A THEOREM OF THE ALTERNATIVE AND A TWO-FUNCTION MINIMAX THEOREM

“Theorem of the alternative” is the generic name of different results used as an important tool in optimization. Though the early versions have been obtained under strong conditions of convexity, for the further extensions, these conditions have been weakened using generalized convexity beyond vector space structure. This is the case of the result proved by Jeyakumar [4], who used an extension of Fan’s convexlike property. Our theorem of the alternative, proved in Section 3, extends the cited result in the case of the usual order of the n-dimensional Euclidean space, and makes use of the weakened convexlike property introduced in Section 2. The two-function minimax theorem extends the classical minimax inequality to the case of two functions. It is motivated both by the equilibrium problem in the noncooperative game theory and by the generalized duality in optimization. The first known result is due to Fan [3]. Several authors, especially in the last two decades, produced different versions of this theorem, employing the basic techniques from single-function minimax theory. We refer the reader to Simons [9] for a valuable survey in this framework. New results, based mainly on special connectedness properties, are obtained by Kindler in [5]. The main theorem of Section 4 is based on the theorem of the alternative proved in Section 3. For both results, one requires neither vector space structures nor topological support.


Introduction
"Theorem of the alternative" is the generic name of different results used as an important tool in optimization.Though the early versions have been obtained under strong conditions of convexity, for the further extensions, these conditions have been weakened using generalized convexity beyond vector space structure.This is the case of the result proved by Jeyakumar [4], who used an extension of Fan's convexlike property.
Our theorem of the alternative, proved in Section 3, extends the cited result in the case of the usual order of the n-dimensional Euclidean space, and makes use of the weakened convexlike property introduced in Section 2.
The two-function minimax theorem extends the classical minimax inequality to the case of two functions.It is motivated both by the equilibrium problem in the noncooperative game theory and by the generalized duality in optimization.The first known result is due to Fan [3].Several authors, especially in the last two decades, produced different versions of this theorem, employing the basic techniques from single-function minimax theory.We refer the reader to Simons [9] for a valuable survey in this framework.New results, based mainly on special connectedness properties, are obtained by Kindler in [5].
The main theorem of Section 4 is based on the theorem of the alternative proved in Section 3.For both results, one requires neither vector space structures nor topological support.subsets of A, and, if E is a subset of a linear space, then co E stands for the convex hull of E. Denote also by ∆ n the standard (n − 1)-dimensional simplex.
We briefly review some notions which extend the classical convexity beyond vector spaces. Let The concept of convexlike functions is due to Fan [3].Weakening this property, König [6] introduced t-convexlike functions (for t = 1/2) in order to extend Fan's minimax theorem.The concept of t-subconvexlike functions originates from Craven and Jeyakumar [2] but, in a particular form, it was introduced in Neumann [7].
Definition 2.9.Ᏺ is t-weakly convexlike on X if Ᏺ is weakly convexlike on X if it is t-weakly convexlike for all t ∈ [0,1].
Remark 2.10.Ᏺ is t-weakly convexlike for some t if and only if coᏲ is t-weakly convexlike.
For the converse implications, consider the following simple counterexamples.
Example 2.14.Let Z * be the set of all nonzero integers.Take One easily verifies that sup On the other hand, for any Thus, both Ᏺ and coᏲ are weakly convexlike.Moreover, since sup for every (t 1 ,t 2 ,...,t n ) ∈ ∆ n and for any x 1 ,x 2 ,...,x n ∈ X, it follows that Ᏺ is affine weakly convexlike.
Example 2.15.Let N * be the set of all positive integers.Take (2.8) Hence, coᏲ (Ᏺ) is weakly concavelike and Ᏺ is affine weakly concavelike.However, Ᏺ is not t-subconcavelike for any t ∈ (0,1).One can verify that there does not exist a dense set D ⊆ [0,1] such that, for every t ∈ D, the inequality admits solutions in X independent of a, if is small enough.For instance, if < 1/12, there are no t in the interval (3 /2,1 − ) satisfying (2.10) for an appropriate x ∈ X and all a ∈ N * .

A general theorem of the alternative
Theorem 3.1.Let Ᏺ be a finite family of real-valued functions defined on a nonvoid set X.If every Ᏼ ∈ σ(coᏲ) is weakly convexlike, then exactly one of the following two situations occurs: The proof of the theorem is based on the following three lemmas.
Proof.As in the proof of Lemma 3.3, choose t and observe that ∀i = 1,2,...,n, Note that condition (3.3) was first used in Stefanescu [10] for proving a general minimax theorem.

Proof of Theorem 3.1. The proof follows by induction on n = |Ᏺ|.
For n = 2, the alternative (3.1) follows directly from Lemmas 3.2 and 3.3.Assume that the theorem holds for every family of at most n − 1 functions, n ≥ 3, and let Ᏺ = { f 1 , f 2 ,..., f n } satisfy max 1≤i≤n f i (x) ≥ 0 for all x ∈ X.
Denote X = {x ∈ X | f i (x) < 0, i = 3,...,n}.If X = ∅, then the conclusion follows from the inductive assumption.Otherwise, max i=1,2 f i (x) ≥ 0 for all x ∈ X .Then, by Lemma 3.4, it follows that { f 1 , f 2 } satisfies the assumptions of Lemma 3.2 with respect to X .Hence, there exists f 0 ∈ co{ f 1 , f 2 } such that f 0 (x) ≥ 0 for all x ∈ X .Moreover, max{ f 0 (x), f 3 ,..., f n (x)} ≥ 0 for all x ∈ X.By induction, there exists f ∈ co{ f 0 , f 3 ,..., By Remark 2.10, coᏲ is weakly convexlike if Ᏺ is weakly convexlike.But unlike in the case of previously known convexlike properties, a subfamily of a weakly convexlike family of functions is not necessarily weakly convexlike too so that in Theorem 3.1 it is not sufficient to assume that Ᏺ is weakly convexlike.However, we can observe that all that we need for proving Theorem 3.1 is the following condition.
The following corollary of Theorem 3.1 easily follows from Remarks 2.13 and 2.5.It is not else than the alternative theorem of Jeyakumar [4] for the usual (partial) order of R n .Corollary 3.5.Let Ᏺ be t-subconvexlike for some t ∈ (0,1).Then the alternative (3.1) holds.
Example 2.14 can be used once more to show that Theorem 3.1 generalizes previously known alternative theorems concerning convexlike-type functions.
It is easy to see that the alternative (3.1) holds for any finite subset of Ᏺ (and of co Ᏺ as well), but the conditions of Corollary 3.5 are not satisfied.Obviously, co Arguing as in Example 2.14, one can verify that any finite subfamily is not t-subconvexlike for any t ∈ (0,1), so that Corollary 3.5 fails.
On the other hand, we can show that Theorem 3.1 can be applied in this example.All that we must do is to verify that any is equivalent to the inequality inf x∈X max 1≤i≤k α i x ≤ max 1≤i≤k α i x and the latter results from a simple calculation.
Remark 3.6.We may ask whether Jeyakumar's alternative theorem can be generalized in the more abstract framework of an order induced by an arbitrary convex cone.The appropriate reformulation of Definition 2.9 in terms of such order is an open problem.Moreover, since our approach involves only quantitative relations, the technique of proof is inadequate in an abstract framework.

A two-function minimax theorem
The main result of this section is obtained via the previous theorem of the alternative.Since X is any finite subset of X, this proves the theorem.
For the case when f (x, y) ≤ g(x, y), for all (x, y) ∈ X × Y , which is usually considered in the two-function minimax theorems, we easily obtain the following theorem.Theorem 4.2.Let X, Y be any nonvoid sets and f , g two real-valued functions defined on the product set X × Y such that f (x, y) ≤ g(x, y) for all (x, y) ∈ X × Y .If ( The proposition also holds if one assumes Y ᾱ( x) to be compact, for some x ∈ X, and Y α (x) closed in X, for all x ∈ X and α ≤ ᾱ.From Proposition 4.3 and Theorems 4.1 and 4.2, we easily obtain the following twofunction minimax theorems.Theorem 4.5.Let X, Y be any nonvoid sets and f , g two real-valued functions defined on X × Y .Assume condition ( * ) to be satisfied.If (4.1) and (4.2) hold, then (4.8) holds.
Theorem 4.6.Let X, Y be any nonvoid sets and f , g two real-valued functions defined on the product set X × Y , such that f (x, y) ≤ g(x, y) for all (x, y) ∈ X × Y .Assume condition ( * ) to be satisfied.If (4.2) and (4.7) hold, then (4.8) holds.
Based on the examples of Section 2, one can see that the above results are independent of most known similar results.
An easy consequence of the last two theorems is a slight generalization of the twofunction minimax theorem proved by Cheng and Lin in [1].
Corollary 4.7.Let X, Y be any nonvoid sets and f , g two real-valued functions defined on the product set X × Y , such that f (x, y) ≤ g(x, y) for all (x, y) ∈ X × Y .Assume that there exists t,s ∈ (0,1) such that Ᏺ is t-subconvexlike and Ᏻ is s-subconcavelike.If condition ( * ) holds, then (4.8) holds.
Theorem 4.1.Let X, Y be any nonvoid sets and f , g two real-valued functions defined on the product set X × Y , such that∀x 1 ,...,x n ∈ X, ∀ t 1 ,...,t n ∈ ∆ n , supProof.Let X ∈ σ(X) and suppose that inf y∈Y max x∈X f (x, y) ≥ α, for some α.Apply the theorem of the alternative to ( f (x,•) − α) x∈X .It follows the existence of the nonnegative numbers t x , x ∈ X , x∈X t x = 1, such that We denoteY α (x) = {y ∈ Y | f (x, y) ≤ α}, x ∈ X, α ∈ R.There exists ᾱ ≥ inf y∈Y sup x∈X f (x, y) such that for all α ≤ ᾱ, the family (Y α (x)) x∈Xhas nonempty intersection whenever it has the finite intersection property.