYD(y, ε)Aε,vnAε for n large enough. This shows thatynK1ε and this is a contradiction. Using these observations and Theorem 3 we get the next result.

Proposition6. LetY be a reflexive Banach space,P, QY be cones such that P, Q are weakly closed and Q has the property (S). If PQ={0} then there exists ε >0 s.t. PQε1={0}.

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3. AN APPLICATION

In this section we present some applications to vector optimization, mainly based on Corollary 2 (i),i.e. on the separation with enlargement cones. Note that several applications based on the separation with dilating cones have been done in [3].

Let K be a proper closed convex cone. It is well known that such a cone induces a partial order relation≤K onY byy1K y2 if and only ify2−y1K and, with respect to this relation, one can consider different types on minimum points. If AY is a nonempty set, then a point yA is called a Pareto minimum point forA with respect to K if (A−y)∩ −K ⊂K. In particular, if K is pointed, this means that (A−y)∩ −K = {0}. Observe that, in fact the condition (A−y)∩ −K ⊂K (resp. (A−y)∩ −K ={0}) is equivalent by cone(A−y)∩ −K ⊂K (resp. cone(A−y)∩ −K ={0}). If intK 6=∅, then a point yA is called weak Pareto minimum point of A with respect to K if (A−y)∩ −intK =∅, i.e. it is a minimum point forA with respect to the cone intK ∪ {0}. One denotes by Min(A |K) (WMin(A |K), respectively) the sets of Pareto (weak Pareto, respectively) minima ofAwith respect toK.

It is well-known that the main difficulty which arise when dealing with Pareto minima consist of the emptiness of the interior of the underlying ordering cone.

Unfortunately, this situation is a common one for the majority of the natural ordering cones of the usual Banach spaces. In contrast, when the cone has nonempty interior one has several tools to handle the (weak) Pareto minima.

One of this tools is the following scalarizing lemma (see [5, Section 2.3] and [4]). The symbol denotes the Fenchel subdifferential of a convex function.

Lemma 1. Let KY be a closed convex cone with nonempty interior and let e∈intK, MY, yM. Define the functional ϕe:Y →Ras

ϕe(y) = inf{λ∈R|yλeK}.

This map is continuous, convex, strictly-intK-monotone,d(e,bd(Q))−1-Lipschitz (where bdQ denotes the topological boundary of Q) and for every λ∈R

{y |ϕe(y)≤λ}=λeK, {y|ϕe(y)< λ}=λe−intK.

Moreover, for everyuY,

∂ϕe(u) ={vK |v(e) = 1, v(u) =ϕe(u)}.

The point y is a weak minimum point for M with respect to K if and only if y is a minimum point for ϕe(· −y) onM.

The other major tools we use in this section are the generalized differentia- tion objects introduced and developed by Mordukhovich and his collaborators (see [9, Vol. I]). We restrict ourselves to the case of Asplund spaces because we shall work in the setting of reflexive Banach spaces, a subset of the collections of Asplund spaces.

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Definition 2. Let X be an Asplund space and SX be a non-empty closed subset ofX and let xS.

(i) The basic (or limiting, or Mordukhovich) normal cone to S atx is NM(S, x) :={xX| ∃xnS x, xnw x, xnNF(S, xn)}

whereNF(S, z)denotes the Fr´echet normal cone toS at a pointzS, given as

NF(S, z) :={xX| lim sup

u∈S,u→z x(u−z)

ku−zk ≤0}.

(ii) Let f :X→R be finite atxX; the Fr´echet subdifferential of f atx is the set

∂f(x) :=ˆ {xX |(x,−1)∈NF(epif,(x, f(x)))}

and the basic (or limiting, or Mordukhovich) subdifferential off at x is

Mf(x) :={xX|(x,−1)∈NM(epif,(x, f(x)))}, where epif denotes the epigraph of f.

On the Asplund spaces one has

Mf(x) = lim sup

x→¯fx

∂fˆ (x),

and, in particular, ˆ∂f(x)Mf(x). Of course, if a functionf attains a local minimum at a point x then 0∈Mf(x).If δ denotes the indicator function associated with a nonempty set Ω⊂X (i.e. δ(x) = 0 if x ∈Ω, δ(x) =∞ if x /∈ Ω ), then for any x ∈ Ω, ∂Mδ(x) = NM(Ω, x). In contrast with the Fr´echet subdifferential, the basic subdifferential satisfies a robust calculus rule:

if X is Asplund, f1 is Lipschitz around x and f2 is l.s.c. around this point, then

(3.1) M(f1+f2)(x)⊂Mf1(x) +Mf2(x).

Theorem 4. Let Y be a reflexive Banach space and QY be a weakly closed cone with bounded base. Let AY be a set and y ∈ Min(A | K) s.t. cone(A−y) is weakly closed. Then there exists ε > 0 such that for every eK\ {0} there exists y ∈ −NM(A, y) with y(e) = 1 and y(y) ≥ (1 +ε)−1εkyk kyk for everyyY.

Proof. In our conditions, cone(A−y)∩K ={0},whence, following Corollary 2 (i), there exists a positive ε s.t. cone(A−y)Kε = {0}, i.e. y ∈ Min(A| Kε) ⊂ WMin(A | Kε). We can apply Lemma 1 for the cone Q := Kε and an element eK \ {0} ⊂ intKε. Then y is a minimum point over A for the functional se(· −y) and then, by the infinite penalization method, y is a minimum point without constraints forse(· −y) +δA.Therefore,

0∈M(se(· −y) +δA)(y)

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and, since the first function is locally Lipschitz and the second one is lower- semicontinuous, we have

0∈M(se(· −y))(y) +∂MδA(y).

Moreover, the functionalse(·−y) is sublinear and hence by using again Lemma 1 and Proposition 4 we obtain

M(se(· −y))(y) =∂se(0) ={y ∈(Kε)|y(e) = 1}

={yY|y(y)≥(1+ε)−1εkyk kyk,∀yY, y(e) = 1}.

On the other hand, MδA(y) =NM(A, y),whence the conclusion.

Note that in this result the original cone K can have empty interior. If we denote by B the base ofK, the propertyy(y)≥(1 +ε)−1εkyk kyk for every yY fulfilled by y yields, in particular,

b∈Binf y(b)≥(1 +ε)−1εkykd(0, B) (compare with [3, Theorem 4.1]).

Following the technique developed in [4], one can derive optimality condi- tions for vector optimization problems governed by single-valued or set-valued maps as well. However, in order to keep this note short, we restrict our atten- tion to the above case only.

REFERENCES

[1] Bednarczuk, E.,Stability analysis for parametric vector optimization problems, Dis- sertationes Mathematicae,442, pp. 1–126, 2006.

[2] Dauer, J.P.andSaleh, O.A.,A characterization of proper minimal points as solutions of sublinear optimization problems, J. Math. Anal. Appl.,178, pp. 227–246, 1993.

[3] Durea, M.andDutta, J.,Lagrange multipliers for Pareto minima in general Banach spaces, Pacific Journal of Optimization, to appear.

[4] Durea, M. and Tammer, C., Fuzzy necessary optimality conditions for vector opti- mization problems, Optimization, to appear.

[5] opfert, A., Riahi, H., Tammer, C. and alinescu, C., Variational Methods in Partially Ordered Spaces, Springer, Berlin, 2003.

[6] Henig, M.I., A cone separation theorem, Journal of Optimization Theory and Appli- cations,36, pp. 451–455, 1982.

[7] Jahn, J.,Vector Optimization: Theory, Applications and Extensions, Springer, Berlin, 2004.

[8] Krasnosel’sk˘ii, M.A.,Positive Solutions on Operator Equations, Noordhoff, Gronin- gen, 1964.

[9] Mordukhovich, B.S., Variational Analysis and Generalized Differentiation, Vol. I:

Basic Theory, Vol. II: Applications, Springer, Berlin, 2006.

[10] Luc, D.T.andPenot J.P.,Convergence of asymptotic directions, Trans. Amer. Math.

Soc.,353, pp. 4095–4121, 2001.

[11] alinescu, C.,Stability for a class of nonlinear optimization problems and applications, in: Nonsmooth Optimization and Related Topics (Erice, 1988), Plenum, New York, pp. 437–458, 1989.

Received by the editors: June 25, 2008.

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