Generalized Convexity and Generalized Monotonicity Proceedings of the 6th International Symposium on Generalized Convexity/Monotonicity, Samos, September 1999

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.