Ivar Ekeland
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Ivar I. Ekeland (born 2 July 1944, Paris) is a French mathematician of Norwegian descent. Ekeland has written influential monographs and textbooks on nonlinear
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, the
calculus of variations The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
, and
mathematical economics Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference an ...
, as well as popular books on mathematics, which have been published in French, English, and other languages. Ekeland is known as the author of
Ekeland's variational principle In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exist nearly optimal solutions to some optimization problems. Ekeland's principle can be used when the lower level set of a ...
and for his use of the
Shapley–Folkman lemma The Shapley–Folkman  lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. It is named after mathematicians Lloyd Shapley and Jon Folkman, but was first published by the economist Ros ...
in
optimization theory Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
. He has contributed to the
periodic solution A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
s of
Hamiltonian system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can b ...
s and particularly to the theory of Kreĭn indices for linear systems (
Floquet theory Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form :\dot = A(t) x, with \displaystyle A(t) a Piecewise#Continuity, piecewise contin ...
).According to D. Pascali, writing for ''
Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...
'' ()
Ekeland helped to inspire the discussion of chaos theory in Michael Crichton's 1990 novel '' Jurassic Park''.


Biography

Ekeland studied at the
École Normale Supérieure École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi ...
(1963–1967). He is a senior research fellow at the
French National Centre for Scientific Research The French National Centre for Scientific Research (french: link=no, Centre national de la recherche scientifique, CNRS) is the French state research organisation and is the largest fundamental science agency in Europe. In 2016, it employed 31,637 ...
(CNRS). He obtained his doctorate in 1970. He teaches mathematics and economics at the
Paris Dauphine University Paris Dauphine University - PSL (french: Université Paris-Dauphine, also known as Paris Dauphine - PSL or Dauphine - PSL) is a public research university based in Paris, France. It is one of the 13 universities formed by the division of the ancie ...
, the
École Polytechnique École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, Savoi ...
, the
École Spéciale Militaire de Saint-Cyr The École spéciale militaire de Saint-Cyr (ESM, literally the "Special Military School of Saint-Cyr") is a French military academy, and is often referred to as Saint-Cyr (). It is located in Coëtquidan in Guer, Morbihan, Brittany. Its motto is ...
, and the
University of British Columbia The University of British Columbia (UBC) is a public university, public research university with campuses near Vancouver and in Kelowna, British Columbia. Established in 1908, it is British Columbia's oldest university. The university ranks a ...
in
Vancouver Vancouver ( ) is a major city in western Canada, located in the Lower Mainland region of British Columbia. As the List of cities in British Columbia, most populous city in the province, the 2021 Canadian census recorded 662,248 people in the ...
. He was the chairman of Paris-Dauphine University from 1989 to 1994. Ekeland is a recipient of the D'Alembert Prize and the Jean Rostand prize. He is also a member of the
Norwegian Academy of Science and Letters The Norwegian Academy of Science and Letters ( no, Det Norske Videnskaps-Akademi, DNVA) is a learned society based in Oslo, Norway. Its purpose is to support the advancement of science and scholarship in Norway. History The Royal Frederick Univer ...
.


Popular science: ''Jurassic Park'' by Crichton and Spielberg

Ekeland has written several books on
popular science ''Popular Science'' (also known as ''PopSci'') is an American digital magazine carrying popular science content, which refers to articles for the general reader on science and technology subjects. ''Popular Science'' has won over 58 awards, incl ...
, in which he has explained parts of
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pe ...
s, chaos theory, and
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expr ...
. These books were first written in French and then translated into English and other languages, where they received praise for their mathematical accuracy as well as their value as literature and as entertainment. Through these writings, Ekeland had an influence on ''
Jurassic Park ''Jurassic Park'', later also referred to as ''Jurassic World'', is an American science fiction media franchise created by Michael Crichton and centered on a disastrous attempt to create a theme park of cloned dinosaurs. It began in 1990 when ...
'', on both the novel and film. Ekeland's ''Mathematics and the unexpected'' and
James Gleick James Gleick (; born August 1, 1954) is an American author and historian of science whose work has chronicled the cultural impact of modern technology. Recognized for his writing about complex subjects through the techniques of narrative nonficti ...
's ''
Chaos Chaos or CHAOS may refer to: Arts, entertainment and media Fictional elements * Chaos (''Kinnikuman'') * Chaos (''Sailor Moon'') * Chaos (''Sesame Park'') * Chaos (''Warhammer'') * Chaos, in ''Fabula Nova Crystallis Final Fantasy'' * Cha ...
'' inspired the discussions of chaos theory in the novel '' Jurassic Park'' by Michael Crichton.In his afterword to ''
Jurassic Park ''Jurassic Park'', later also referred to as ''Jurassic World'', is an American science fiction media franchise created by Michael Crichton and centered on a disastrous attempt to create a theme park of cloned dinosaurs. It began in 1990 when ...
'', acknowledges the writings of Ekeland (and Gleick). Inside the novel,
fractal In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
s are discussed on two pages, , and
chaos theory Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have co ...
o
eleven pages, including pages 75, 158, and 245

When the novel was adapted for the film ''
Jurassic Park ''Jurassic Park'', later also referred to as ''Jurassic World'', is an American science fiction media franchise created by Michael Crichton and centered on a disastrous attempt to create a theme park of cloned dinosaurs. It began in 1990 when ...
'' by Steven Spielberg, Ekeland and Gleick were consulted by the actor Jeff Goldblum as he prepared to play the mathematician specializing in chaos theory.:


Research

Ekeland has contributed to
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, particularly to
variational calculus The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
and
mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
.


Variational principle

In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exist a nearly optimal solution to a class of
optimization problem In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions. Optimization problems can be divided into two categories, depending on whether the variables ...
s. Ekeland's variational principle can be used when the lower
level set In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is: : L_c(f) = \left\~, When the number of independent variables is two, a level set is calle ...
of a minimization problem is not
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
, so that the Bolzano–Weierstrass theorem can not be applied. Ekeland's principle relies on the completeness of the metric space. Ekeland's principle leads to a quick proof of the
Caristi fixed point theorem In mathematics, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the Banach fixed-point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem modifies the ''ε''- va ...
. Ekeland was associated with the
University of Paris , image_name = Coat of arms of the University of Paris.svg , image_size = 150px , caption = Coat of Arms , latin_name = Universitas magistrorum et scholarium Parisiensis , motto = ''Hic et ubique terrarum'' (Latin) , mottoeng = Here and a ...
when he proposed this theorem.


Variational theory of Hamiltonian systems

Ivar Ekeland is an expert on
variational analysis In mathematics, the term variational analysis usually denotes the combination and extension of methods from convex optimization and the classical calculus of variations to a more general theory. This includes the more general problems of optimizatio ...
, which studies
mathematical optimization Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
of spaces of functions. His research on
periodic solution A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to desc ...
s of
Hamiltonian system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can b ...
s and particularly to the theory of Kreĭn indices for linear systems (
Floquet theory Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form :\dot = A(t) x, with \displaystyle A(t) a Piecewise#Continuity, piecewise contin ...
) was described in his monograph.


Additive optimization problems

Ekeland explained the success of methods of convex minimization on large problems that appeared to be non-convex. In many optimization problems, the
objective function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
 f are ''separable'', that is, the sum of ''many'' summand-functions each with its own argument: : f(x) = f(x_1,\dots,x_N) = \sum_n f_n(x_n). For example, problems of
linear optimization Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, li ...
are separable. For a separable problem, we consider an optimal solution : x_\min = (x_1,\dots,x_N)_\min with the minimum value  For a separable problem, we consider an optimal solution (''x''min, ''f''(''x''min)) to the "''convexified problem''", where convex hulls are taken of the graphs of the summand functions. Such an optimal solution is the
limit of a sequence As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit ...
of points in the convexified problem : (x_j, f(x_j)) \in \mathrm(\mathrm(f_n)). \, An application of the
Shapley–Folkman lemma The Shapley–Folkman  lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. It is named after mathematicians Lloyd Shapley and Jon Folkman, but was first published by the economist Ros ...
represents the given optimal-point as a sum of points in the graphs of the original summands and of a small number of convexified summands. This analysis was published by Ivar Ekeland in 1974 to explain the apparent convexity of separable problems with many summands, despite the non-convexity of the summand problems. In 1973, the young mathematician
Claude Lemaréchal Claude Lemaréchal is a French applied mathematician, and former senior researcher (''directeur de recherche'') at INRIA near Grenoble, France. In mathematical optimization, Claude Lemaréchal is known for his work in numerical methods for nonlin ...
was surprised by his success with
convex minimization Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization pro ...
method Method ( grc, μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a prescribed process for completing a task. It may refer to: *Scien ...
s on problems that were known to be non-convex.: . Lemaréchal's experiments were discussed in later publications:
:

:
: Published in the first English edition of 1976, Ekeland's appendix proves the Shapley–Folkman lemma, also acknowledging Lemaréchal's experiments on page 373. Ekeland's analysis explained the success of methods of convex minimization on ''large'' and ''separable'' problems, despite the non-convexities of the summand functions.:
and also considered the ''
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
''  closure of a problem of non-convex minimization—that is, the problem defined by the
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
 
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
hull of the epigraph of the original problem. Their study of duality gaps was extended by Di Guglielmo to the ''
quasiconvex In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set. For a function of a single ...
'' closure of a non-convex minimization problem—that is, the problem defined by the
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
 
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...
hull of the
lower Lower may refer to: *Lower (surname) *Lower Township, New Jersey *Lower Receiver (firearms) *Lower Wick Lower Wick is a small hamlet located in the county of Gloucestershire, England. It is situated about five miles south west of Dursley, eight ...
level sets:

:
The Shapley–Folkman lemma has encouraged the use of methods of convex minimization on other applications with sums of many functions.: acknowledging on page 374 and on page 381:


describes an application of Lagrangian dual methods to the
scheduling A schedule or a timetable, as a basic time-management tool, consists of a list of times at which possible task (project management), tasks, events, or actions are intended to take place, or of a sequence of events in the chronological order ...
of electrical power plants (" unit commitment problems"), where non-convexity appears because of integer constraints:

:


Bibliography


Research

* (Corrected reprinting of the 1976 North-Holland () ed.) ::The book is cited over 500 times in
MathSciNet MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal ''Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links ...
. * * * (Reprint of the 1984 Wiley () ed.)


Exposition for a popular audience

* * *


See also

* Jonathan M. Borwein ("smooth" variational principle) * Robert R. Phelps (a "grandfather" of variational principles) *
David Preiss David Preiss FRS (born January 21, 1947) is a Czech and British mathematician, specializing in mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, s ...
("smooth" variational principle)


Notes


External links

*
Ekeland's webpage at CEREMADE

Conference on "Economics and Mathematics" by Ivar Ekeland, held at Canal U (2000)


{{DEFAULTSORT:Ekeland, Ivar Variational analysts Functional analysts Mathematical economists 20th-century French mathematicians 21st-century French mathematicians Canadian mathematicians French people of Norwegian descent Canadian people of Norwegian descent Canadian people of French descent 1944 births Living people École Normale Supérieure alumni University of British Columbia faculty Canada Research Chairs Textbook writers