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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."
Calculus Without Derivatives expounds the foundations and recent
advances in nonsmooth analysis, a powerful compound of mathematical
tools that obviates the usual smoothness assumptions. This textbook
also provides significant tools and methods towards applications,
in particular optimization problems. Whereas most books on this
subject focus on a particular theory, this text takes a general
approach including all main theories. In order to be
self-contained, the book includes three chapters of preliminary
material, each of which can be used as an independent course if
needed. The first chapter deals with metric properties, variational
principles, decrease principles, methods of error bounds, calmness
and metric regularity. The second one presents the classical tools
of differential calculus and includes a section about the calculus
of variations. The third contains a clear exposition of convex
analysis.
This textbook covers the main results and methods of real analysis
in a single volume. Taking a progressive approach to equations and
transformations, this book starts with the very foundations of real
analysis (set theory, order, convergence, and measure theory)
before presenting powerful results that can be applied to concrete
problems. In addition to classical results of functional analysis,
differential calculus and integration, Analysis discusses topics
such as convex analysis, dissipative operators and semigroups which
are often absent from classical treatises. Acknowledging that
analysis has significantly contributed to the understanding and
development of the present world, the book further elaborates on
techniques which pervade modern civilization, including wavelets in
information theory, the Radon transform in medical imaging and
partial differential equations in various mechanical and physical
phenomena. Advanced undergraduate and graduate students, engineers
as well as practitioners wishing to familiarise themselves with
concepts and applications of analysis will find this book useful.
With its content split into several topics of interest, the book's
style and layout make it suitable for use in several courses, while
its self-contained character makes it appropriate for self-study.
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