On minimax theorems for sets closed in measure

The paper is dedicated to the memory of Yuri Abramovich, who was dear friend for both of us.

This article is devoted to the К у Fan minimax theorem for convex sets closed in measure in L1. In general, these sets do not carry any formal compactness properties for any reasonable topology.

Famous Fan’s minimax theorem, stated in its generality, claims that for a. function Ф(ж, у) defined on X x Y (where X and Y have no linear or convex structures) which satisfies a. mild convex-concave-like conditions, the minimax equality holds provided the set X is compact (see [5] for a. simple proof which we will use later for our generalization).

Bukhvalov and Lozanovsky invented in [7] the Optimization Without Compactness (OWC) technique for sets in L¹ (see [6] for a. survey of the current state of art; an elementary exposition is given in [11]). Roughly speaking norm bounded convex sets in L¹, which are closed with respect to convergence in measure, have many properties usually associated with compact sets only.

Our main goal is to derive a. minimax theorem without compactness (Section 1) and further to extend it with milder convexity assumptions (Section 2). Section 3 briefly describes possible applications.

1.    Main Theorem

Our assumptions will be the following (all sets below are expected to be non empty). Let (T, S,/z) be a. cr-finite measure space, and let X be a. convex, norm-bounded set in VfpY Let Y be an arbitrary set and Ф : X x У -> R be a function such that

• (a)    for each у E Y the function ж H Ф(ж,у) is convex;

• (b)    for each t E [0,1] and for every yi,y₂ G U there exists уз EY with

Ф(ж,у3) > tФ(ж,yl) + (1 - ^Ф(ж,у2)

for every ж E X.

Condition (b) is more general than concavity. The latter is too restrictive for many applications. It is said that Ф(ж,у) is concave-like in y. This class has a. much wider area, of applications (see below).

Theorem 1. Suppose that X, Y and Ф satisfy the conditions above. Suppose that X is closed in measure and Ф(-,у) is lower semicontinuous in measure on X for each у eY. Then min sup Ф(ж, y) = sup min Ф(ж, у).                         (1)

$GX y^y         y^y xEX

We will start by justifying ‘min’ instead of ‘inf in (1). The following well-known consequence of [7] is stated in [13], [11].

Lemma 1. Let X be a convex norm-bounded set in L¹^^ which is closed in measure. Let /: X —> JR be a. quasi-convex function which is lower semicontinuous in measure. Then / attains its minimum in X.

Lemma 2. Let X, Y and Ф be as in Theorem 1. Then the function

/(ж) := зир{Ф(ж,у): у G У} is quasi-convex and lower semicontinuous in measure on X.

Proof of Theorem 1. Let p := min sup Ф(ж, у).

If p = — oo there is nothing to prove since the inequality inf зирФ(ж,у) > sup inf Ф(ж,у) тех y^Y         yEY x^X always holds.

Let a be any real such that а < p. By our assumptions each set

C(y) := {ж E X: Ф(ж,у) < a} is norm bounded, convex and closed in measure for any y. Since the assumption а < p implies that the whole intersection of such sets is empty, from Theorem 1.3 in [6] we deduce that there exist yi,... ,y„ G У such that Cjyi) A С(у%) П • • • П C*(yn) = 0. If we assume that a > min sup Ф(ж,уг), then there should exist жд E X such that a > Ф(жд,уг) for all yy г = l,...,n. Hence we would obtain that жд G C^y^, i = 1,... ,n. This contradiction shows that a < min sup Ф(ж,уг). жСХ | у,у,,

Consider now the set

E := {(z, r) E K"⁺¹ : (3 ж E X ) Ф(ж,yj <  г + zy i = 1,... , n}.

The set E is convex. Indeed, if

Ф(ж,уг) ^ r + Zj, г = 1,... ,n,

Ф(ж,уг) < г + г = 1,... ,П- then, for t G [0,1] and for ж = tx + (1 — £)ж we have

Ф(ж,Уг) = Ф(tж + (1 - t^X,^ < Ef^X,^ + (1 - ^Ф(ж,Уг)

< t(r + z^ + (1 - t^r + z^ = tr + (1 - t)r + tZi + (1 - t^Zi . ч.^^^^,^,^^^™^  ^S'^^^^^—*V^^^^^—*^/

Then, the point

(

0,1 + max Ф(ж, у,) iE«Cn /

G >”+1

is interior to E for any ж G X. Indeed, if r < e and \z^\ < E, i = 1,... , n, then, putting ж = maxi^^n Ф(ж, yj ^we get Ф(^ yj ^ 1 + Ф(ж, у,) + ^ + r provided ^ + r| <  2e ^ 1. Also, by construction (0,a) ^ E. Indeed, Ф(ж, у,) < a, i = 1,... , n, for some ж G X implies that

min sup Ф(ж,у_г) <  а ®EX i^i^_n

which contradicts the choice of a.

Now we can apply the Separation Theorem to the point (0, a) ^ E and to the convex set E in the finite dimensional space ]Rⁿ⁺¹: note that it is not difficult to prove that the set E is also closed, though we do not need this to apply the Separation Theorem.

So we can find a vector (Ai,... , An,r) ^ 0 with

n

E X^z^ + rr^ra i=l

((z,r) e Ey

It is clear that E + >"⁺¹ С E. If it were r < 0 then, taking r > 0 one would come to a contradiction with (2). This proves that r > 0. Analogously, if А_г< 0 for some г, then taking the corresponding entry ^ > 0 we would come to a contradiction with (2). Again this proves that А_г > 0 for each г = l,...,n. Note that actually r > 0 as the point (0,1 + maxi^^n Ф(ж, y,)) lies in the interior of E. Indeed, if r = 0, then 52"₌₁ A^ ^ 0 for all (z,^ G E. Taking (z,^ with z^ negative, and sufficiently close to 0, we find

n

E Xa < 0 i=l

since A, > 0, although (z,r) is still a point in E.

The point (Ф(ж,у_г) + r, —r) lies in E for all ж G X and all r G R. Indeed, Ф(ж,у_г) < Ф(ж,уг) Tv — r. Then, the inequality (2) implies that

У2 А, (ф(ж, Уг) + r) + r( —r) >  та. г=1

Since r > 0, dividing (3) by r we have

E ^"ф(ж,Уг) + г=1 Г

for all ж G X and all r G Ж.

Since г E Ж is arbitrary, we derive that ^^ тг = 1. So, (4) can be rewritten as i=i Г

— Ф(ж,Уг) > а, i=l Г where on the left-hand side we have a convex combination of the n points Ф(ж,yj. Then condition (b) implies, by induction, that there exists у E У such that Ф(ж,у) > a for all ж E A, whence sup min Ф(ж, у) > а.

y£Y -'-С А

Since a is arbitrary underneath p, we achieve min sup Ф(ж, у) = sup min Ф(ж, у). > жСХ y^Y y^y xEX

In Section 3 we will discuss some possible application of Theorem 1; for instance, it is usual to connect minimax equality with results concerning the consistency of a system of inequalities. We explicitly present a result in this direction, which can be derived from (the dual version of) Theorem 1.

Proposition 1. Let ft be a. convex set of concave functionals defined on a set X C L¹^^ which is closed, norm-bounded and closed in measure. Also assume that each / E ft is upper semicontinuous in measure on X. Suppose that for every f Gft there exists Xf E X such that f(,x^ ^ 0. Then there exists a point xq G X such that /(жд) > 0 for all / E ft.

REMARKS. (1) Since OWC technique is true for more general situation then L1, e. g., for perfect Banach Function Spaces and their vector-valued generalizations when the corresponding Banach space of values is reflexive (see [6]), then the results here and below extend to that more general setting.

• (2)    Using OWC technique, Levin derived a minimax theorem (see [6; Theorem 4.1]) but he confined himself with convex-concave setting, which is both easier for the proof and not much interesting in applications. Actually, only the approach from [5] gave us the tools for the current general result.

2.    Weaker Convexity Conditions

In the literature connected with minimax relationships and related topics, many generalizations of the notion of convexity have been proposed.

DEFINITION 1. A function Ф : X x Y —> > is finite convex-like with respect to ж if, for each finite set {yi,... ,y„} C Y, each pair ж₁₅ж₂ E X and every t E [0,1], an element жз E X exists, such that

Ф(жз,Уг) < ^Ф(жъуг) + (1 - ^Ф(ж2,Уг)                      (5)

for every г = 1,... , n.

This definition has been introduced by Granas and Liu [9] in the following more general way:

DEFINITION 2. A function Ф : X x Y —> JR is midpoint finite convex-like with respect to ж if, for each finite set {yx,..., y„} C Y and each pair жх, ж₂ E X. an element жз E X exists, such that

Ф(жз,Уг) < 2 [Ф(Ж1’Уг) + Ф(ж2,Уг)] ,                          (6)

for every i = 1,... , n.

It is quite clear that if condition (a) of Section 1 is replaced by

(ai) Ф is finitely convex-like with respect to ж;

(a₂) for every у E Y the map ж H Ф(ж,у) is quasi-convex, then Theorem 1 is still true.

It is less trivial to note that the result remains true if (ai) is replaced by the midpoint finite convex-likeness with respect to ж. In fact the following result holds:

Proposition 2. Let X be a convex, norm-bounded set in L¹^^, Y any non-empty set and Ф: X x Y —> JR be such that

• (i)    for each у E У, the map ж н Ф(ж, у) is lower semicontinuous in measure;

• (ii)    for each у G Y, the map ж н Ф(ж,у) is quasi-convex;

• (iii)    Ф is finitely midpoint convex-like with respect to ж.

Then Ф is finitely convex-like with respect to ж.

< Let {yi,... , y„} C Y and жх, ж₂ E X be fixed.

Claim 1: For every q E Q(2) (the set of dyadic rational numbers in [0,1]) there exists ж_д G X such that

Wq^ < уФ(жх,у^) + (1 - д)Ф(ж_д,у,)                   (7)

for each г = 1,... , n.

This is a standard iterative argument. The proof is that of ([17; Lemma 3.2]).

Claim 2: Let ^C_n^_n be a non increasing sequence of convex subsets of L¹^^ which are closed in measure. Then

Q c_n = {ж E L¹^: ж = (у)-Нтж„, ж. E C_n} =: В. n

The inclusion Q_n C_u С В is immediate. Conversely, since the sequence is decreasing, and each set C_n is closed in measure, it is clear that В C C_n for every n.

Claim 3: Ф is finitely convex-like with respect to ж.

Let t E [0,1] be fixed, and let (£&)& be a sequence in Q(2) with t^ —t t. From Claim 1 we can determine ж^ E X such that

Ф(^,Уг) < 1д.Ф(ж1,уг) + (1 - ^)Ф(ж2,Уг)

for every i = 1,... , n and every к E N.

Since (ж^.) С X, and X is by assumption norm bounded, by Theorem 1.4 in [6] there are:

a sequence of integers 1 = ki < k^ < . . . , a sequence of non-negative numbers (A_p)_p with ^i + l —1                                                        ^i + 1—1

E

Xj = 1 and such that the sequence g_n =

E ^з

converges p-а. e. to an element

3=k ж G E.

3=ki

For each i = 1,... , n and every p, define the number

M('i,p) = тах{^Ф(жх, yj + (1 - ^)Ф(ж2,уг), ; = kp,... , kp+1 - 1} and note that, since tj —> t, then

Нт[^Ф(жх,Уг) + (1 - ^)Ф(ж₂,уг)] = 1Ф(ж_х,у_г) + (1 - 1)Ф(ж₂,у_г) = : Тг 3

for every i = 1,... , п. Hence lim_p M^i,p) = ту for i = 1,... , n.

Set now

Cp_p = C(M(i,p)) = ^EX: Ф(ж,у) <  M(i,p), i = l,...,n, p£^

and observe that (i) and (ii) ensure that each Cp_p is convex and closed in measure. Hence, since for each integer к in \к_р,к_рдд — 1] the corresponding ж/; E Ср_р for г = 1,... ,n then g_p G Сф_р for i = 1,... , n and p E N.

By a diagonal argument, we can find a subsequence {p_r,r E N} such that (Af(/,p)) is monotone for i = 1,... , n. Let

Ji = ^i E {1,... , n} such that (Af(/,p_r))_r is non decreasing}, J2 = ^ E {1,... , n} such that (Af(/,p_r))_r is non increasing}.

Now, for г E Ji the sequence (C'gpjr is non decreasing with respect to r, and Ср_Рт С С(т_г). Hence g_Pr E С^тф. Since С^тф is closed in measure, and ^g_Pr //-converges to ж, ж E С(т_г), namely,

ф(ж, yj <  T_t = ^(ж_Х, yj + (1 - ^Ф(ж₂, Уг)

for г Е Ji-

Рог г Е J₂, the sequence (G^jr is non increasing; as g_Pr E Ci_iPr1 by Claim 2, ж E П_г Cp_Pr. But, as it is easily seen, in this case