Minimax theorems in a fully non-convex setting

TL;DR

This paper proves two minimax theorems for functions without the usual quasi-concavity assumption, broadening the theoretical framework for non-convex optimization problems.

Contribution

It introduces minimax theorems applicable to fully non-convex functions, expanding the scope of minimax theory beyond traditional convexity constraints.

Findings

Established two minimax theorems without quasi-concavity assumptions

Extended minimax theory to fully non-convex functions

Presented applications of the new theorems

Abstract

In this paper, we establish two minimax theorems for functions $f:X\times I\to {\bf R}$, where $I$ is a real interval, without assuming that $f(x,\cdot)$ is quasi-concave. Also, some related applications are presented.

Equations

$$\underset{Y}{\sup }\underset{X}{\inf }f=\underset{X}{\inf }\underset{Y}{\sup }f$$

$$\underset{Y}{\sup }\underset{X}{\inf }=\underset{X}{\inf }\underset{Y}{\sup }f.$$

$$X=I=\{(t,s)\in {\mathbf{R}}^2:t^2+s^2=1\}$$

$$f(t,s,u,v)=tu+sv$$

$$\underset{X}{\sup }\underset{I}{\inf }f=-1<1=\underset{X}{\inf }\underset{I}{\sup }f.$$

$$\underset{Y_n}{\sup }\underset{X}{\inf }f=\underset{X}{\inf }\underset{Y_n}{\sup }f$$

$$\underset{Y}{\sup }\underset{X}{\inf }f=\underset{X}{\inf }\underset{Y}{\sup }f.$$

$$F(y)=\left\{u\in X:f(u,y)=\underset{x\in X}{\inf }f(x,y)\right\}.$$

$$\left\{x\in X:\underset{v\in V}{\inf }f(x,v)<\rho \right\}\subseteq K.$$

$$\left\{x\in X:\underset{v\in V}{\inf }f(x,v)<\rho \right\}=\bigcup _{v\in V}\left\{x\in X:f(x,v)<\rho \right\}$$

$$\bigcup _{v\in V}\left\{x\in X:f(x,v)<\rho \right\}\subseteq K.$$

$$f(\tilde{u},\tilde{y})\le \underset{\alpha }{\mathrm{limsup}}f(u_{\alpha },y_{\alpha }).$$

$$\underset{\alpha }{\mathrm{limsup}}f(u_{\alpha },y_{\alpha })<r<f(\tilde{u},\tilde{y}).$$

$$f(u_{\alpha },y_{\alpha })<r$$

$$r<f(u_{{\alpha }_3},y_{{\alpha }_3})$$

$$f(\tilde{u},\tilde{y})\le \underset{\alpha }{\mathrm{limsup}}f(u_{\alpha },y_{\alpha })\le \underset{\alpha }{\lim }f(x,y_{\alpha })=f(x,\tilde{y}).$$

$$\underset{I_n}{\sup }\underset{X}{\inf }f=\underset{X}{\inf }\underset{I_n}{\sup }f.$$

$$F(\lambda )=\left\{u\in X:f(u,\lambda )=\underset{x\in X}{\inf }f(x,\lambda )\right\}$$

$$\Phi (x)=\left\{\mu \in I_n:f(x,\mu )=\underset{\lambda \in I_n}{\sup }f(x,\lambda )\right\}$$

$$f(\tilde{x},\tilde{\lambda })=\underset{x\in X}{\inf }f(x,\tilde{\lambda })=\underset{\lambda \in I_n}{\sup }f(\tilde{x},\lambda ).$$

$$\underset{I}{\sup }\underset{X}{\inf }f<r<\underset{X}{\inf }\underset{I}{\sup }f.$$

$$G(\lambda )=\{x\in X:f(x,\lambda )<r\}$$

$$\Phi (x)=\left\{\mu \in I:f(x,\mu )=\underset{\lambda \in I}{\sup }f(x,\lambda )\right\}$$

$$f(\hat{x},\hat{\lambda })<r<\underset{X}{\inf }\underset{I}{\sup }f\le \underset{\lambda \in I}{\sup }f(\hat{x},\lambda )=f(\hat{x},\hat{\lambda })$$

$$X=\bigcup _{\lambda \in \mathbf{R}}\lambda K.$$

$$\underset{x\in X}{\inf }\Vert f(x)\Vert <c<\Vert f(0)\Vert ,$$

$$\{x\in X:x=\tilde{\lambda }f(x)\}$$

$$\phi (x,\lambda )=\Vert x-\lambda f(x){\Vert }^2-c^2{\lambda }^2+2\alpha \lambda$$

$$\phi (x,\lambda )=\Vert x{\Vert }^2+(\Vert f(x){\Vert }^2-c^2){\lambda }^2-2(⟨x,f(x)⟩-\alpha )\lambda .$$

Full text

Minimax theorems in a fully non-convex setting

Dedicated to Professor Wataru Takahashi, with esteem and friendship, on his 75th birthday

BIAGIO RICCERI

Abstract. In this paper, we establish two minimax theorems for functions $f:X\times I\to{\bf R}$, where $I$ is a real interval, without assuming that $f(x,\cdot)$ is quasi-concave. Also, some related applications are presented.

Keywords. Minimax theorem; Connectedness; Real interval; Global extremum.

2010 Mathematics Subject Classification. 49J35; 49K35; 49K27; 90C47.

The most known minimax theorem ([7]) ensures the occurrence of the equality

[TABLE]

for a function $f:X\times Y\to{\bf R}$ under the following assumptions: $X$, $Y$ are convex sets in Hausdorff topological vector spaces, one of them is compact, $f$ is lower semicontinuous and quasi-convex in $X$, and upper semicontinuous and quasi-concave in $Y$.

In the past years, we provided some contributions to the subject where, keeping the assumption of quasi-concavity on $f(x,\cdot)$, we proposed alternative hypotheses on $f(\cdot,y)$. Precisely, in [2], we assumed the inf-connectedness of $f(\cdot,y)$ and, the same time, that $Y$ is a real interval, while, in [5], we assumed the inf-compactness and uniqueness of the global minimum of $f(\cdot,y)$.

In the present paper, we offer a new contribution where the hypothesis that $f(x,\cdot)$ is quasi-concave is no longer assumed.

Let $T$ be a topological space. A function $g:T\to[-\infty,+\infty[$ is said to be relatively inf-compact if, for each $r\in{\bf R}$, there exists a compact set $K\subseteq T$ such that $g^{-1}(]-\infty,r[)\subseteq K$. Moreover, $g$ is said to be inf-connected if, for each $r\in{\bf R}$, the set $g^{-1}(]-\infty,r[)$ is connected. For the basic notions on multifunctions, we refer to [1].

Our main results are as follows:

THEOREM 1. - Let $X$ be a topological space, let $I$ be a real interval and let $f:X\times I\to{\bf R}$ be a continuous function such that, for each $\lambda\in I$, the set of all global minima of the function $f(\cdot,\lambda)$ is connected. Moreover, assume that there exists a non-decreasing sequence of compact intervals, $\{I_{n}\}$, with $I=\cup_{n\in{\bf N}}I_{n}$, such that, for each $n\in{\bf N}$, the following conditions are satisfied:

$(a_{1})$*  the function $\inf_{\lambda\in I_{n}}f(\cdot,\lambda)$ is relatively inf-compact ;*

$(b_{1})$*  for each $x\in X$, the set of all global maxima of the restriction of the function $f(x,\cdot)$ to $I_{n}$ is connected.*

Then, one has

[TABLE]

THEOREM 2. - Let $X$ be a topological space, let $I$ be a compact real interval and let $f:X\times I\to{\bf R}$ be an upper semicontinuous function such that $f(\cdot,\lambda)$ is continuous for all $\lambda\in I$. Assume that:

$(a_{2})$*  there exists a set $D\subseteq I$, dense in $I$, such that the function $f(\cdot,\lambda)$ is inf-connected for all $\lambda\in D$ ;*

$(b_{2})$*  for each $x\in X$, the set of all global maxima of the function $f(x,\cdot)$ is connected.*

Then, one has

[TABLE]

REMARK 1. - We want to remark that, in both Theorems 1 and 2, it is essential that $I$ be a real interval. To see this, consider the following example. Take

[TABLE]

and define $f:X\times I\to{\bf R}$ by

[TABLE]

for all $(t,s),(u,v)\in X$. Clearly, $f$ is continuous, $f(\cdot,\cdot,u,v)$ is inf-connected and has a unique global minimum, and $f(t,s,\cdot,\cdot)$ has a unique global maximum. However, we have

[TABLE]

The common key tool in our proofs of Theorems 1 and 2 is provided by the following general principle:

THEOREM A ([2], Theorem 2.2). - Let $X$ be a topological space, let $I$ be a compact real interval and let $S\subseteq X\times I$ be a connected set whose projection on $I$ is the whole of $I$.

Then, for every upper semicontinuous multifunction $\Phi:X\to 2^{I}$, with non-empty, closed and connected values, the graph of $\Phi$ intersects $S$.

Another known proposition which is used in the proof of Theorem 1 is as follows:

PROPOSITION A ([5], Proposition 2.1). - Let $X$ be a topological space, $Y$ a non-empty set, $y_{0}\in Y$ and $f:X\times Y\to{\bf R}$ a function such that $f(\cdot,y)$ is lower semicontinuous for all $y\in Y$ and relatively inf-compact for $y=y_{0}$. Assume also that there is a non-decreasing sequence of sets $\{Y_{n}\}$, with $Y=\cup_{n\in{\bf N}}Y_{n}$, such that

[TABLE]

for all $n\in{\bf N}$.

Then, one has

[TABLE]

A further result which is used in the proofs of Theorems 1 and 2 is provided by the following proposition which, in the given generality, is new:

PROPOSITION 1. - Let $X,Y$ be two topological spaces and let $f:X\times Y\to{\bf R}$ be a lower semicontinuous function such that $f(x,\cdot)$ is continuous for all $x\in X$. Moreover, assume that, for each $y\in Y$, there exists a neighbourhood $V$ of $y$ such that the function $\inf_{v\in V}f(\cdot,v)$ is relatively inf-compact. For each $y\in Y$, set

[TABLE]

Then, the multifunction $F$ is upper semicontinuous.

PROOF. Let $C\subseteq X$ be a closed set. We have to prove that $F^{-}(C)$ is closed. So, let $\{y_{\alpha}\}_{\alpha\in D}$ be a net in $F^{-}(C)$ converging to some $\tilde{y}\in Y$. For each $\alpha\in D$, pick $u_{\alpha}\in F(y_{\alpha})\cap C$. By assumption, there is a neighbourhood $V$ of $\tilde{y}$ such that the function $\inf_{v\in V}f(\cdot,v)$ is relatively inf-compact. Since the function $\inf_{x\in X}f(x,\cdot)$ is upper semicontinuous, we can assume that it is bounded above on $V$. Fix $\rho>\sup_{V}\inf_{X}f$. Then, there is a compact set $K\subseteq X$ such that

[TABLE]

But

[TABLE]

and so

[TABLE]

Let $\alpha_{1}\in D$ be such that $y_{\alpha}\in V$ for all $\alpha\geq\alpha_{1}$. Consequently, by $(1)$, $u_{\alpha}\in K$ for all $\alpha\geq\alpha_{1}$. By compactness, the net $\{u_{\alpha}\}_{\alpha\in D}$ has a cluster point $\tilde{u}\in K$. Clearly, $(\tilde{u},\tilde{y})$ is a cluster point in $X\times Y$ of the net $\{(u_{\alpha},y_{\alpha})\}_{\alpha\in D}$. We claim that

[TABLE]

Arguing by contradiction, assume the contrary and fix $r$ so that

[TABLE]

Then, there would be $\alpha_{2}\in D$ such that

[TABLE]

for all $\alpha\geq\alpha_{2}$. On the other hand, since, by assumption, the set $f^{-1}(]r,+\infty[)$ is open, there would be $\alpha_{3}\geq\alpha_{2}$ such that

[TABLE]

which gives a contradiction. Now, fix $x\in X$. Then, since $u_{\alpha}\in F(y_{\alpha})$, we have

[TABLE]

That is, $\tilde{u}\in F(\tilde{y})$. Since $C$ is closed, $\tilde{u}\in C$. Hence, $\tilde{y}\in F^{-}(C)$ and this ends the proof. $\bigtriangleup$

We now can prove Theorems 1 and 2.

Proof of Theorem 1. Fix $n\in{\bf N}$. Let us prove that

[TABLE]

Consider the multifunction $F:I_{n}\to 2^{X}$ defined by

[TABLE]

for all $\lambda\in I_{n}$. Thanks to Proposition 1, $F$ is upper semicontinuous and, by assumption, its values are non-empty, compact and connected. As a consequence, by Theorem 7.4.4 of [1], the graph of $F$ is connected. Let $S$ denote the graph of the inverse of $F$. So, $S$ is connected as it is homeomorphic to the graph of $F$. Now, consider the multifunction $\Phi:X\to 2^{I_{n}}$ defined by

[TABLE]

for all $x\in X$. By Proposition 1 again, the multifunction $\Phi$ is upper semicontinuous and, by assumption, its values are non-empty, closed and connected. After noticing that the projection of $S$ on $I_{n}$ is the whole of $I_{n}$, we can apply Theorem A. Therefore, there exists $(\tilde{x},\tilde{\lambda})\in S$ such that $\tilde{\lambda}\in\Phi(\tilde{x})$. That is

[TABLE]

Clearly, $(2)$ follows from $(3)$. Now, the conclusion is a direct consequence of Proposition A. $\bigtriangleup$ Proof of Theorem 2. Arguing by contradiction, assume the contrary and fix a constant $r$ so that

[TABLE]

Let $G:I\to 2^{X}$ be the multifunction defined by

[TABLE]

for all $\lambda\in I$. Notice that $G(\lambda)$ is non-empty for all $\lambda\in I$ and connected for all $\lambda\in D$. Moreover, the graph of $G$ is open in $X\times I$ and so $G$ is lower semicontinuous. Then, by Proposition 5.7 of [3], the graph of $G$ is connected and so the graph of the inverse of $G$, say $S$, is connected too. Consider the multifunction $\Phi:X\to 2^{I}$ defined by

[TABLE]

for all $x\in X$. Notice that $\Phi(x)$ is non-empty, closed and connected, in view of $(b_{2})$. By Proposition 1, the multifunction $\Phi$ is upper semicontinuous. Now, we can apply Theorem A. So, there exists $(\hat{x},\hat{\lambda})\in S$ such that $\hat{\lambda}\in\Phi(\hat{x})$. This implies that

[TABLE]

which is absurd. $\bigtriangleup$ Here is an application of Theorem 1.

THEOREM 3. - Let $(H,\langle\cdot,\cdot\rangle)$ be a real inner product space, let $K\subset H$ be a compact and convex set, with $0\notin K$, and let $f:X\to K$ be a continuous function, where

[TABLE]

Assume that there are two numbers $\alpha,c$, with

[TABLE]

such that:

$(a)$*  $\{x\in X:\langle x,f(x)\rangle=\alpha\}\subset\{x\in X:\|f(x)\|<c\}$ ;*

$(b)$*  *$\{x\in X:c^{2}\langle x,f(x)\rangle=\alpha\|f(x)\|^{2}\}\subset\{x\in X:\|f(x)\|\geq c\}\ .$

Then, there exists $\tilde{\lambda}\in{\bf R}$ such that the set

[TABLE]

is disconnected.

PROOF. Consider the function $\varphi:X\times{\bf R}\to{\bf R}$ defined by

[TABLE]

for all $(x,\lambda)\in X\times{\bf R}$. Notice that

[TABLE]

Further, observe that, when $\|f(x)\|\geq c$, in view of $(a)$, we have

[TABLE]

as well as

[TABLE]

for all $\lambda>0$. When $\|f(x)\|\geq c$ again, the function $\varphi(x,\cdot)$ is convex and so, by $(6)$, for each $\lambda>0$, its restriction to $[-\lambda,\lambda]$ it has a unique global maximum. Clearly, $\varphi(x,\cdot)$ has the same uniqueness property also when $\|f(x)\|<c$. Now, observe that, for each $\lambda\in{\bf R}$, the function $\lambda f$ has a fixed point in $X$, in view of the Schauder theorem. Hence, we have

[TABLE]

We claim that

[TABLE]

First, observe that, since $0\notin K$, every closed and bounded subset of $X$ is compact. This easily implies that, for each $\mu>0$, the function $x\to\inf_{|\lambda|\leq\mu}\varphi(x,\lambda)$ is relatively inf-compact. Consequently, the sublevel sets of the function $x\to\sup_{\lambda\in{\bf R}}\varphi(x,\lambda)$ (which is finite if $\|f(x)\|<c$) are compact. Therefore, there exists $\tilde{x}\in X$ such that

[TABLE]

So, by $(5)$, one has $\|f(\tilde{x})\|<c$. Clearly, we also have

[TABLE]

Let us prove that

[TABLE]

After some manipulations, one realizes that $(11)$ is equivalent to

[TABLE]

Now, for each $y\in X\setminus\{0\}$, $t\in{\bf R}$, set

[TABLE]

Consider the inequality

[TABLE]

After some manipulations, one realizes that $(13)$ is equivalent to

[TABLE]

So, $(13)$ is satisfied if and only if

[TABLE]

Observe that

[TABLE]

Therefore, if $(14)$ is satisfied, for each $x\in I(y,t)\cap X$, in view of $(13)$ and $(15)$, we have

[TABLE]

At this point, taking into account that $c^{2}\langle\tilde{x},f(\tilde{x})\rangle\neq\alpha\|f(\tilde{x})\|^{2}$ (by $(b)$), we draw $(12)$ from $(16)$ since $\tilde{x}\in I(f(\tilde{x}),\langle\tilde{x},f(\tilde{x})\rangle)$. Summarizing: taking $I={\bf R}$ and $I_{n}=[-n,n]$ ($n\in{\bf N}$), the continuous function $\varphi$ satisfies $(a_{1})$ and $(b_{1})$ of Theorem 1, but, in view of $(7)-(11)$, it does not satisfy the conclusion of that theorem. As a consequence, there exists $\tilde{\lambda}\in{\bf R}$ such that the set of all global minima of $\varphi(\cdot,\tilde{\lambda})$ is disconnected. But such a set agrees with the set of all solutions of the equation $x=\tilde{\lambda}f(x)$, and the proof is complete. $\bigtriangleup$

REMARK 2. - We do not know whether Theorem 3 is still true when $0\in K$ and $(b)$ is (necessarily) changed in

[TABLE]

However, the proof of Theorem 3 shows that the following is true:

THEOREM 4. - Let $(X,\langle\cdot,\cdot\rangle)$ be a finite-dimensional real Hilbert space and let $f:X\to X$ be a continuous function with bounded range. Assume that there are two numbers $\alpha,c$, with

[TABLE]

such that:

$(a^{\prime})$*  $\{x\in X:\langle x,f(x)\rangle=\alpha\}\subset\{x\in X:\|f(x)\|<c\}$ ;*

$(b^{\prime})$*  *$\{x\in X:f(x)\neq 0,\hskip 3.0ptc^{2}\langle x,f(x)\rangle=\alpha\|f(x)\|^{2}\}\subset\{x\in X:\|f(x)\|\geq c\}\ .$

Then, there exists $\tilde{\lambda}\in{\bf R}$ such that the set

[TABLE]

is disconnected.

Finally, we present two applications of Theorem 2.

THEOREM 5. - Let $X$ be a Banach space, let $\varphi\in X^{*}\setminus\{0\}$ and let $\psi:X\to{\bf R}$ be a Lipschitzian functional whose Lipschitz constant is equal to $\|\varphi\|_{X^{*}}$. Moreover, let $[a,b]$ be a compact real interval, $\gamma:[a,b]\to[-1,1]$ a convex (resp. concave) and continuous function, with $\hbox{\rm int}(\gamma^{-1}(\{-1,1\}))=\emptyset$, and $c\in{\bf R}$. Assume that

[TABLE]

for all $x\in X$ such that $\psi(x)>0$ (resp. $\psi(x)<0$).

Then (with the convention $\sup\emptyset=-\infty$), one has

[TABLE]

PROOF. Consider the continuous function $f:X\times[a,b]\to{\bf R}$ defined by

[TABLE]

for all $(x,\lambda)\in X\times[a,b]$. By Theorem 2 of [4], for each $\lambda\in\gamma^{-1}(]-1,1[)$, the function $f(\cdot,\lambda)$ is inf-connected and unbounded below. Also, notice that $\gamma^{-1}(]-1,1[)$, by assumption, is dense in $[a,b]$. Now fix $x\in X$. If $\psi(x)>0$ (resp. $\psi(x)<0$) the function $f(x,\cdot)$ is convex and, by assumption, $f(x,a)\neq f(x,b)$. As a consequence, the unique global maximum of this function is either $a$ or $b$. If $\psi(x)\leq 0$, the function is concave and so, obviously, the set of all its global maxima is connected. Now, the conclusion follows directly from Theorem 2. $\bigtriangleup$

Let $(T,{\cal F},\mu)$ be a $\sigma$-finite measure space, $E$ a real Banach space and $p\geq 1$.

As usual, $L^{p}(T,E)$ denotes the space of all (equivalence classes of) strongly $\mu$-measurable functions $u:T\rightarrow E$ such that $\int_{T}\parallel u(t)\parallel^{p}d\mu<+\infty$, equipped with the norm

[TABLE]

A set $D\subseteq L^{p}(T,E)$ is said to be decomposable if, for every $u,v\in D$ and every $A\in{\cal F}$, the function

[TABLE]

belongs to $D$, where $\chi_{A}$ denotes the characteristic function of $A$.

A real-valued function on $T\times E$ is said to be a Caratéodory function if it is measurable in $T$ and continuous in $E$.

THEOREM 6. - Let $(T,{\cal F},\mu)$ be a $\sigma$-finite non-atomic measure space, $E$ a real Banach space, $p\in[1,+\infty[$, $X\subseteq L^{p}(T,E)$ a decomposable set, $[a,b]$ a compact real interval, $\gamma:[a,b]\to{\bf R}$ a convex (resp. concave) and continuous function. Moreover, let $\varphi,\psi,\omega:T\times E\to{\bf R}$ be three Carathéodory functions such that, for some $M\in L^{1}(T)$, $k\in{\bf R}$, one has

[TABLE]

for all $(t,x)\in T\times E$ and

[TABLE]

for all $u\in X$ such that $\int_{T}\psi(t,u(t))d\mu>0$ (resp. $\int_{T}\psi(t,u(t))d\mu<0$).

Then, one has

[TABLE]

[TABLE]

PROOF. The proof goes on exactly as that of Theorem 5. So, one considers the function $f:X\times[a,b]\to{\bf R}$ defined by

[TABLE]

for all $(u,\lambda)\in X\times[a,b]$, and realizes that it satisfies the hypotheses of Theorem 2. In particular, for each $\lambda\in[a,b]$, the inf-connectedness of the function $f(\cdot,\lambda)$ is due to [6], Théorème 7. $\bigtriangleup$

Acknowledgement. The author has been supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by the Università degli Studi di Catania, “Piano della Ricerca 2016/2018 Linea di intervento 2”.

References

[1]  E. KLEIN and A. C. THOMPSON, Theory of correspondences, John Wiley $\&$ Sons, 1984. [2]  B. RICCERI, Some topological mini-max theorems via an alternative principle for multifunctions, Arch. Math. (Basel), 60 (1993), 367-377.

[3]  B. RICCERI, Nonlinear eigenvalue problems, in “Handbook of Nonconvex Analysis and Applications” D. Y. Gao and D. Motreanu eds., 543-595, International Press, 2010.

[4]  B. RICCERI, On the infimum of certain functionals, in “Essays in Mathematics and its Applications - In Honor of Vladimir Arnold”, Th. M. Rassias and P. M. Pardalos eds., 361-367, Springer, 2016.

[5]  B. RICCERI, On a minimax theorem: an improvement, a new proof and an overview of its applications, Minimax Theory Appl., 2 (2017), 99-152.

[6]  J. SAINT RAYMOND, Connexité des sous-niveaux des fonctionnelles intégrales, Rend. Circ. Mat. Palermo, 44 (1995), 162-168.

[7]  M. SION, On general minimax theorems, Pacific J. Math., 8 (1958), 171-176.

Department of Mathematics and Informatics

University of Catania

Viale A. Doria 6

95125 Catania, Italy

e-mail address: ricceri@dmi.unict.it