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Mathematical optimization or mathematical programming is the selection of a best element,with regard to some criterion,from some set of available alternatives. Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics,and the development of solution methods has been of interest in mathematics for centuries.
Gradient descent is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient of the function at the current point,because this is the direction of steepest descent. Conversely,stepping in the direction of the gradient will lead to a local maximum of that function;the procedure is then known as gradient ascent.
In mathematics and mathematical optimization,the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation,Fenchel transformation,or Fenchel conjugate. It allows in particular for a far reaching generalization of Lagrangian duality.
Pierre-Louis Lions is a French mathematician. He is known for a number of contributions to the fields of partial differential equations and the calculus of variations. He was a recipient of the 1994 Fields Medal and the 1991 Prize of the Philip Morris tobacco and cigarette company.
In mathematics,a variational inequality is an inequality involving a functional,which has to be solved for all possible values of a given variable,belonging usually to a convex set. The mathematical theory of variational inequalities was initially developed to deal with equilibrium problems,precisely the Signorini problem:in that model problem,the functional involved was obtained as the first variation of the involved potential energy. Therefore,it has a variational origin,recalled by the name of the general abstract problem. The applicability of the theory has since been expanded to include problems from economics,finance,optimization and game theory.
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets. Many classes of convex optimization problems admit polynomial-time algorithms,whereas mathematical optimization is in general NP-hard.
The Instituto Nacional de Matemática Pura e Aplicada is widely considered to be the foremost research and educational institution of Brazil in the area of mathematics. It is located in the city of Rio de Janeiro,and was formerly known simply as Instituto de Matemática Pura e Aplicada,hence its official abbreviation.
Guido Stampacchia was a 20th-century Italian mathematician,known for his work on the theory of variational inequalities,the calculus of variation and the theory of elliptic partial differential equations.
Victor L. Klee,Jr. was a mathematician specialising in convex sets,functional analysis,analysis of algorithms,optimization,and combinatorics. He spent almost his entire career at the University of Washington in Seattle.
Boris Mordukhovich is an American mathematician recognized for his research in the areas of nonlinear analysis,optimization,and control theory. Mordukhovich is one of the founders of modern variational analysis and generalized differentiation. Currently he is Distinguished University Professor and Lifetime Scholar of the Academy of Scholars at Wayne State University.
Paul Tseng was a Chinese-American and Canadian applied mathematician and a professor at the Department of Mathematics at the University of Washington,in Seattle,Washington. Tseng was recognized by his peers to be one of the leading optimization researchers of his generation. On August 13,2009,Paul Tseng went missing while kayaking in the Yangtze River in the Yunnan province of China and is presumed dead.
Roger Jean-Baptiste Robert Wets is a "pioneer" in stochastic programming and a leader in variational analysis who publishes as Roger J-B Wets. His research,expositions,graduate students,and his collaboration with R. Tyrrell Rockafellar have had a profound influence on optimization theory,computations,and applications. Since 2009,Wets has been a distinguished research professor at the mathematics department of the University of California,Davis.
Ralph Tyrrell Rockafellar is an American mathematician and one of the leading scholars in optimization theory and related fields of analysis and combinatorics. He is the author of four major books including the landmark text "Convex Analysis" (1970),which has been cited more than 27000 times according to Google Scholar and remains the standard reference on the subject,and "Variational Analysis" for which the authors received the Frederick W. Lanchester Prize from the Institute for Operations Research and the Management Sciences (INFORMS).
Augmented Lagrangian methods are a certain class of algorithms for solving constrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem by a series of unconstrained problems and add a penalty term to the objective;the difference is that the augmented Lagrangian method adds yet another term,designed to mimic a Lagrange multiplier. The augmented Lagrangian is related to,but not identical with the method of Lagrange multipliers.
Ivar I. Ekeland is a French mathematician of Norwegian descent. Ekeland has written influential monographs and textbooks on nonlinear functional analysis,the calculus of variations,and mathematical economics,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 and for his use of the Shapley–Folkman lemma in optimization theory. He has contributed to the periodic solutions of Hamiltonian systems and particularly to the theory of Kreĭn indices for linear systems. Ekeland helped to inspire the discussion of chaos theory in Michael Crichton's 1990 novel Jurassic Park.
Fernando Codádos Santos Cavalcanti Marques is a Brazilian mathematician working mainly in geometry,topology,partial differential equations and Morse theory. He is a professor at Princeton University. In 2012,together with AndréNeves,he proved the Willmore conjecture.
Gheorghe Moroșanu is a Romanian mathematician known for his works in Ordinary and Partial Differential Equations,Operator Theory,Integral Equations,Calculus of Variations and Optimal Control,Fluid Mechanics,Difference Equations,and other branches of Mathematics. He earned his Ph.D. in 1981 from the Alexandru Ioan Cuza University in Iași.
Djairo Guedes de Figueiredo is a Brazilian mathematician noted for his researches on differential equations,elliptic operators,and calculus of variations. He is considered the greatest analyst from Brazil. He was the president of the Brazilian Mathematical Society from 1977 to 1979.
Artur Oscar Lopes is a Brazilian writer and mathematician working on dynamical systems and ergodic theory. He is a professor at UFRGS,Porto Alegre.
Regina Sandra Burachik is an Argentine mathematician who works on optimization and analysis. Currently,she is a professor at the University of South Australia.
References
- 1 2 "Academia Brasileira de Ciências". Archived from the original on 2015-05-22. Retrieved 2015-05-22.
- 1 2 Alfredo Noel Iusem at the Mathematics Genealogy Project
- ↑ "Portal do Ministério da Ciência, Tecnologia e Inovação". mct.gov.br. Archived from the original on 2015-04-18.
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