Minimax Theorems for Extended Real-Valued Abstract Convex–Concave Functions

In this paper, we provide sufficient and necessary conditions for the minimax equality for extended real-valued abstract convex–concave functions. As an application, we get sufficient and necessary conditions for the minimax equality for extended real-valued convex–concave functions.

The first minimax theorem was proved by von Neumann in [5]. Since then there have been many generalizations of the original result, an exhaustive survey is given, e.g., in [6]; see also [7]. To the best of our knowledge, the existing minimax theorems are restricted to real-valued functions. Extended real-valued functions appear in standard constructions in variational analysis. It is a natural question what are the conditions under which the minimax equality for such functions holds. In this paper, we use the tools developed in [8] and [9] to provide minimax theorems for extended real-valued abstract convex-concave functions.

Preliminaries
We provide sufficient and necessary conditions for the minimax equality where X is nonempty set, Y is a real vector space and a : X × Y →R := R ∪ {± ∞} is an extended real-valued Φ-convex (abstract convex) function with respect to x and concave with respect to y. Now, we recall some definitions related to Φ-convexity. For any f, g : X →R, f ≤ g if and only if f (x) ≤ g(x) for all x ∈ X .
Let Φ be defined as follows where L is an arbitrary class of functions : X →R. The class L is called a set of abstract linear functions. The class Φ is called a set of abstract affine functions if it is stable by adding constants. Note that, if L is the set of real-valued linear functions defined on X , then Φ is the set of real-valued affine functions defined on X .
The set supp f ⊂ Φ, defined as Let us note that a Φ-convex function, taking the value − ∞, has an empty support or must contain in its support functions admitting − ∞. In the case where, Φ contains only real-valued functions ϕ : X → R, the Φ-convex function f : X →R is proper, if and only if supp f = ∅ and dom f = ∅.
For any ϕ : X →R and Z ⊂ X we define the strict lower level set of ϕ at the level α ∈ R as Minimax theorems for functions a : X × Y → R ∪ {+ ∞}, where for each y ∈ Y the function a(·, y) : X → R ∪ {+ ∞} is Φ-convex, are based on the following intersection property introduced in [8] and investigated in [10] and [9]. Definition 2.1 Let ϕ 1 , ϕ 2 : X → R be any two functions and α ∈ R. The functions ϕ 1 and ϕ 2 are said to have the intersection property on X at the level α ∈ R iff for every t ∈ [0, 1] (1) where X is a normed space, and X * is a topological dual to X , have the intersection property on X at the level α if and only if For more results along this line see [9].
Let us note that by Rubinov's theorem ( [3], Example 6.6) a proper lower semicontinuous function f : The following theorem has been proved in [10].
The following conditions are equivalent: (i) for every α ∈ R, α < inf x∈X sup y∈Y a(x, y), there exist y 1 , y 2 ∈ Y and ϕ 1 ∈ supp a(·, y 1 ), ϕ 2 ∈ supp a(·, y 2 ) such that the intersection property holds for ϕ 1 , ϕ 2 on X at the level α. (ii) sup y∈Y inf x∈X a(x, y) = inf x∈X sup y∈Y a(x, y).
The proof of this fact will be given as a corollary of Theorem 5.1.

Generalized Intersection Property
Throughout this paper, we use the following convention In [11], the above addition was called inf-addition (the authors considered also supaddition).
To derive necessary and sufficient conditions for the minimax equality for extended real-valued Φ-convex-concave functions we use the generalized intersection property introduced below. Definition 3.1 Let ϕ 1 , ϕ 2 : X →R be any two functions and α ∈ R. The functions ϕ 1 and ϕ 2 are said to have the generalized intersection property on X at the level α ∈ R iff (2) hold: (2) and (3) or (4) or (5) hold: Proof Assume that ϕ 1 , ϕ 2 have the intersection property on X at the level α. It is obvious that (2) holds. Then, the following situations may occur: then ϕ 1 , ϕ 2 have the generalized intersection property on X at the level α.
Below we give examples related to generalized intersection property.
It is easy to see that functions ϕ 1 , ϕ 2 have the generalized intersection property at every level α ∈ R. 2. Let ϕ 1 , ϕ 2 : R →R be defined as follows It is easy to see that for ϕ 1 , ϕ 2 condition (5) does not hold, thus functions ϕ 1 , ϕ 2 do not have the generalized intersection property at any level α.

An Auxiliary Lemma
Now we prove a lemma which is crucial for our results.
Proof Assume that ϕ 1 , ϕ 2 have the generalized intersection property on X at the level α. If [ϕ 1 < α] = ∅, then ϕ 1 (x) ≥ α for all x ∈ X , thus, in view of convention adopted, and [ϕ 2 < α] are nonempty and define the sets T 1 , T 2 , Consider the following cases: Clearly, it must be ϕ 1 (x) < + ∞ and ϕ 2 (x) < + ∞. By (5)a, it cannot be Then, for every t ∈ [t − ε,t + ε], we have t ∈ T 1 . To show that T 2 is open, take anyt ∈ T 2 , i.e., there existsx ∈ X such that Clearly, ϕ 1 (x) and ϕ 2 (x) are finite. Let Since T 1 , T 2 are nonempty and disjoint, and we showed that they are open, we get To show the converse implication, assume now that (6) holds. If t 0 = 0, then Consider now the case where (6) holds with t 0 ∈ ]0, 1[. It is easy to see that (5) is true for t 0 .

Main Results
Now we are in a position to prove necessary and sufficient conditions for the minimax equality for extended real-valued Φ-convex-concave functions. We say that a function f : X →R is concave in the sense of Ky Fan [12] if for any x 1 , x 2 ∈ X and t ∈ [0, 1] there exists

Theorem 5.1 Let X be a set and Y be a real vector space. Let a
If (i) for every α ∈ R, α < inf x∈X sup y∈Y a(x, y), there exist y 1 , y 2 ∈ Y and ϕ 1 ∈ supp a(·, y 1 ), ϕ 2 ∈ supp a(·, y 2 ) such that the generalized intersection property holds for ϕ 1 , ϕ 2 on X at the level α. always holds, we get the required conclusion. Let α < inf x∈X sup y∈Y a(x, y). By Lemma 4.1, there exists t 0 ∈ [0, 1] such that where ϕ 1 ∈ supp a(·, y 1 ), ϕ 2 ∈ supp a(·, y 2 ), y 1 , y 2 ∈ Y . Hence By concavity of a(x, ·) and by (9), there exists y 0 ∈ Y such that a(x, y 0 ) ≥ α for all x ∈ X. (ii) ⇒ (i) We need only to consider the case inf x∈X sup y∈Y a(x, y) > − ∞. Let α < inf x∈X sup y∈Y a(x, y). By the equality, sup y∈Y inf x∈X a(x, y) = inf x∈X sup y∈Y a(x, y), we get sup y∈Y inf x∈X a(x, y) > α.
Thus, the functionφ := α belongs to the support set supp a(·,ȳ). By the fact that [φ < α] = ∅, we get Then, for all ϕ ∈ Φ, the functionsφ and ϕ have the generalized intersection property on X at the level α.

Proof of Theorem 2.1
The proof follows directly from Remark 3.1a and Theorem 5.1.
Let us note that in the proof of Theorem 5.1 the roles of a(·, y) and a(x, ·) are not symmetric, i.e., one cannot get the conclusion of Theorem 5.1 under the assumption that a(x, ·) is -concave for a certain class .
By examining the proof of Theorem 5.1, we see that the fact, that functions a(·, y) are pointwise suprema of functions from Φ is not used. What is needed in the proof is that, for α < inf x∈X sup y∈Y a(x, y) there exist y 1 , y 2 ∈ Y and any functions ϕ 1 , ϕ 2 : X →R, ϕ 1 ≤ a(·, y 1 ), ϕ 2 ≤ a(·, y 2 ), satisfying the generalized intersection property on X at the level α.
This allows to formulate the following theorem.

Theorem 5.2 Let X be a set and Y be a real vector space.
Let a : X × Y →R be such that: -for any x ∈ X the function a(x, ·) : Y →R is concave on Y .
Proof The same as the proof of Theorem 5.1.
In Theorem 5.1 we can change the assumptions on function a(·, ·) symmetrically. It is possible to assume that a(·, ·) is convex as a function of X and Φ-concave as a function of y, i.e., is equal to the pointwise infimum of all functions ϕ grater than or equal to a(·, ·). In such case condition (i) of Theorem 5.1 has to hold for all α > sup y∈Y inf x∈X a(x, y) and definition of generalized intersection property with opposite inequalities.

Φ-convexity of Convex Functions
In this section, we present a class Φ such that all convex functions defined on R n are Φ-convex (Theorem 6.1). This class is used in next section to provide minimax theorem for convex-concave functions.
We recall that x = (x 1 , . . . , x n ) T ∈ R n is said to be "lexicographically less" than y = (y 1 , . . . , y n ) T ∈ R n , denoted x < L y, if x = y and for k = min{i ∈ {1, . . . , n} | x i = y i } we have x k < y k .
For k ∈ {0, 1, . . . , n}, we denote by L(R n , R k ) the set of all linear mappings u : R n → R k .
Following [13], for any u ∈ L(R n , R k ), with rank u = k, z ∈ R k , x * ∈ (R n ) * and d ∈ R, we define ϕ u,z,x * ,d : R n →R by In the example, below we investigate functions ϕ 1 , ϕ 2 : R →R, ϕ 1 , ϕ 2 ∈Φ which have the generalized intersection property on R at the given level α.
In this setting we have k = 0 or k = 1, then either u 1 ≡ 0 or u 1 is linear function from R to R.

Minimax Theorems for Convex-Concave Functions
Taking into account Theorem 6.1, we can formulate the minimax theorem for convexconcave functions.
Theorem 7.1 Let Y be a real vector space. Let a : R n × Y →R be such that -for any y ∈ Y the function a(·, y) : R n →R is convex on R n , -for any x ∈ R n the function a(x, ·) : Y →R is concave on Y .
The following conditions are equivalent: (i) for every α ∈ R, α < inf x∈R n sup y∈Y a(x, y), there exist y 1 , y 2 ∈ Y and ϕ 1 ∈ supp a(·, y 1 ), ϕ 2 ∈ supp a(·, y 2 ) such that the generalized intersection property holds for ϕ 1 , ϕ 2 on R n at the level α. (ii) sup y∈Y inf x∈R n a(x, y) = inf x∈R n sup y∈Y a(x, y).
Proof Follows from Theorems 6.1 and 5.1.

Conclusions
We provide minimax theorems for extended real-valued Φ-convex-concave functions (or symmetrically for convex-Φ-concave functions). A distinguished feature of these results is that we do not need any topological structure on the spaces involved. On the other hand, the results obtained are applicable to functions being pointwise suprema of abstract affine functions. In particular, we obtain minimax theorems for extended real-valued convex-concave functions which do not have to be proper or lower (upper) semicontinuous.