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A famous saying (due toHerriot)definescultureas "what remainswhen
everythingisforgotten ." One couldparaphrase thisdefinitionin
statingthat generalizedconvexity iswhat remainswhen convexity has
been dropped . Of course, oneexpectsthatsome
convexityfeaturesremain.For functions, convexity ofepigraphs(what
is above thegraph) is a simplebut strong assumption.It leads
tobeautifulpropertiesand to a field initselfcalled convex analysis.
In several models, convexity is not presentandintroducing genuine
convexityassumptionswouldnotberealistic. A simple extensionof
thenotionof convexity consists in requiringthatthe sublevel sets
ofthe functionsare convex (recall thata sublevel set offunction a
is theportionof thesourcespaceon which thefunctiontakesvalues below
a certainlevel).Its first use is usuallyattributed to deFinetti, in
1949. This propertydefinesthe class ofquasiconvexfunctions, which
is much larger thanthe class of convex functions: a non
decreasingor nonincreasingone variablefunctionis quasiconvex, as
well asanyone-variable functionwhich is nonincreasingon
someinterval(-00, a] or(-00, a) and nondecreasingon its
complement.Many otherclasses ofgeneralizedconvexfunctionshave been
introduced, often fortheneeds ofvariousapplications: algorithms,
economics, engineering, management science, multicriteria
optimization, optimalcontrol, statistics .Thus, theyplay
animportantrole in severalappliedsciences . A monotonemappingF from
aHilbertspace to itself is a mappingfor which the angle between
F(x) - F(y) and x- y isacutefor anyx, y. It is well-known
thatthegradientof a differentiable convexfunctionis monotone.The
class of monotonemappings(and theclass
ofmultivaluedmonotoneoperators) has remarkableproperties.This class
has beengeneralizedin various direc tions, withapplicationsto
partialdifferentialequations, variationalinequal ities,
complementarity problemsand more generally, equilibriumproblems.
The classes ofgeneralizedmonotonemappingsare more or lessrelatedto
the classes ofgeneralizedfunctionsvia differentiation or
subdifferentiation procedures.They are also link edvia
severalothermeans."
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