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\begin{document}
> > >
\begin{center}
>{\bf ARE GENERALIZED DERIVATIVES USEFUL FOR } > >{\bf GENERALIZED CONVEX
FUNCTIONS?}\bigskip\ > >Jean-Paul PENOT > >University of Pau > >\vskip .5in >
\end{center}
> >{\bf Abstract.} We present a review of some ad hoc subdifferentials which
>have been devised for the needs of generalized convexity such as the
>quasi-subdifferentials of Greenberg-Pierskalla, the tangential of Crouzeix,
>the lower subdifferential of Plastria, the infradifferential of
>Guti\'{e}rrez, the subdifferentials of Martinez-Legaz-Sach, Penot-Volle,
>Thach. We compare them to some all-purpose subdifferentials used in
>nonsmooth analysis. We give some hints about their uses. We also point out
>links with duality theories. > >
\section{Introduction}
> >The fields of generalized convexity and of nonsmooth analysis do not fit
>well with the image of mathematics as a well-ordered building. The notions
>are so abundant and sometimes so exotic that these two fields evoke the
>richness of a luxuriant nature rather than the purity of classical
>architecture (see for instance \cite{C-C}, \cite{S-Z}, \cite{Pini Singh}
and >its rich bibliography for generalized convexity and \cite{E-T}, \cite
{{Gwinner}, \cite{Ioffe Fermat}, \cite{Treiman 2}, \cite{Treiman 3}, \cite }{%
Gwinner}, \cite{Ioffe Fermat}, \cite{Treiman 2}, \cite{Treiman 3}, \cite
{{Views}, \cite{Japan} for nonsmooth analysis). Therefore, mixing both
topics }{Views}, \cite{Japan} for nonsmooth analysis). Therefore, mixing
both topics >brings the risk of increasing the complexity of the picture. >
>However, we try to put some order and to delineate lines of thought around
>these two fields. It appears that the different concepts usually have
>comparable strengths. Moreover, to a certain extend, they can be treated in
>a somewhat unified way. > >A first interplay between generalized convexity
and nonsmooth analysis deals >with generalized directional derivatives.
Their comparison being reduced to >inequalities, it suffices to give bounds
for the derivatives one can use. >Among the different concepts we present,
we bring to the fore the incident >derivative $f^i$ (also called
intermediate derivative, inner epi-derivative, >upper epi-derivative,
adjacent derivative). It is always a lower >semicontinuous (l.s.c.) function
of the direction; we show that it is also a >quasiconvex function of the
direction when the function is quasiconvex. This >is an important feature of
this derivative since Crouzeix has shown how to >decompose such positively
homogeneous functions into two convex parts. We >present a variant of his
decomposition which also preserves lower >semicontinuity. > >For what
concerns subdifferentials, one already has the disposal of >axiomatic
approaches which capture the main properties of usual >subdifferentials (%
\cite{A-C-L}, \cite{Ioffe 96}, \cite{I-P}, \cite{Catalan}>...). These
subdifferentials may be used for characterizing various >generalized
convexity properties. We recall such characterizations in >sections 3 and 4,
omiting important cases such as strong quasiconvexity (>\cite{C-F-Z}),
invexity (\cite{Craven}, \cite{Craven-Glover}, \cite{H}, \cite{{Martin},
\cite{Y-C}-\cite{Y3}...) log-concavity, rank-one convexity, rough }{Martin},
\cite{Y-C}-\cite{Y3}...) log-concavity, rank-one convexity, rough >convexity
(\cite{Phu}-\cite{Phu 4}) and several others, but adding the case >of
paraconvexity. > >Other subdifferentials exist which have been devised for
the special needs >of generalized convexity. We compare these specific
notions (and add a few >other variants) in section 5. We also look for links
with all-purpose >subdifferentials (section 6). > >These comparisons are not
just made for the sake of curiosity. The >relationships we exhibit enable
one to deduce properties of a concept from >known properties of another
concept by using sufficient conditions to get >equality or inclusion between
the two subdifferentials. They also ease the >choice of an appropriate
subdifferential for a specific problem. In several >instances the choice is
dictated by the nature of the problem or by the >duality theory which is
avalaible. This fact is parallel to what occurs in >nonsmooth analysis where
the structure of the space in which the problem can >be set influences the
choice of the appropriate subdifferential. We evoke >shortly in section 8
the links with duality, leaving to other contributions >the task of being
more complete on this topic and on the other ones we >tackle. Since
subdifferentials would be of poor use if no calculus rule were >avalaible,
we give a short account of such rules in section 7. We close the >paper with
a short account of a new proposal of Martinez-Legaz and P.H. Sach >\cite
{ML-S} which has a special interest because it is small enough, close >to
usual subdifferentials and still adapted to generalized convexity. > >Our
study focuses on subdifferentials, so that other tools of nonsmooth
>analysis such as tangent cones, normal cones, coderivatives remain in the
>shadow. It is probably regretful, but we tried to keep a reasonable size to
>our study. Still, since coderivatives are the adapted tool for studying
>multimappings (correspondences or relations) they certainly have a role to
>play in a field in which the sublevel set multimapping $r\rightrightarrows
>F(r):=[f\leq r]:=f^{-1}(]-\infty ,r])$ associated with a function $f$ plays
>a role which is more important than the role played by the epigraph of $f.$
>An explanation also lies in the freshness of the subject (see \cite{Ioffe
96}>, \cite{I-P}, \cite{Mordu-S}, \cite{Japan} and their references for
>instance). Applications to algorithms are not considered here; we refer to >%
\cite{Plastria}, \cite{P-Q}, \cite{Tao}, \cite{Tuy} for some illustrations
>and numerous references. An application to well-posedness and conditioning
>will be treated elsewhere. For applications to mathematical economics we
>refer to \cite{Crou2}, \cite{Mar}, \cite{M'}, \cite{M-S}, \cite{Maz}, \cite
{{S-Z}...A nice application of lower subdifferentials to time optimal
control }{S-Z}...A nice application of lower subdifferentials to time
optimal control >problems is contained in \cite{ML}, \cite{ML time}. A
recent interplay >between quasiconvexity and Hamilton-Jacobi equations is
revealed in \cite{{Volle 97}. }{Volle 97}. > >We hope that the reader will
draw from the present study the conclusion that >it is possible to stand
outside the lost paradises of convexity and >smoothness, and, hopefully, to
go forward. > >
\section{Generalized directional derivatives and their uses}
> >A natural idea for generalizing convexity of differentiable functions
>expressed in terms of monotonicity of the derivative consists in replacing
>the derivative by a generalized derivative, so that nondifferentiable
>functions can be considered. A number of choices can be made. Let us recall
>some of them, the first one, the dag derivative, being a rather special
>notion introduced in \cite{Dijon} whose interest seems to be limited to the
>fact that it is the largest possible notion which can be used in this
>context. In the sequel $f$ is an extended real-valued function on the
n.v.s. >$X$ which is finite at some $x\in X$ and $v$ is a fixed vector of $%
X. $ The >closed unit ball of $X$ with center $x$ and radius $r$ is denoted
by $B(x,r). >$ The closure of a subset $S$ of $X$ is denoted by cl($S$). >
>Thus the {\it dag derivative} of $f$ is > >
\[
>f^{\dagger }(x,v):=\lim \sup_{(t,y)\rightarrow (0_{+},x)}\frac
>1t(f(y+t(v+x-y))-f(y)). >
\]
>which majorizes both the {\it upper radial} (or upper Dini) derivative >
\[
>f_{+}^{\prime }(x,v):=\lim \sup_{t\rightarrow 0_{+}}\frac 1t(f(x+tv)-f(x))
>
\]
> >\noindent and the {\it Clarke-Rockafellar}{\bf \ }derivative or
>circa-derivative >
\[
>f^{\uparrow }(x,v):=\inf_{r>0}\lim \!\sup_{\stackunder{f(y)\rightarrow f(x)%
}{>(t,y)\rightarrow (0_{+},x)}}\;\inf_{w\in B(v,r)}\frac 1t(f(y+tw)-f(y)). >
\]
>When $f$ is Lipschitzian, $f^{\dagger }$ coincides with {\it Clarke's
>derivative} $f^{\circ }$: >
\[
>f^{\circ }(x,v):=\lim \sup_{(t,y,w)\rightarrow (0_{+},x,v)}\frac
>1t(f(y+tw)-f(y)). >
\]
> >\noindent The {\it contingent derivative} (or lower epiderivative or
lower >Hadamard derivative) >
\[
>f^{!}(x,v):=\lim \inf_{(t,u)\rightarrow (0_{+},v)}\frac 1t(f(x+tu)-f(x)) >
\]
>can also be denoted by $f^{\prime }(x,v)$ in view of its importance. The >%
{\it incident derivative} (or inner epiderivative) >
\[
>f^i(x,v):=\sup_{r>0}\lim \sup_{t\searrow 0}\;\inf_{u\in B(v,r)}\frac
>1t(f(x+tu)-f(x)) >
\]
>is intermediate between the contingent derivative and the circa-derivative
>and also the upper Hadamard derivative (or upper hypo-derivative) >
\[
>f^{\sharp }(x,v):=\lim \sup_{(t,w)\rightarrow (0_{+},v)}\frac
>1t(f(x+tw)-f(x))=-(-f)^{!}(x,v). >
\]
>These derivatives can be ranked. Moreover, in the most useful cases such as
>one-variable functions, convex nondifferentiable functions, convex
composite >functions, finite maxima of functions of class $C^1,$ these
different >notions coincide. Several of the preceding derivatives are such
that their >epigraphs are tangent cones (in a related sense) to the epigraph
of the >function. Unlike the convex case, such a geometrical interpretation
does not >bring much for generalized convex functions because their
epigraphs are not >as important as their sublevel sets. > >The importance of
the incident derivative stems from the following results: >other derivatives
share some properties such as lower semi-continuity (this >is the case for
the contingent and the circa-derivatives) or quasi-convexity >(this is the
case for the radial upper derivative, as shown in \cite{Cr}) or >accuracy,
but not all. For instance Crouzeix has given in \cite{C 1} an >example of a
quasiconvex function such that the upper radial derivative is >not l.s.c.. >
>
\begin{proposition}
>If $f$ is quasiconvex and finite at $x$ then the incident derivative $%
>f^i(x,\cdot )$ is l.s.c. and quasiconvex. >
\end{proposition}
> >{\it Proof.} A simple direct proof can be given using the definitions: $%
>f^i(x,v)\leq r$ iff for any sequence $(t_n)\rightarrow 0_{+}$ there exist
>sequences $(r_n)\rightarrow r,$ $(v_n)\rightarrow v$ such that $%
>f(x+t_nv_n)\leq f(x)+t_nr_n;$ thus, if $u,w$ are such that $f^i(x,u)\leq
>r,\;f^i(x,w)\leq r,$ then any $v$ in the interval $[u,w]$ satisfies $%
>f^i(x,v)\leq r.$ > >A more elegant proof follows from the expression given
in \cite{Volle Cras} >Th\'{e}or\`{e}me 7 of the sublevel sets of the
epi-limit superior $q$ of a >family $(q_t)$ of functions on $X$ parametrized
by $t>0:$ >
\[
>\lbrack q\leq r]=\bigcap_{s>r}\lim \sup_{t\rightarrow 0}[q_t\leq s]. >
\]
>This formula shows that $q$ is quasiconvex whenever the functions $q_t$ are
>quasiconvex. Since $q:=f^i(x,\cdot )$ is the epi-limit superior of the
>family of quotients $q_t$ given by $q_t(u):=t^{-1}(f(x+tu)-f(x))$ which are
>obviously quasiconvex, the result follows.$\Box $ > >Now we will make use
of the following result which is a simple variant of >results of Crouzeix (%
\cite{Crou}, \cite{C 1}, \cite{Cr}). > >
\begin{proposition}
>Suppose $h$ is a positively homogeneous quasiconvex extended real-valued
>function on $X.$ Then each of the following two assumptions ensures that $f$
>is convex : > >(a) $h$ is non negative ; > >(b) there exists a nonempty
dense subset $D$ of the domain $D_h$ of $h$ on >which $h$ is negative. >
\end{proposition}
> >{\it Proof}. Assertion (a) is proved in \cite{Cr}. In order to prove
>assertion (b), using \cite{Cr} Theorem 10 it suffices to show that for each
$>y\in X^{*}$ the Crouzeix function $F$ given by >
\[
>F\left( y,r\right) =\sup \left\{ \langle y,x\rangle :x\in \left[ h\leq
>r\right] \right\} >
\]
>is concave in its second variable. It is obviously nondecreasing and since $%
0 >$ belongs to the closure of $D$ we have $F\left( y,0\right) \geq 0,$
hence $>F\left( y,1\right) \geq 0.$ As $h$ is positively homogeneous, $%
F\left( >y,.\right) $ is also positively homogeneous. When $F\left(
y,1\right) =0$ we >have $F\left( y,-1\right) \leq 0$ and $F\left( y,.\right)
$ is concave. When >$F\left( y,1\right) >0$ we can find $x\in \left[ h\leq
1\right] $ with $>\langle y,x\rangle >0.$ Then there exists a sequence $%
\left( x_n\right) $ in >$D$ with limit $x$ ; we may suppose there exists $%
r>0 $ such that $\langle >x_n,y\rangle >r$ for each $n.$ Since $h\left(
x_n\right) <0$ we can find a >sequence $\left( t_n\right) $ of positive
numbers with limit $+\infty $ such >that $h\left( t_nx_n\right) \leq -1$ for
each $n.$ Then $F\left( y,-1\right) >\geq \langle t_nx_n,y\rangle
\rightarrow \infty $ and we get $F\left( >y,.\right) \equiv +\infty ,$ a
concave function. $\Box $ > >We will use jointly the preceding proposition
and a decomposition of an >arbitrary l.s.c. positively homogeneous function $%
h$ which takes a special >form when $h$ is quasiconvex. Then, it differs
from the Crouzeix's >decomposition by the fact that its two terms are l.s.c.
sublinear functions. >Namely, let us set for an arbitrary l.s.c. positively
homogeneous function $>h,$>
\[
>D=\left[ h<0\right] ,\,\,\,\,\,\,\,\,\overline{D}=clD, >
\]
>
\[
>h^{<}\left( x\right) =\left\{ >
\begin{tabular}{r}
>$h\left( x\right) \,\,\,\,\,\,\,\,\,x\in \overline{D}$ \\
>$+\infty \;\;\;\;\;x\in X\backslash \overline{D}$>
\end{tabular}
>\right. \;\;\;\;\;\;h^{\geq }\left( x\right) =\left\{ >
\begin{array}{c}
>0\,\,\,\,\,\,x\in \overline{D} \\
>h\left( x\right) \,\,\,\,\,x\in X\backslash \overline{D}, >
\end{array}
>\right. >
\]
>so that $\overline{D}$ replaces $D$ in the Crouzeix's construction. Since $%
h $ >is l.s.c. $h\left( x\right) =0$ for each $x\in \overline{D}\backslash D$%
, >and $h^{\geq }$ coincides with the function $h^{+}$ of the Crouzeix's
>decomposition, which is exactly the positive part of $h.$ However $h^{<}$
>differs from the corresponding term $h^{-}$ of the Crouzeix's decomposition
>(which is not the negative part of $h$) on $\overline{D}\backslash D$ since
$>h^{-}\mid X\backslash D=\infty $ whereas $h^{<}\left( x\right) =0$ on $>%
\overline{D}\backslash D$ as observed above$.$ > >The proof of the following
statement is immediate from what precedes since >
\begin{eqnarray*}
>\lbrack h^{<} &\leq &r]=[h\leq r]\;\text{for }r<0,\;[h^{<}\leq r]=\overline{%
D>}\text{ for }r\geq 0, \\
>\lbrack h^{\geq } &\leq &r]=\emptyset \;\text{for }r<0,\;[h^{\geq }\leq
>r]=[h^{<}\leq r]\text{ for }r\geq 0. >
\end{eqnarray*}
> >
\begin{theorem}
>\label{decomposition}The functions $h^{<}$ and $h^{\geq }$ are l.s.c.; they
>are convex when $h$ is quasiconvex and >
\[
>h=\min \left( h^{<},h^{\geq }\right) . >
\]
>
\end{theorem}
> >We observe that when $h=f^i\left( x,.\right) ,$ the incident derivative
at $>x,$ the set $\overline{D}$ is contained in the tangent set $T\left(
>S,x\right) $ to the sublevel $S:=\left[ fat $x$. In fact, for any $v\in D$ and any sequence $\left( t_n\right)
>\searrow 0$ there exists a sequence $\left( v_n\right) \rightarrow v$ with $%
>\stackunder{n}{\lim \sup }t_n^{-1}\left( f\left( x+t_nv_n\right) -f\left(
>x\right) \right) <0,$ so that $x+t_nv_n\in S$ for $n$ large enough : $v\in
>T\left( S,x\right) $ ; as $T\left( S,x\right) $ is closed we also have $>%
\overline{D}\subset T\left( S,x\right) .$ > >We have proved the first part
of the following statement. For the second >part, we adapt the arguments of
\cite{Cr} Prop. 18 in order to identify $>\overline{D}.$ > >
\begin{proposition}
>When $h=f^i\left( x,.\right) ,$ the set $\overline{D}$ is contained in the
>tangent set $T\left( S,x\right) .$ If $f$ is quasiconvex, $D$ is nonempty
>and if $S$ is open (in particular if $f$ is upper semicontinuous on $S$)
one >has $]0,\infty [(S-x)\subset D$ and $\overline{D}=T\left( S,x\right) .$
>
\end{proposition}
> >We observe that the assumption that $D$ is nonempty cannot be replaced
with >the assumption that $S$ is nonempty (consider the function $f$ on $%
I\!\!R$ >given by $f(r)=r^3$ and take $x=0$). > >{\it Proof. }As $h$ is
quasiconvex $T\left( S,x\right) $ is the closure of $>]0,\infty [\left(
S-x\right) $, and it remains to prove that $]0,\infty >[\left( S-x\right) $
is contained in $D$ or equivalently that $S-x\subset D.$ >We may suppose $%
x=0.$ Let $u\in S-x$ and let $w\in D.$ Since $S$ is open, we >can find $s\in
]0,1[$ and $v\in S-x$ such that >
\[
>u=\left( 1-s\right) w+sv. >
\]
>Given $t>0,$ let $p\left( t\right) :=\left( 1-st\right) ^{-1}(1-s)t$ and
let >$\left( w_t\right) \rightarrow w$ be such that >
\[
>\stackunder{t\rightarrow 0}{\lim \sup }\frac 1{p\left( t\right) }\left(
>f\left( x+p\left( t\right) w_t\right) -f\left( x\right) \right) <0. >
\]
>Let us define $u_t$ by >
\[
>u_t:=\left( 1-s\right) w_t+sv >
\]
>so that $\left( u_t\right) \rightarrow u.$ We can write >
\[
>tu_t=\left( 1-st\right) p\left( t\right) w_t+stv >
\]
>so that >
\[
>f\left( x+tu_t\right) \leq \max \left( f\left( x+p\left( t\right)
w_t\right) >,f\left( x+v\right) \right) >
\]
>and >
\[
>\frac{f\left( x+tu_t\right) -f\left( x\right) }t\leq \frac{f\left(
x+p\left( >t\right) w_t\right) -f\left( x\right) }t >
\]
>since $t^{-1}\left( f\left( x+v\right) -f\left( x\right) \right)
\rightarrow >-\infty $ as $t\downarrow 0.$ As $t^{-1}p\left( t\right)
\rightarrow 1$ we >get >
\[
>\stackunder{t\searrow 0}{\lim \sup }\frac{f\left( x+tu_t\right) -f\left(
>x\right) }t\leq \stackunder{t\searrow 0}{\lim \sup }\frac{p\left( t\right) }%
>t.\frac 1{p\left( t\right) }\left( f\left( x+p\left( t\right) w_t\right)
>-f\left( x\right) \right) <0. >
\]
>Therefore $f^i\left( x,u\right) <0$ and $u\in D.$ $\Box $ > >
\section{Characterizations via directional derivatives}
> >Let us recall some answers to the question: is it possible to
characterize >the various sorts of generalized convexity with the help of
generalized >derivatives? We limit our presentation to quasi-convexity and
>pseudo-convexity. We refer to \cite{G-K}, \cite{G-M}, \cite{K1}, \cite{K2},
>\cite{K5}, \cite{Dijon} for other cases, further details and proofs. > >
\begin{theorem}
>Let $f$ be a l.s.c. function with domain $C$ and let $f^{?}$be an arbitrary
>bifunction on $C\times X.$\ Among the following statements one has the
>implication $(b)\Rightarrow (a)$; when $f^{?}\leq f^{\dagger },$ or when $f$
>is continuous and $f^{?}\leq f^{\circ }$ one has $(a)\Rightarrow (b)$; when
$>f^{?}\leq f^{\dagger }$ and $f^{?}$ is l.s.c. in its second variable one
has >$(b)\Rightarrow (c);$ when for each $x\in X$ the function $%
f^{?}(x,\cdot )$ >is positively homogeneous, l.s.c. and minorized by $%
f^{!}(x,\cdot )$ one has >$(c)\Rightarrow (b).$ > >$(a)$ $f$ is quasiconvex
; > >$(b)$ $f$ is $f^{?}$-quasiconvex i.e. satisfies the condition >
\[
>(Q^{?})\;(x,z)\in C\times X,\;f^{?}(x,z-x)>0\Rightarrow \forall y\in \left[
>x,z\right] \ f\left( z\right) \geq f\left( y\right) ; >
\]
> >$(c)$ $f^{?}$ is quasimonotone, i.e. satisfies the relation min$%
>(f^{?}(x,y-x),f^{?}(y,x-y))\leq 0$ for any $x,y\in C.$\ \ >
\end{theorem}
> >The following definition generalizes a well known notion to non
>differentiable functions. > >
\begin{definition}
>The function $f$ is said to be $f^{?}$-pseudoconvex if >
\[
>(P^{?})\text{\ \ }x,y\in C,\text{ }f^{?}(x,y-x)\geq 0\Longrightarrow
>f(y)\geq f(x). >
\]
>
\end{definition}
> >
\begin{theorem}
\TEXTsymbol{>}{\bf \ }Let $f$ be l.s.c. with $f^{?}$ l.s.c. in its second
variable. \TEXTsymbol{>} \TEXTsymbol{>}$(a)$ Suppose that for each local
minimizer $y$ of $f$ one has $>f^{?}(y,w)\geq 0$ for any $w\in X$ and
suppose that $f^{?}\leq f^{\dagger }$ \TEXTsymbol{>}. Then, if $f$ is $f^{?}$%
-pseudoconvex, $f^{?}$ is pseudomonotone, i.e. for \TEXTsymbol{>}any $x,y\in
C$ with $x\neq y$ one has $f^{?}(y,x-y)<0$ whenever $>f^{?}(x,y-x)>0.$
\TEXTsymbol{>} \TEXTsymbol{>}$(b)$ Conversely, suppose $f^{?}$ is
pseudomonotone, sublinear in its second \TEXTsymbol{>}variable with $%
f^{?}\geq f^{!}$. Then, if $f$ is continuous, it is \TEXTsymbol{>}%
pseudoconvex. \TEXTsymbol{>}
\end{theorem}
> >
\section{Subdifferential characterizations}
> >Let us now consider the use of another generalization of derivatives,
namely >subdifferentials. We first consider subdifferentials which have
sense for >any type of function.
\end{document}