Journal of Convex Analysis Volume 4 (1997), No. 1, 109–127

A Systematization of Convexity Concepts for Sets and Functions

Wolfgang W. Breckner, Gábor Kassay

Babeş-Bolyai University, Faculty of Mathematics and Computer Science, Str. Kogălniceanu Nr. 1, 3400 Cluj-Napoca, Romania. e-mail: {breckner, kassay}@math.ubbcluj.ro

Received September 12, 1994 Revised manuscript received January 3, 1996

The paper surveys several convexity concepts, referring to sets and to functions respectively, with the purpose to put them in some kind of order according to the same principle. First it examines the connections between six convexity concepts regarding sets in topological linear spaces and points out the most general concept among these concepts. On the basis of this analysis it is then revealed that twelve convexity concepts concerning functions, that take values in topological linear spaces, can be naturally defined by reduction to the investigated convexities for sets. The most general convexity concept for functions is also found. It is applied to establish an alternative theorem as well as necessary optimality conditions for weak multiobjective optimization problems.

Keywords: Generalized convex sets, generalized convex functions, necessary optimality conditions

1991 Mathematics Subject Classification: 52A07, 26B25, 90C29, 46N10.

1. Introduction

The investigations of scalar optimization problems in topological linear spaces use above all ordinary convex sets, but also two types of generalized convex sets, called nearly convex sets and closely convex sets, respectively. Multiobjective optimization requires additional convexity concepts for sets. Tackling decision problems with multiobjectives, P. L. Yu [23] has given a generalization of the convex sets by introducing sets that are convex with respect to another set. The point of his idea was to consider the translate of the original set by a given set K. In the present paper we show that both the nearly convex sets and the closely convex sets can be generalized in the same way. In consequence, the number of convexity concepts induced by a given set will increase to three. The main purpose of the present paper is to reveal the connections between the three basic convexity concepts mentioned at the beginning, on the one hand, and the three convexity concepts induced by a given set K, on the other hand. The major results highlighting these connections are proved in Section 2. They show that, if K is a convex set, then the concept of a closely convex set with respect to K is the most general concept among the six convexity concepts considered here for sets. Some interesting characterizations of

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the sets, that are closely convex with respect to a convex cone with nonempty interior, are also established in Section 2. As an application of these sets we derive in Section 3 a characterization of the elements that are weakly minimal with respect to a convex cone. After that we leave the convexity concepts regarding sets and turn in Section 4 to a discussion of convexity concepts regarding functions that take values in topological linear spaces. In order to define a convexity concept for such a function we start either from the graph or from the range of the function. Then we apply the six convexity concepts, examined in Section 2, to each of these two particular sets associated with the function and obtain in this way in each case six convexity concepts for functions. The names that we attribute to these convexities concerning functions are in such a manner chosen that they distinctly emphasize the relationship with one of the six convexity concepts concerning sets. It will be seen that this procedure of introducing convexity concepts for functions by reduction to sets allows to find easily the connections between the twelve classes of functions we have obtained. The connections that we will reveal will show that if K is a convex set, then the functions that are called closely convexlike with respect to K are the most general ones among all the functions considered in this paper. By using closely convexlike functions with respect to a convex cone with nonempty interior we state in the last section of the paper an alternative theorem as well as necessary conditions for the solutions of weak multiobjective optimization problems. It should be mentioned that most of the convexities concerning functions, that occur in the present paper, are not new. They have already been introduced in earlier papers, mostly separately by other approaches and under other less logical names. But here, for the first time, there is given a simple unitary scheme, based on convexities for sets, in which they all can be naturally arranged and which clearly emphasizes the relations between them. Besides it is shown that the familiar technique of separating convex sets by closed hyperplanes can be directly used to derive some results, that relate to classes of generalized convex sets or generalized convex functions, considered in this paper, and that are applicable in nonconvex multiobjective optimization.

Throughout the paper X and Y denote topological linear spaces over the field ℝ of real numbers. The zero-elements in X and Y are denoted by o_{X} and o_{Y}, respectively. The topological dual of Y is denoted by Y′ and its zero-element by o′. The addition and scalar multiplication on the family of subsets of Y are defined by

aM + bN = {z ∈ Y | ∃(x, y) ∈ M × N : z = ax + by},

where M, N are subsets of Y, and a, b are real numbers. If M is any set in a topological space, then int M denotes the interior of M, while cl M denotes the closure of M.

2. Convexity Concepts for Sets

We start by recalling the definitions of the basic convexity concepts concerning subsets of topological linear spaces.

A subset S of Y is said to be:

(a) convex if (1 − a)S + aS ⊆ S for all a ∈ ]0, 1[;

(b) nearly convex if there is an a ∈ ]0, 1[ such that (1 − a)S + aS ⊆ S;

(c) closely convex if cl S is a convex set.

Introduced in ancient times for the needs of geometry, in the last century the convex sets have demonstrated their utility also in analysis, functional analysis and optimization theory (see, for instance, [9], [19] and [5]). Unlike convex sets, nearly convex sets have not been frequently used. Properties of the sets of this type may be found in [1], [14] and [20]. From the three convexity concepts defined above, that of a closely convex set is the most recent. It has been introduced in [2].

Within the class of convex sets the convex cones play an important role. We say that a subset S of Y is a convex cone if it is not empty, convex and satisfies aS ⊆ S for all a ∈ ]0, ∞[.

A simple example of a convex cone is the set ℝ_{+}^{n} consisting of all vectors (α_{1}, …, α_{n}) in ℝ^{n} with α_{i} ≥ 0 for each i ∈ {1, …, n}. In particular, the set ℝ_{+} of all nonnegative real numbers is a convex cone. If S is a subset of Y, then

S^{∗} = {y′ ∈ Y′ | ∀y ∈ S : y′(y) ≥ 0}

is a convex cone in Y′ called the dual cone of S.

Next let us fix a subset K of Y. By means of this set and the three basic convexity concepts we can define three other convexity concepts for sets as follows.

A subset S of Y is said to be: (a′) convex with respect to K (or shortly K-convex) if S + K is a convex set; (b′) nearly convex with respect to K (or shortly nearly K-convex) if S + K is a nearly

convex set; (c′) closely convex with respect to K (or shortly closely K-convex) if S + K is a closely

convex set.

Obviously, a set S in Y is convex (respectively nearly convex, closely convex) if and only if it is convex (respectively nearly convex, closely convex) with respect to {o_{Y} }. Under the assumption that K is a convex cone, the K-convex sets have been introduced in [23]. So far as we know, the other two convexity concepts induced by the set K have not been explored before.

In what follows we discuss the connections between the six convexity concepts mentioned in this section. In order to simplify the formulations, we denote by

C(Y), NC(Y) and CC(Y)

the families of all convex sets, nearly convex sets, and closely convex sets in Y, respectively. Similarly, we denote by

C_{K}(Y), NC_{K}(Y) and CC_{K}(Y)

the families of all K-convex sets, nearly K-convex sets, and closely K-convex sets in Y, respectively.

Theorem 2.1. If K is a convex subset of Y and S is a subset of Y, then the implications indicated by the arrows in the diagram below are true:

S ∈ C(Y) ⇒ S ∈ NC(Y) ⇒ S ∈ CC(Y)
⇓ ⇓ ⇓
S ∈ C_{K}(Y) ⇒ S ∈ NC_{K}(Y) ⇒ S ∈ CC_{K}(Y).

(2.1)

Proof. We establish this theorem in a sequence of five steps. Step 1. The implication S ∈ C(Y) ⇒ S ∈ NC(Y) follows immediately from the definitions of the convex and nearly convex sets, while the implication S ∈ NC(Y) ⇒ S ∈ CC(Y) is a consequence of Corollary 2.3 stated in [1]. Consequently, the implications in the first row of diagram (2.1) are true. Step 2. By applying the implications of the first row of diagram (2.1) to the set S + K instead of S, we obtain the implications of the second row of the above diagram. Step 3. Suppose that S ∈ C(Y). Because K is also convex, it follows that

(1 − a)(S + K) + a(S + K) ⊆ (1 − a)S + aS + (1 − a)K + aK ⊆ S + K  (2.2)

for all a ∈ ]0, 1[. Hence S + K is a convex set. Thus the implication S ∈ C(Y) ⇒ S ∈ C_{K}(Y) is true. Step 4. Suppose that S ∈ NC(Y). Then there exists a number a ∈ ]0, 1[ such that (1 − a)S + aS ⊆ S. Consequently, this number satisfies (2.2). Hence S + K is a nearly convex set. Thus the implication S ∈ NC(Y) ⇒ S ∈ NC_{K}(Y) is true. Step 5. Suppose that S ∈ CC(Y). This means that cl S is a convex set. On the other hand, taking into account that K is a convex set, it follows by Step 1 that cl K is a convex set. Since both the sets cl S and cl K are convex, it follows that the set cl S + cl K is also convex. By applying again Step 1, we conclude that cl (cl S + cl K) is a convex set. But, in view of

S + K ⊆ cl S + cl K ⊆ cl (S + K),

we have

cl (S + K) = cl (cl S + cl K).

Consequently, cl (S + K) must be a convex set. Thus the implication S ∈ CC(Y) ⇒ S ∈ CC_{K}(Y) is true.

Corollary 2.2. If K and S are subsets of Y and L is a convex subset of Y, then the following implications are true:

S ∈ C_{K}(Y) ⇒ S ∈ C_{K+L}(Y), S ∈ NC_{K}(Y) ⇒ S ∈ NC_{K+L}(Y),
S ∈ CC_{K}(Y) ⇒ S ∈ CC_{K+L}(Y).

Proof. Apply the vertical implications of diagram (2.1).

By invoking Corollary 2.2 we get the next corollary.

Corollary 2.3. Let K be a subset of Y containing o_{Y}, let L be a convex cone in Y such that K ⊆ L, and let S be a subset of Y. Then the following implications are true:

S ∈ C_{K}(Y) ⇒ S ∈ C_{L}(Y), S ∈ NC_{K}(Y) ⇒ S ∈ NC_{L}(Y),
S ∈ CC_{K}(Y) ⇒ S ∈ CC_{L}(Y).

Proof. Note that K + L = L, and then apply Corollary 2.2.

Remark 2.4. By reversing the arrows in diagram (2.1) we obtain implications that are not generally true. This claim results from the following simple examples involving only subsets of ℝ^{2}.

1. The set M = {(x, y) ∈ ℝ^{2} | y > 0} ∪{(x, 0) ∈ ℝ^{2} | x ∈ ℚ} is nearly convex, but it is not convex. Hence, the implication S ∈ NC(Y) ⇒ S ∈ C(Y) is not always true. Consequently, the implication S ∈ NC_{K}(Y) ⇒ S ∈ C_{K}(Y) is not always true either.

2. The set M = {(x, y) ∈ ℝ^{2} | y > 0} ∪{(x, 0) ∈ ℝ^{2} | x ∈ ℝ \ ℚ} is closely convex. Next note that, for every a ∈ ]0, 1[, the irrational numbers

x_{1} = \{ \begin{matrix}\sqrt{2} & if a ∈ ℚ \\ 1/(a − 1) & if a ∈ ℝ \ ℚ \end{matrix} and x_{2} = (1 − 1/a)x_{1}

satisfy (1 − a)x_{1} + ax_{2} = 0. Thus there is no a ∈ ]0, 1[ such that

(1 − a)(ℝ \ ℚ) + a(ℝ \ ℚ) ⊆ ℝ \ ℚ.

This means that ℝ \ ℚ is not nearly convex. Hence the set M is not nearly convex either. Consequently, the implication S ∈ CC(Y) ⇒ S ∈ NC(Y) is not always true. Of course, the implication S ∈ CC_{K}(Y) ⇒ S ∈ NC_{K}(Y) is not always true either.

3. The set M = {(x, 0) ∈ ℝ^{2} | x ≥ 0} ∪{(0, y) ∈ ℝ^{2} | y ≥ 0} is ℝ_{+}^{2}-convex, because M + ℝ_{+}^{2} = ℝ_{+}^{2}. Taking into account the implications in the second row of diagram (2.1), it follows that the set M is also nearly ℝ_{+}^{2}-convex and closely ℝ_{+}^{2}-convex. But, obviously, M is not closely convex. Thus, due to the implications in the first row of diagram (2.1), the set M cannot be neither convex nor nearly convex. In consequence, the implications

S ∈ C_{K}(Y) ⇒ S ∈ C(Y), S ∈ NC_{K}(Y) ⇒ S ∈ NC(Y),
S ∈ CC_{K}(Y) ⇒ S ∈ CC(Y)

are not always true.

Provided that K is a convex set, it follows from Theorem 2.1 and Remark 2.4 that the concept of a closely K-convex set is the most general concept among the six convexities concerning sets that we have considered in this section. In the case when K is a convex cone with nonempty interior we can give some nice characterizations of the closely Kconvex sets. For establishing these characterizations we need the following lemma.

Lemma 2.5. Let K be a convex cone in Y with int K ≠ ∅, and let S be a subset of Y. Then the following equalities hold:

int[cl (S + K)] = S + int K;  (2.3)
cl (S + K) = cl (S + int K).  (2.4)

Proof. Let y be any point in int[cl (S + K)]. Then there exists a neighbourhood U of o_{Y} such that

y − U ⊆ cl (S + K).  (2.5)

Now we choose a point k_{0} ∈ int K. Since U is absorbing, we can find an a ∈ ]0, ∞[ such that ak_{0} ∈ U. In view of (2.5) it follows that the point z = y − ak_{0} lies in cl (S + K). On the other hand, y − int K is a neighbourhood of z, because this set is open and contains z. Consequently, we have

(S + K) ∩ (y − int K) ≠ ∅.

Thus there exist s ∈ S and k ∈ K such that s + k ∈ y − int K. From this we get

y ∈ s + k + int K ⊆ S + K + int K ⊆ S + int K.

Since y was arbitrarily chosen in int[cl (S + K)], we have proved that

int[cl (S + K)] ⊆ S + int K.  (2.6)

Next note that

S + int K ⊆ S + K ⊆ cl (S + K).

Taking into account that the set S + int K is open, it follows that

S + int K ⊆ int[cl (S + K)].  (2.7)

From (2.6) and (2.7) we obtain (2.3). Now let y be any point in S + K. Further let V be any neighbourhood of y. From y ∈ S + K it follows that y − s ∈ K for some s ∈ S. Next we choose a point k_{0} ∈ int K. Since V is a neighbourhood of y and

lim_{a→1} [s + (1 − a)k_{0} + a(y − s)] = y,

we can choose a number a_{0} ∈ [0, 1[ such that the point z = s + (1 − a_{0})k_{0} + a_{0}(y − s) lies in V. On the other hand, a well-known property of the convex sets in topological linear spaces (see, for instance, formula (11.1) in [9, p. 59]) implies that

(1 − a_{0})k_{0} + a_{0}(y − s) ∈ int K.

Therefore z lies also in S + int K. Consequently, we have V ∩ (S + int K) ≠ ∅. Since y was arbitrarily chosen in S + K and V was an arbitrary neighbourhood of y, we have shown that

S + K ⊆ cl (S + int K).

Taking into account that the set cl (S + int K) is closed, it follows that

cl (S + K) ⊆ cl (S + int K).  (2.8)

Finally note that S + int K ⊆ S + K implies

cl (S + int K) ⊆ cl (S + K).  (2.9)

From (2.8) and (2.9) we obtain (2.4).

By applying this lemma, we now can obtain the announced characterizations of the closely K-convex sets.

Theorem 2.6. Let K be a convex cone in Y with int K ≠ ∅, let L = int K, and let S be a subset of Y. Then the implications indicated by the arrows in the diagram below are true:

S ∈ CC_{K}(Y) ⇒ S ∈ C_{L}(Y)
⇑  ⇓
S ∈ CC_{L}(Y) ⇐ S ∈ NC_{L}(Y).

Proof. The key observation for the proof is the well-known fact that the interior of a convex set in a topological linear space is a convex set (see, for instance, [19, Lemma 2, p. 13]). Suppose that S ∈ CC_{K}(Y). Thus the set cl (S + K) is convex, and hence int[cl (S + K)] is also convex. By applying (2.3), we conclude that S + int K is convex. In other words, we have S ∈ C_{L}(Y). Consequently, the implication S ∈ CC_{K}(Y) ⇒ S ∈ C_{L}(Y) is true. Next we note that the implications

S ∈ C_{L}(Y) ⇒ S ∈ NC_{L}(Y) ⇒ S ∈ CC_{L}(Y)

are true in virtue of Theorem 2.1. Finally, suppose that S ∈ CC_{L}(Y). Thus cl (S + int K) is a convex set. Taking into account (2.4), it follows that cl (S + K) is also convex, i.e. S ∈ CC_{K}(Y). Therefore the implication S ∈ CC_{L}(Y) ⇒ S ∈ CC_{K}(Y) is true.

3. A Characterization of Weakly K-Minimal Elements

Since their introduction, the K-convex sets have been successfully applied in multiobjective optimization (see [23], [8], [11]). Here we will illustrate that the more general closely K-convex sets, we dealt with in the preceding section, are also useful in optimization. When K is a convex cone in Y with int K ≠ ∅, then we intend to characterize those points s_{0} of a given closely K-convex subset S of Y that satisfy

(s_{0} − int K) ∩ S = ∅.  (3.1)

Any element s_{0} ∈ S satisfying (3.1) is said to be weakly minimal with respect to K (or shortly weakly K-minimal).

Lemma 3.1. Let K be a convex cone in Y with int K ≠ ∅, let S be a nonempty subset of Y, and let s_{0} be an element of Y. Then the following assertions are true:

1. If s_{0} satisfies (3.1) and S is closely K-convex, then there exists a y′_{0} ∈ K^{∗} \ {o′} such

that

y′_{0}(s_{0}) ≤ y′_{0}(s) for all s ∈ S.  (3.2)

2. If there exists a y′_{0} ∈ K^{∗} \ {o′} satisfying (3.2), then (3.1) holds.

Proof. 1. First we observe that

(s_{0} − int K) ∩ (S + K) = ∅.  (3.3)

Indeed, if we assume the contrary, then there exist elements s ∈ S and k ∈ K such that s + k ∈ s_{0} − int K. From this we obtain s ∈ s_{0} − (k + int K) ⊆ s_{0} − int K. Consequently, we have s ∈ (s_{0} − int K) ∩ S, which contradicts (3.1). Hence (3.3) must hold.

From (3.3) it follows that

S + K ⊆ Y \ (s_{0} − int K).  (3.4)

Since s_{0} − int K is an open set, it follows that Y \ (s_{0} − int K) is a closed set. Therefore (3.4) implies

cl (S + K) ⊆ Y \ (s_{0} − int K),

i.e.

(s_{0} − int K) ∩ [cl (S + K)] = ∅.

Since the sets s_{0} − int K and cl (S + K) are convex, we can apply a well-known separation theorem (see [9, p. 63]) and conclude that there is a y′_{0} ∈ Y′ \ {o′} such that

sup {y′_{0}(s_{0} − k) | k ∈ K} ≤ inf {y′_{0}(y) | y ∈ cl (S + K)}.  (3.5)

Now let s be any element in S, and let k be any element in K. Since K is a cone, we have ak ∈ K for all a ∈ ]0, ∞[. Consequently, (3.5) implies

y′_{0}(s_{0} − k) ≤ y′_{0}(s + \frac{1}{a}k) for all a ∈ ]0, ∞[,

on the one hand, and

y′_{0}(s_{0} − ak) ≤ y′_{0}(s + ak) for all a ∈ ]0, ∞[,

on the other hand. These inequalities can be rewritten as follows:

y′_{0}(k) ≥ a[y′_{0}(s_{0} − k) − y′_{0}(s)] for all a ∈ ]0, ∞[,
y′_{0}(s) ≥ y′_{0}(s_{0}) − 2ay′_{0}(k) for all a ∈ ]0, ∞[.

Letting a → 0, we then obtain y′_{0}(k) ≥ 0 and y′_{0}(s) ≥ y′_{0}(s_{0}). Hence y′_{0} lies in K^{∗} and satisfies (3.2).

2. Let y′_{0} ∈ K^{∗} \ {o′} satisfy (3.2). Suppose that there is an element s ∈ S such that s − s_{0} ∈ − int K. Then we can choose a neighbourhood U of o_{Y} which satisfies

s − s_{0} + U ⊆ − K.  (3.6)

Next we select an element y ∈ Y for which y′_{0}(y) > 0. Since y′_{0} ≠ o′, we can find such an element. Then take into consideration that U is absorbing and choose a number a ∈]0, ∞[ such that ay ∈ U. From (3.6) it follows that s − s_{0} + ay lies in − K, and therefore we have y′_{0}(s − s_{0} + ay) ≤ 0. This inequality implies

y′_{0}(s) ≤ y′_{0}(s_{0}) − ay′_{0}(y) < y′_{0}(s_{0}),

which contradicts (3.2). Thus (3.1) must hold.

By applying Lemma 3.1 we obtain the following characterization theorem for the weakly K-minimal elements.

Theorem 3.2. Let K be a convex cone in Y with int K ≠ ∅, let S be a nonempty subset of Y which is closely K-convex, and let s_{0} be an element of S. Then s_{0} is weakly K-minimal if and only if there exists a y′_{0} ∈ K^{∗} \ {o′} such that

y′_{0}(s_{0}) = min {y′_{0}(s) | s ∈ S}.

This theorem generalizes Theorem 1 given in [21, p. 104].

4. Convexity Concepts for Functions

Let D be a nonempty subset of X. The convexity concepts regarding sets, studied in Section 2, allow us to introduce convexity concepts for functions defined on D and taking values in Y. The procedure is quite simple. It is based on assigning a determined subset S(f) of a topological linear space to each function f : D → Y. If the set S(f) has one of the six convexity properties defined in Section 2, then we say that the function f has that one same convexity property. Consequently, the set-valued mapping f ⟼ S(f) will yield six convexity concepts regarding functions. We illustrate this technique, which can be called defining by reduction to sets, in two cases, namely when the set S(f) assigned to f is the graph

gr f = {(x, f(x)) | x ∈ D}

and the range

rng f = {f(x) | x ∈ D}

of f, respectively.

Let K be a subset of Y. Set K_{0} = {o_{X}} × K. Depending on the convexity properties of gr f in the topological linear space X × Y, we say that f : D → Y is: (a_{1}) convex if gr f ∈ C(X × Y); (b_{1}) nearly convex if gr f ∈ NC(X × Y); (c_{1}) closely convex if gr f ∈ CC(X × Y); (a′_{1}) convex with respect to K (or shortly K-convex) if gr f ∈ C_{K_{0}}(X × Y);

(b′_{1}) nearly convex with respect to K (or shortly nearly K-convex) if gr f ∈ NC_{K_{0}}(X ×Y);

(c′_{1}) closely convex with respect to K (or shortly closely K-convex) if gr f ∈ CC_{K_{0}}(X×Y).

Similarly, depending on the convexity properties of rng f in the topological linear space Y, we say that f : D → Y is: (a_{2}) convexlike if rng f ∈ C(Y); (b_{2}) nearly convexlike if rng f ∈ NC(Y); (c_{2}) closely convexlike if rng f ∈ CC(Y); (a′_{2}) convexlike with respect to K (or shortly K-convexlike) if rng f ∈ C_{K}(Y); (b′_{2}) nearly convexlike with respect to K (or shortly nearly K-convexlike) if rng f ∈

NC_{K}(Y); (c′_{2}) closely convexlike with respect to K (or shortly closely K-convexlike) if rng f ∈

CC_{K}(Y).

In virtue of the definitions (a′_{1}), (b′_{1}) and (c′_{1}) we have to check whether the set gr f + K_{0} is convex, nearly convex or closely convex, so that we can decide whether f is K-convex, nearly K-convex or closely K-convex. This set gr f + K_{0} is called the epigraph of f with respect to K and denoted by epi_{K} f. Obviously, we have

epi_{K} f ={(x, y) ∈ X × Y | x ∈ D, y ∈ f(x) + K}.

Similarly, in virtue of the definitions (a′_{2}), (b′_{2}) and (c′_{2}) we have to examine whether the set rng f + K is convex, nearly convex or closely convex, so that we can decide whether f is K-convexlike, nearly K-convexlike or closely K-convexlike. This set rng f + K is called the epirange of f with respect to K.

By analogy with the notations introduced in Section 2 for the families of all subsets of Y having the same convexity property, we denote by

C(D, Y), NC(D, Y), CC(D, Y), C_{K}(D, Y), NC_{K}(D, Y), CC_{K}(D, Y)

the families of all functions from D into Y that are convex, nearly convex, closely convex, K-convex, nearly K-convex and closely K-convex, respectively. Similarly, we denote by

CL(D, Y), NCL(D, Y), CCL(D, Y), CL_{K}(D, Y), NCL_{K}(D, Y), CCL_{K}(D, Y)

the families of all functions from from D into Y that are convexlike, nearly convexlike, closely convexlike, K-convexlike, nearly K-convexlike and closely K-convexlike, respectively. According to the definitions of the above introduced twelve convexity concepts regarding functions, Theorem 2.1 implies the following theorem proclaiming some connections between these concepts.

Theorem 4.1. If K is a convex subset of Y and f is a function from D into Y, then the implications indicated by the arrows in the diagrams below are true:

f ∈ C(D, Y) ⇒ f ∈ NC(D, Y) ⇒ f ∈ CC(D, Y)
⇓ ⇓ ⇓
f ∈ C_{K}(D, Y) ⇒ f ∈ NC_{K}(D, Y) ⇒ f ∈ CC_{K}(D, Y);

(4.1)

f ∈ CL_{K}(D, Y) ⇒ f ∈ NCL_{K}(D, Y) ⇒ f ∈ CCL_{K}(D, Y)
⇑ ⇑ ⇑
f ∈ CL(D, Y) ⇒ f ∈ NCL(D, Y) ⇒ f ∈ CCL(D, Y).

(4.2)

The relationship between the diagrams (4.1) and (4.2) is given by the next theorem.

Theorem 4.2. Let K be a subset of Y, and let f be a function from D into Y. Then the following implications are true:

f ∈ C_{K}(D, Y) ⇒ f ∈ CL_{K}(D, Y), f ∈ NC_{K}(D, Y) ⇒ f ∈ NCL_{K}(D, Y),
f ∈ CC_{K}(D, Y) ⇒ f ∈ CCL_{K}(D, Y).

Proof. Suppose that f ∈ C_{K}(D, Y). We claim that under this assumption the set rng f + K is convex. To prove this, let y_{1} and y_{2} be elements of rng f + K, and let a be in ]0,1[. We can associate with y_{1} and y_{2} ordered pairs (x_{1}, k_{1}), (x_{2}, k_{2}) ∈ D × K such that y_{i} = f(x_{i}) + k_{i} for i ∈ {1, 2}. Then we have (x_{i}, y_{i}) ∈ epi_{K} f for i ∈ {1, 2}. Taking into account that the set epi_{K} f is convex, it follows that

((1 − a)x_{1} + ax_{2}, (1 − a)y_{1} + ay_{2}) = (1 − a)(x_{1}, y_{1}) + a(x_{2}, y_{2}) ∈ epi_{K} f.

Thus

(1 − a)x_{1} + ax_{2} ∈ D and (1 − a)y_{1} + ay_{2} ∈ f((1 − a)x_{1} + ax_{2}) + K.

These relations show that (1 − a)y_{1} + ay_{2} ∈ rng f + K. Therefore rng f + K is convex, as claimed. In other words, the function f is K-convexlike. Consequently, the implication f ∈ C_{K}(D, Y) ⇒ f ∈ CL_{K}(D, Y) is true. Now suppose that f ∈ NC_{K}(D, Y). This means that the set epi_{K} f is nearly convex. Hence there exists a number a ∈ ]0, 1[ such that

(1 − a) epi_{K} f + a epi_{K} f ⊆ epi_{K} f.

By reasoning as in the proof of the implication f ∈ C_{K}(D, Y) ⇒ f ∈ CL_{K}(D, Y), we immediately see that this number a satisfies

(1 − a)(rng f + K) + a(rng f + K) ⊆ rng f + K.

Therefore rng f +K is nearly convex, and whence f is nearly K-convexlike. Consequently, the implication f ∈ NC_{K}(D, Y) ⇒ f ∈ NCL_{K}(D, Y) is also true. Finally, suppose that f ∈ CC_{K}(D, Y). We claim that in this case the set cl (rng f + K) is convex. To prove this, let y_{1} and y_{2} be elements of cl (rng f + K), and let a be in ]0,1[. Further, fix any neighbourhood U of the element y = (1 − a)y_{1} + ay_{2}. Since the mapping

(t, u) ∈ Y × Y ⟼ (1 − a)t + au ∈ Y

is continuous at (y_{1}, y_{2}), we can choose an open neighbourhood V_{1} of y_{1} and an open neighbourhood V_{2} of y_{2} such that

(1 − a)V_{1} + aV_{2} ⊆ U.  (4.3)

Taking into account that y_{1}, y_{2} ∈ cl (rng f + K), it results that

(rng f + K) ∩ V_{i} ≠ ∅ for i ∈ {1, 2}.

Thus there exist x_{1}, x_{2} ∈ D and k_{1}, k_{2} ∈ K such that

z_{1} = f(x_{1}) + k_{1} ∈ V_{1} and z_{2} = f(x_{2}) + k_{2} ∈ V_{2}.

Now, note that (x_{1}, z_{1}) and (x_{2}, z_{2}) lie in cl (epi_{K} f). Then it follows that

((1 − a)x_{1} + ax_{2}, (1 − a)z_{1} + az_{2}) ∈ cl (epi_{K} f),

because the set cl (epi_{K} f) is convex. Since W = X ×[(1−a)V_{1} +aV_{2}] is a neighbourhood of the element ((1 − a)x_{1} + ax_{2}, (1 − a)z_{1} + az_{2}), we must have (epi_{K} f) ∩ W ≠ ∅. Hence there exist an x_{0} ∈ D and a k_{0} ∈ K such that

f(x_{0}) + k_{0} ∈ (1 − a)V_{1} + aV_{2}.  (4.4)

From (4.3) and (4.4) it results that (rng f + K) ∩ U ≠ ∅. Since U was an arbitrary neighbourhood of y, we have y ∈ cl (rng f +K). This shows that cl (rng f +K) is a convex set. Hence the function f is closely K-convexlike. Consequently, the last implication of our theorem, namely f ∈ CC_{K}(D, Y) ⇒ f ∈ CCL_{K}(D, Y), is also true.

Remark 4.3. By reversing the arrows in the diagrams (4.1) and (4.2) we obtain implications that are not necessarily true. This assertion results from the following examples.

1. Let D be a nonempty subset of X. Define φ : D → X by φ(x) = x. Then it is immediately seen that the following equivalences are true:

D ∈ C(X) ⇔ φ ∈ C(D, X);
D ∈ NC(X) ⇔ φ ∈ NC(D, X);
D ∈ CC(X) ⇔ φ ∈ CC(D, X).

These equivalences together with Remark 2.4 show that the implications that result from reversing the arrows in the first row of diagram (4.1) are not always true. Consequently, the implications that result from reversing the arrows in the second row of diagram (4.1) are not always true either.

2. Let ϕ : ℝ → ℝ be the function defined by ϕ(x) = x^{2}. This function is ℝ_{+}-convex. Taking into account the implications in the second row of diagram (4.1), it follows that ϕ is also nearly ℝ_{+}-convex and closely ℝ_{+}-convex. But ϕ is not closely convex. Thus, due to the implications in the first row of diagram (4.1), the function ϕ cannot be neither convex nor nearly convex. Consequently, the implications

f ∈ C_{K}(D, Y) ⇒ f ∈ C(D, Y), f ∈ NC_{K}(D, Y) ⇒ f ∈ NC(D, Y),
f ∈ CC_{K}(D, Y) ⇒ f ∈ CC(D, Y)

are not generally true.

3. Consider again the function φ, defined in example 1. For this function we have rng φ = D. Therefore the following equivalences are obvious:

D ∈ C(X) ⇔ φ ∈ CL(D, X);
D ∈ NC(X) ⇔ φ ∈ NCL(D, X);
D ∈ CC(X) ⇔ φ ∈ CCL(D, X).

Moreover, if K is a subset of X, then we also have

D ∈ C_{K}(X) ⇔ φ ∈ CL_{K}(D, X);
D ∈ NC_{K}(X) ⇔ φ ∈ NCL_{K}(D, X);
D ∈ CC_{K}(X) ⇔ φ ∈ CCL_{K}(D, X).

Taking into account all these equivalences and Remark 2.4, it follows that in diagram (4.2) the reversed implications are not always true.

Remark 4.4. The implications that are obtained by reversing the arrows in the implications stated in Theorem 4.2 do not always hold either. To see this, consider the function ϕ : ℝ → ℝ, defined by ϕ(x) = x^{3}. This function is ℝ_{+}-convexlike, because rng ϕ+ℝ_{+} = ℝ. Taking into account the implications in the first row of diagram (4.2), it follows that the function ϕ is also nearly ℝ_{+}-convexlike and closely ℝ_{+}-convexlike. But ϕ is not closely ℝ_{+}-convex, because the closed set

gr ϕ + {0} × ℝ_{+} = {(x, y) ∈ ℝ^{2} | x^{3} ≤ y}

is not convex. Thus, due to the implications in the second row of diagram (4.1), the function ϕ cannot be neither ℝ_{+}-convex nor nearly ℝ_{+}-convex. Consequently, the implications

f ∈ CL_{K}(D, Y) ⇒ f ∈ C_{K}(D, Y), f ∈ NCL_{K}(D, Y) ⇒ f ∈ NC_{K}(D, Y),
f ∈ CCL_{K}(D, Y) ⇒ f ∈ CC_{K}(D, Y)

are not generally true.

When K is a convex subset of Y, then the Theorems 4.1 and 4.2 together with the Remarks 4.3 and 4.4 show that the concept of a closely K-convexlike function is the most general concept among the twelve convexities concerning functions studied in this section. From Theorem 2.6 we immediately obtain the following characterizations of the closely K-convexlike functions.

Theorem 4.5. Let K be a convex cone in Y with int K ≠ ∅, let L = int K, and let f be a function from D into Y. Then the implications indicated by the arrows in the diagram below are true:

f ∈ CCL_{K}(D, Y) ⇒ f ∈ CL_{L}(D, Y)
⇑  ⇓
f ∈ CCL_{L}(D, Y) ⇐ f ∈ NCL_{L}(D, Y).

Next we deal with a weakened convexity for functions that has been discussed in [20]. It is defined as follows. If K is a subset of Y, then a function f : D → Y is called K-subconvexlike if there exists a k_{0} ∈ int K such that for any a ∈ ]0, 1[ and any b ∈ ]0, ∞[ one has

(1 − a) rng f + a rng f + bk_{0} ⊆ rng f + K.  (4.5)

In the special case when Y = ℝ^{n}, this concept has been introduced in [13]. It has been applied not only in [13], but also in [22]. Its relationship with the convexity concepts investigated in the present paper is given by our next result.

Corollary 4.6. Let K be a convex cone in Y with int K ≠ ∅, and let f be a function from D into Y. Then the following assertions are equivalent:

1. f is K-subconvexlike.

2. There exist a k_{0} ∈ int K and an a ∈ ]0, 1[ such that (4.5) holds for all b ∈ ]0, ∞[.

3. f is closely K-convexlike.

Proof. Obviously, the implication 1. ⇒ 2. is true. Next we assume that 2. holds. For short we set L = int K. Let y_{1} and y_{2} be in rng f +L. Then there exist elements k_{1}, k_{2} ∈ L such that y_{i} ∈ rng f +k_{i} for i ∈ {1, 2}. The convexity of L implies that (1−a)k_{1}+ak_{2} ∈ L. But, in addition, the set L is open. Hence there exists a neighbourhood U of o_{Y} such that (1 − a)k_{1} + ak_{2} − U ⊆ L. Since U is absorbing, we can find a number b ∈ ]0, ∞[ such that bk_{0} ∈ U. Therefore we have

(1 − a)k_{1} + ak_{2} − bk_{0} ∈ L.  (4.6)

From (4.5) and (4.6) we obtain

(1 − a)y_{1} + ay_{2} ∈ (1 − a) rng f + a rng f + (1 − a)k_{1} + ak_{2}
= [(1 − a) rng f + a rng f + bk_{0}] + [(1 − a)k_{1} + ak_{2} − bk_{0}]
⊆ rng f + K + L
⊆ rng f + L.

Since y_{1} and y_{2} were arbitrarily chosen in rng f + L, we have shown that rng f + L is a nearly convex set. In other words, f is nearly L-convexlike. By Theorem 4.5 it follows that f is closely K-convexlike. In conclusion, the implication 2. ⇒ 3. is true. Finally, if the function f is closely K-convexlike, then by Theorem 4.5 it is also L-con-vexlike, where as before L = int K. Hence rng f + L is a convex set. We choose any k_{0} ∈ L. Then we have

(1 − a) rng f + a rng f + bk_{0} = (1 − a)(rng f + bk_{0}) + a(rng f + bk_{0})
⊆ (1 − a)(rng f + L) + a(rng f + L)
⊆ rng f + L
⊆ rng f + K

for all a ∈ ]0, 1[ and all b ∈ ]0, ∞[. Thus f is K-subconvexlike. In conclusion, the implication 3. ⇒ 1. is also true.

From Corollary 4.6, which completes the results of Section 3 of the paper [20], we conclude that the class CCL_{K}(D, Y) of all closely K-convexlike functions f : D → Y coincides with the class of all K-subconvexlike functions f : D → Y when K is a convex cone in Y with int K ≠ ∅. Finally, it should be remarked that some of the convexities regarding functions, defined here by reduction to sets, can also be defined otherwise. This is shown by the following elementary theorem.

Theorem 4.7. Let K be a convex cone in Y containing o_{Y}, and let f be a function from D into Y. Then the following assertions are true:

1. f ∈ C_{K}(D, Y) if and only if for all a ∈ ]0, 1[ and all x_{1}, x_{2} ∈ D the conditions

(1 − a)x_{1} + ax_{2} ∈ D,  (4.7)
(1 − a)f(x_{1}) + af(x_{2}) ∈ f((1 − a)x_{1} + ax_{2}) + K  (4.8)

are satisfied.

2. f ∈ NC_{K}(D, Y) if and only if there exists a number a ∈ ]0, 1[ such that for all x_{1}, x_{2} ∈ D the conditions (4.7) and (4.8) are satisfied.

3. f ∈ CL_{K}(D, Y) if and only if for all a ∈ ]0, 1[ and all x_{1}, x_{2} ∈ D the condition

(1 − a)f(x_{1}) + af(x_{2}) ∈ rng f + K  (4.9)

is satisfied.

4. f ∈ NCL_{K}(D, Y) if and only if there exists a number a ∈ ]0, 1[ such that for all x_{1}, x_{2} ∈ D the condition (4.9) is satisfied.

Proof. 1. Necessity. Let x_{1} and x_{2} be points in D. Since o_{Y} ∈ K, we have

(x_{i}, f(x_{i})) ∈ gr f ⊆ epi_{K} f for i ∈ {1, 2}.

Taking into account that the set epi_{K} f is convex, it follows that

((1 − a)x_{1} + ax_{2}, (1 − a)f(x_{1}) + af(x_{2})) = (1 − a)(x_{1}, f(x_{1})) + a(x_{2}, f(x_{2})) ∈ epi_{K} f

for all a ∈ ]0, 1[. Thus (4.7) and (4.8) hold for all a ∈ ]0, 1[. Sufficiency. Let (x_{1}, y_{1}) and (x_{2}, y_{2}) be elements of epi_{K} f. Then (4.7) holds for all a ∈ ]0, 1[. On the other hand, taking into account that y_{i} ∈ f(x_{i}) + K for i ∈ {1, 2} and that (4.8) holds for all a ∈ ]0, 1[, we obtain

\begin{aligned}(1 − a)y_{1} + ay_{2} & ∈ (1 − a)f(x_{1}) + af(x_{2}) + (1 − a)K + aK \\ & ⊆ f((1 − a)x_{1} + ax_{2}) + K \end{aligned}

for all a ∈ ]0, 1[. Thus

(1 − a)(x_{1}, y_{1}) + a(x_{2}, y_{2}) = ((1 − a)x_{1} + ax_{2}, (1 − a)y_{1} + ay_{2}) ∈ epi_{K} f

for all a ∈ ]0, 1[. Consequently, the set epi_{K} f is convex. This means that f is K-convex.

2. By reasoning as in the proof of assertion 1., but for a fixed number a ∈ ]0, 1[, we immediately see that assertion 2. is true.

3. Necessity. Let x_{1} and x_{2} be points in D. Since o_{Y} ∈ K, we have

f(x_{i}) ∈ rng f ⊆ rng f + K for i ∈ {1, 2}.

Due to the convexity of the set rng f + K it follows that (4.9) holds for all a ∈ ]0, 1[. Sufficiency. Let y_{1} and y_{2} be points in rng f + K. Then there exist elements x_{1}, x_{2} ∈ D such that y_{i} ∈ f(x_{i}) + K for i ∈ {1, 2}. In view of (4.9) it follows that

\begin{aligned}(1 − a)y_{1} + ay_{2} & ∈ (1 − a)f(x_{1}) + af(x_{2}) + (1 − a)K + aK \\ & ⊆ rng f + K \end{aligned}

for all a ∈ ]0, 1[. Consequently, the set rng f + K is convex. This means that the function f is K-convexlike.

4. By reasoning as in the proof of assertion 3., but again for a fixed number a ∈ ]0, 1[, we see that assertion 4. is true.

Corollary 4.8. Let f be a function from D into Y. Then the following assertions are true:

1. f ∈ C(D, Y) if and only if for all a ∈ ]0, 1[ and all x_{1}, x_{2} ∈ D the conditions (4.7)

and

(1 − a)f(x_{1}) + af(x_{2}) = f((1 − a)x_{1} + ax_{2})  (4.10)

are satisfied.

2. f ∈ NC(D, Y) if and only if there exists a number a ∈ ]0, 1[ such that for all x_{1}, x_{2} ∈ D the conditions (4.7) and (4.10) are satisfied.

Remark 4.9. The assertions 3. and 4. of Theorem 4.7 are mentioned without proofs in [20, Lemma 3.2].

Remark 4.10. The definitions (a_{2}), (b_{2}), (c_{2}), (a′_{2}), (b′_{2}) and (c′_{2}) as well as the definition of the K-subconvexlike functions do not require that D is a subset of the topological linear space X. All these definitions and the results referring to the functions corresponding to these definitions remain valid even if D is any nonempty set.

5. Characterizations of Weakly K-Minimizers

The K-convex functions as well as their generalizations that we have called here K-con-vexlike and nearly K-convexlike functions, respectively, have been employed for deriving minimax theorems, theorems of the alternative, optimality conditions and duality results (e.g. see [4], [16], [3], [7], [6], [24], [12], [17], [14], [18], [15], [10]). Some of the results involving such functions are also valid for the closely K-convexlike functions. To see this it suffices to transpose the results, stated for sets in Section 3, to functions defined on a nonempty set D and taking values in Y. Indeed, Lemma 3.1 yields the following theorem of the alternative of Gordan type.

Theorem 5.1. Let K be a convex cone in Y with int K ≠ ∅, let D be a nonempty set, and let f : D → Y be a closely K-convexlike function. Then exactly one of the following assertions is true:

1. There exists an x ∈ D such that −f(x) ∈ int K.

2. There exists a y′ ∈ K^{∗} \ {o′} such that y′(f(x)) ≥ 0 for all x ∈ D.

Proof. Take S = rng f and s_{0} = o_{Y}. Then apply Lemma 3.1.

This theorem has been stated in [15]. According to Corollary 4.6 it generalizes Theorem 2.1 proved in [13]. In virtue of Theorem 4.1 it is also a generalization of several other results of the same type (Theorem 2.5.1 in [3], Lemma 2.1 in [7], Lemma 3 in [25], Theorem 4.1 in [10]).

When K is a convex cone in Y with int K ≠ ∅ and f is a function from D into Y, then we say that an element x_{0} ∈ D is a weak K-minimizer of f over a subset E of D if x_{0} ∈ E and if f(x_{0}) is a weakly K-minimal element of the set f(E). By taking into account this definition, Theorem 3.2 yields the next theorem which generalizes several results given in the literature (see, for instance, [21, Theorem 2, p. 105], [17, Theorem 2.10, p. 91] and [22, Theorem 3.1]).

Theorem 5.2. Let K be a convex cone in Y with int K ≠ ∅, let D be a nonempty set, let f : D → Y be a closely K-convexlike function, and let x_{0} be an element of D. Then x_{0} is a weak K-minimizer of f over D if and only if there exists a y′_{0} ∈ K^{∗} \ {o′} such that

y′_{0}(f(x_{0})) = min {y′_{0}(f(x)) | x ∈ D}.

Lemma 3.1 can also be used to derive necessary conditions for weak solutions of constrained multiobjective optimization problems. To see this, let Y_{1} and Y_{2} be real topological linear spaces. Let Y′_{1} and Y′_{2} be their topological duals, respectively. The zero-elements of the spaces Y_{1}, Y_{2}, Y′_{1} and Y′_{2} are denoted by o_{1}, o_{2}, o′_{1} and o′_{2}, respectively.

Theorem 5.3. Let K_{1} ⊆ Y_{1} and K_{2} ⊆ Y_{2} be convex cones with

int K_{1} ≠ ∅ and int K_{2} ≠ ∅,

let D be a nonempty set, let f = (f_{1}, f_{2}) : D → Y_{1} × Y_{2} be a closely K_{1} × K_{2}-convexlike function, and let x_{0} ∈ D be a weak K_{1}-minimizer of f_{1} over the set

E = {x ∈ D | − f_{2}(x) ∈ K_{2}}.  (5.1)

Then the following assertions are true:

1. There exists a (y′_{1}, y′_{2}) ∈ K_{1}^{∗} × K_{2}^{∗} \ {(o′_{1}, o′_{2})} such that

y′_{1}(f_{1}(x_{0})) = min {y′_{1}(f_{1}(x)) + y′_{2}(f_{2}(x)) | x ∈ D}.  (5.2)

2. If (rng f_{2}) ∩ (− int K_{2}) ≠ ∅, then there exists a (y′_{1}, y′_{2}) ∈ K_{1}^{∗} × K_{2}^{∗} with y′_{1} ≠ o′_{1} such that (5.2) holds.

Proof. 1. If we set y_{0} = (f_{1}(x_{0}), o_{2}) and K = K_{1} × K_{2}, then we have

(y_{0} − int K) ∩ rng f = ∅.

By applying assertion 1. of Lemma 3.1 in the space Y_{1} × Y_{2}, we conclude that there is a y′_{0} ∈ K^{∗} \ {o′} such that

y′_{0}(y_{0}) ≤ y′_{0}(y) for all y ∈ rng f.  (5.3)

Now we define the functions y′_{1} : Y_{1} → ℝ and y′_{2} : Y_{2} → ℝ by

y′_{1}(y_{1}) = y′_{0}(y_{1}, o_{2}) and y′_{2}(y_{2}) = y′_{0}(o_{1}, y_{2}),

respectively. Then we have

y′_{0}(y) = y′_{1}(y_{1}) + y′_{2}(y_{2}) for all y = (y_{1}, y_{2}) ∈ Y_{1} × Y_{2}.  (5.4)

In view of this equality, (5.3) can be rewritten as follows:

y′_{1}(f_{1}(x_{0})) ≤ y′_{1}(f_{1}(x)) + y′_{2}(f_{2}(x)) for all x ∈ D.  (5.5)

On the other hand, due to (5.4) it is also seen that

(y′_{1}, y′_{2}) ∈ K_{1}^{∗} × K_{2}^{∗} \ {(o′_{1}, o′_{2})}.

Since −f_{2}(x_{0}) ∈ K_{2} and y′_{2} ∈ K_{2}^{∗}, we have y′_{2}(f_{2}(x_{0})) ≤ 0. Taking into account this inequality, it follows from (5.5) that

y′_{1}(f_{1}(x_{0})) + y′_{2}(f_{2}(x_{0})) ≤ y′_{1}(f_{1}(x_{0})) ≤ y′_{1}(f_{1}(x)) + y′_{2}(f_{2}(x))

for all x ∈ D. Consequently, (5.2) must hold.

2. According to assertion 1. there is a (y′_{1}, y′_{2}) ∈ K_{1}^{∗} × K_{2}^{∗} \ {(o′_{1}, o′_{2})} satisfying (5.2). We must have y′_{1} ≠ o′_{1}. Indeed, if we suppose that y′_{1} = o′_{1}, then we have y′_{2} ≠ o′_{2} as well as

0 = min {y′_{2}(f_{2}(x)) | x ∈ D}.

By applying assertion 2. of Lemma 3.1 it follows that

(rng f_{2}) ∩ (− int K_{2}) = ∅,

which is absurd. Hence we have y′_{1} ≠ o′_{1}, as claimed.

Remark 5.4. If x_{0} is an element of the set E, defined by (5.1), and if there is a (y′_{1}, y′_{2}) ∈ K_{1}^{∗} × K_{2}^{∗} with y′_{1} ≠ o′_{1} such that (5.2) holds, then x_{0} is a weak K_{1}-minimizer of the function f_{1} over E even when the function f = (f_{1}, f_{2}) : D → Y_{1} × Y_{2} is not closely K_{1} ×K_{2}-convexlike. Indeed, for any x ∈ E we have −f_{2}(x) ∈ K_{2}, and thus y′_{2}(f_{2}(x)) ≤ 0. Therefore (5.2) implies

y′_{1}(f_{1}(x_{0})) ≤ y′_{1}(y) for all y ∈ f_{1}(E).

By applying assertion 2. of Lemma 3.1, it follows that

[f_{1}(x_{0}) − int K_{1}] ∩ f_{1}(E) = ∅.

This means that x_{0} is a weak K_{1}-minimizer of f_{1} over E.

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