Table Of ContentEmergence, Complexity and Computation ECC
Yaroslav D. Sergeyev
Renato De Leone Editors
Numerical
Infinities and
Infinitesimals
in Optimization
Emergence, Complexity and Computation
Volume 43
SeriesEditors
IvanZelinka,TechnicalUniversityofOstrava,Ostrava,CzechRepublic
AndrewAdamatzky,UniversityoftheWestofEngland,Bristol,UK
GuanrongChen,CityUniversityofHongKong,HongKong,China
EditorialBoard
AjithAbraham,MirLabs,USA
AnaLucia,UniversidadeFederaldoRioGrandedoSul,PortoAlegre,Rio
GrandedoSul,Brazil
JuanC.Burguillo,UniversityofVigo,Spain
SergejCˇelikovský,AcademyofSciencesoftheCzechRepublic,Czech
Republic
MohammedChadli,UniversityofJulesVerne,France
EmilioCorchado,UniversityofSalamanca,Spain
DonaldDavendra,TechnicalUniversityofOstrava,CzechRepublic
AndrewIlachinski,CenterforNavalAnalyses,USA
JouniLampinen,UniversityofVaasa,Finland
MartinMiddendorf,UniversityofLeipzig,Germany
EdwardOtt,UniversityofMaryland,USA
LinqiangPan,HuazhongUniversityofScienceandTechnology,Wuhan,China
GheorghePa˘un,RomanianAcademy,Bucharest,Romania
HendrikRichter,HTWKLeipzigUniversityofAppliedSciences,Germany
JuanA.Rodriguez-Aguilar ,IIIA-CSIC,Spain
OttoRössler,InstituteofPhysicalandTheoreticalChemistry,Tübingen,
Germany
YaroslavD.Sergeyev,UniversityofCalabria,Italy
VaclavSnasel,TechnicalUniversityofOstrava,CzechRepublic
IvoVondrák,TechnicalUniversityofOstrava,CzechRepublic
HectorZenil,KarolinskaInstitute,Sweden
The Emergence, Complexity and Computation (ECC) series publishes new
developments,advancementsandselectedtopicsinthefieldsofcomplexity,
computationandemergence.Theseriesfocusesonallaspectsofreality-based
computation approaches from an interdisciplinary point of view especially
from applied sciences, biology, physics, or chemistry. It presents new ideas
andinterdisciplinaryinsightonthemutualintersectionofsubareasofcompu-
tation,complexityandemergenceanditsimpactandlimitstoanycomputing
basedonphysicallimits(thermodynamicandquantumlimits,Bremermann’s
limit, Seth Lloyd limits…) as well as algorithmic limits (Gödel’s proof
and its impact on calculation, algorithmic complexity, the Chaitin’s Omega
numberandKolmogorovcomplexity,non-traditionalcalculationslikeTuring
machineprocessanditsconsequences,…)andlimitationsarisinginartificial
intelligence.Thetopicsare(butnotlimitedto)membranecomputing,DNA
computing, immune computing, quantum computing, swarm computing,
analogiccomputing,chaoscomputingandcomputingontheedgeofchaos,
computational aspects of dynamics of complex systems (systems with
self-organization, multiagent systems, cellular automata, artificial life,…),
emergence of complex systems and its computational aspects, and agent
based computation. The main aim of this series is to discuss the above
mentioned topics from an interdisciplinary point of view and present new
ideascomingfrommutualintersectionofclassicalaswellasmodernmethods
ofcomputation.Withinthescopeoftheseriesaremonographs,lecturenotes,
selectedcontributionsfromspecializedconferencesandworkshops,special
contributionfrominternationalexperts.
IndexedbyzbMATH.
More information about this series at https://link.springer.com/bookseries/
10624
·
Yaroslav D. Sergeyev Renato De Leone
Editors
Numerical Infinities
and Infinitesimals
in Optimization
Editors
YaroslavD.Sergeyev RenatoDeLeone
DipartimentodiIngegneria ScuoladiScienzeeTecnologie
Informatica,Modellistica,Elettronicae UniversitàdegliStudidiCamerino
Sistemistica Camerino,Italy
UniversityofCalabria
Rende,Italy
LobachevskyUniversity
NizhnyNovgorod,Russia
ISSN2194-7287 ISSN2194-7295 (electronic)
Emergence,ComplexityandComputation
ISBN978-3-030-93641-9 ISBN978-3-030-93642-6 (eBook)
https://doi.org/10.1007/978-3-030-93642-6
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringer
NatureSwitzerlandAG2022
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The editors thank their families for their
love, support, and really infinite patience
during the years of preparation of this
book on ➀ infinity.
To Adriana and Ekaterina
To Chiara, Dmitriy, and Robert
Preface
Thisbookisdedicatedtotwotopicsthat,uptotherecenttimes,werebelieved
incompatible.Infact,infinitiesandinfinitesimalsforcenturieswereconsid-
ered within highly theoretical research areas whereas optimization is a part
of applied mathematics and deals with algorithms and codes implemented
oncomputers.Thepossibilitytojointhesetwotopicsunderthesamecover
isaconsequenceofthefactthatinfinitiesandinfinitesimalsconsideredhere
are not traditional symbolic entities but numerical ones, namely, they can
be represented on a new supercomputer patented in several countries and
called the Infinity Computer. This computer is able to execute numerical
(i.e.,floating-point)operationswithnumbersthatcanhavedifferentinfinite,
finite, and infinitesimal parts represented in the positional numeral system
withaninfinitebasecalledgrossoneexpressedbythenumeral➀.Thenumer-
ical character of these infinite and infinitesimal numbers is one of the key
differenceswithrespecttotraditionaltheoriesofinfinityandinfinitesimals.
This novel point of view opens completely new horizons not only in pure
mathematicsbutalsoincomputerscienceandappliedmathematics.
Duringthelastyearstherewasanincreasinginterestinthisnewcomputa-
tionalparadigm.In2013,2016,and2019therewereinternationalconferences
“NUMTA—Numerical Computations: Theory and Algorithms” where the
InfinityComputeranditsapplicationswereamongthemaintopics.Several
special issues in leading scientific journals have been published. The last
NUMTAconferencein2019hadmorethan200participantsfrom30coun-
tries. Three special issues and proceedings in volumes 11973 and 11974 of
the Springer book series “Lecture Notes in Computer Science” have been
published.
Theinterestoftheinternationalscientificcommunitytothisparadigmis
also due to the fact that, for the first time, infinities and infinitesimals were
vii
viii Preface
involvednotonlyinscientificfieldsrelatedtopuremathematicssuchasfoun-
dationsofmathematics,logic,andphilosophybutalsoincomputerscience
andappliedmathematics.Moreover,thereexistseveralsoftwaresimulatorsof
theInfinityComputerthatareactivelyusedbyresearchersintheirworkwith
numerical methods. Nowadays among the fields of application of the new
computational paradigm we find: numerical differentiation and numerical
solution of ordinary differential equations, game theory, probability theory,
cellularautomata,Euclideanandhyperbolicgeometry,percolation,fractals,
infiniteseriesandtheRiemannzetafunction,thefirstHilbertproblem,Turing
machines,teaching,etc.AhugebulkofresultsusingtheInfinityComputer
paradigm in optimization (linear, non-linear, global, and multi-criteria) is
collectedinthisvolumeco-authoredby22leadingexpertsworkinginthese
fields. Thus, the reader will have a unique opportunity to find in the same
book the state-of-the-art knowledge and practices in optimization using the
InfinityComputingmethodology.
The first two chapters of the book written by the editors have an intro-
ductorycharacteranddescribetheInfinityComputermethodology(Chapter
“ANewComputationalParadigmUsingGrossone-BasedNumericalInfini-
ties and Infinitesimals”, written by Yaroslav D. Sergeyev) and some of
the most important results in unconstrained and constrained optimization
(Chapter “Nonlinear Optimization: A Brief Overview”, written by Renato
De Leone). Chapter “The Role of grossone in Nonlinear Programming
andExactPenaltyMethods”,alsowrittenbyRenatoDeLeone,isdedicatedto
exactpenaltymethodsforsolvingconstrainedoptimizationproblems.Using
penaltyfunctions,theoriginalconstrainedoptimizationproblemcanbetrans-
formed in an unconstrained one. The chapter shows how ➀ can be utilized
in constructing exact continuously differentiable penalty functions for the
caseofonlyequalityconstraints,thegeneralcaseofequalityandinequality
constraints, and quadratics problems. It is also discussed how these new
penaltyfunctionsallowonetorecovertheexactsolutionoftheunconstrained
problemfromtheoptimalsolutionoftheunconstrainedproblem.Moreover,
Lagrangian duals associated to the constraints can also be automatically
obtainedthankstothe➀-basedpenaltyfunctions.
Chapter “Krylov-Subspace Methods for Quadratic Hypersurfaces:
aGrossone–basedPerspective”iswrittenbyGiovanniFasanowhostudiesthe
roleof➀todealwithtworenownedKrylov-subspacemethodsforsymmetric
(possiblyindefinite)linearsystems.First,theauthorexploresarelationship
between the Conjugate Gradient method and the Lanczos process, along
with their role of yielding tridiagonal matrices which retain large infor-
mation on the original linear system matrix. Then, he shows that coupling
Preface ix
➀ with the Conjugate Gradient method provides clear theoretical improve-
ments. Furthermore, reformulating the iteration of this algorithm using this
➀-based framework allows the author to encompass also a certain number
of Krylov-subspace methods relying on conjugacy among vectors. The last
generalizationremarkablyjustifiestheuseofa➀-basedreformulationofthe
Conjugate Gradient method to solve also indefinite linear systems. Finally,
pairing this method with the algebra of grossone easily provides relevant
geometricpropertiesofquadratichypersurfaces.
Marco Cococcioni, Alessandro Cudazzo, Massimo Pappalardo, and
Yaroslav D. Sergeyev show in Chapter “Multi-objective Lexicographic
Mixed-IntegerLinearProgramming:AnInfinityComputerApproach”how
a lexicographic multi-objective linear programming problem can be trans-
formed into an equivalent single-objective problem by using the grossone
methodology. Then, the authors provide a simplex-like algorithm, called
GrossSimplex, able to solve the original problem using a single run of the
algorithm. In the second part of the chapter, the authors tackle the mixed-
integerlexicographicmulti-objectivelinearprogrammingproblemandsolve
it in an exact way, by using a ➀-version of the Branch-and-Bound scheme.
Afterprovingthetheoreticalcorrectnessoftheassociatedpruningrulesand
terminatingconditions,theypresentafewexperimentalresultsexecutedon
anInfinityComputersimulator.
In Chapter “The Use of Infinities and Infinitesimals for Sparse Classi-
fication Problems”, Renato De Leone, Nadaniela Egidi, and Lorella Fatone
discusstheuseofgrossoneindeterminingsparsesolutionsforspecialclasses
of optimization problems. In fact, in various optimization and regression
problems, and in solving overdetermined systems of linear equations it is
oftennecessarytodetermineasparsesolution,thatisasolutionwithasmany
aspossiblezerocomponents.Theauthorsshowhowcontinuouslydifferen-
tiable concave approximations of the pseudoâe“norm can be constructed
using ➀ and discuss properties of some new approximations. Finally, the
authors present some applications in elastic net regularization and Sparse
SupportVectorMachine.
Chapter “The Grossone-Based Diagonal Bundle Method” written by
Manlio Gaudioso, Giovanni Giallombardo, and Marat S. Mukhametzhanov
discussesafruitfulimpactoftheInfinityComputingparadigmonthepractical
solutionofconvexnonsmoothoptimizationproblems.Theauthorsconsider
aclassofmethodsbasedonavariablemetricapproach:theuseoftheInfinity
Computingtechniquesallowsthemtonumericallydealwithquantitieswhich
cantakearbitrarilysmallorlargevalues,asaconsequenceofnonsmoothness.
Inparticular,bychoosingadiagonalmatrixwithpositiveentriesasametric,
the authors modify the well-known Diagonal Bundle algorithm by means
x Preface
of matrix updates based on the Infinity Computing paradigm and provide
computationalresultsobtainedonasetofbenchmarktestproblems.
Chapter “On the Use of Grossone Methodology for Handling Priorities
in Multi-objective Evolutionary Optimization” written by Leonardo Lai,
Lorenzo Fiaschi, Marco Cococcioni, and Kalyanmoy Deb describes a new
class of problems, called mixed Pareto-lexicographic multi-objective opti-
mizationproblems,asuitablemodelforscenarioswheresomeobjectiveshave
priorityoversomeothers.Tworelevantsubclassesofthisproblemareconsid-
ered:prioritychainsandprioritylevels.ItisshownthattheInfinityComputing
methodologyallowsonetohandleprioritiesefficiently.Itisremarkablethat
this technique can be easily embedded in most of the existing evolutionary
algorithms,withoutalteringtheircorelogic.Threealgorithmsaredescribed
and tested on benchmark problems, including some real-world problems.
The experiments show that the algorithms using the Infinity Computing
methodologyareabletoproducemoresolutionsandofhigherquality.
Chapter “Exact Numerical Differentiation on the Infinity Computer
and Applications in Global Optimization” written by Maria Chiara Nasso
and Yaroslav D. Sergeyev shows how exact numerical differentiation can
be executed on the Infinity Computer and efficiently applied in Lipschitz
globaloptimizationforcomputingderivatives.Optimizationalgorithmsusing
smooth piece-wise quadratic support functions to approximate the global
minimumarealsodiscussedandtheirconvergenceconditionsareprovided.
It is shown that all the methods can be implemented both in the tradi-
tional floating-point arithmetic and in the novel Infinity Computing frame-
work.Numericalexperimentsconfirmthatmethodsusinganalyticderivatives
and derivatives computed numerically on the Infinity Computer exhibit the
identicalbehavior.
Chapter “Comparing Linear and Spherical Separation Using
Grossone-Based Numerical Infinities in Classification Problems” authored
by Annabella Astorino and Antonio Fuduli investigates the role played
by the linear and spherical separations in binary supervised learning and
in Multiple Instance Learning (MIL), in connection with the use of the
grossone-based numerical infinities. While in classical binary supervised
learning the objective is to separate two sets of samples, a binary MIL
problemconsistsinseparatingtwodifferenttypeofsets(positiveandnega-
tive), each of them constituted by a finite number of samples. The authors
focusonthepossibilitytoconstructbinarysphericalclassifierscharacterized
by an infinitely far center adopting the Infinity Computing methodology.
They show that this approach allows them to obtain a good performance in
terms of average testing correctness and to manage very easily numerical
computationswithoutanytuningofthe“bigM”parameter.
