Mathematics is an area of knowledge that includes such topics as numbers, formulas and related structures (algebra), shapes and the spaces in which they are contained (geometry), and quantities and their changes. Most mathematical activity involves the use of pure reason to discover or prove the properties of abstract objects, which consist of either abstractions from nature or—in modern mathematics—entities that are stipulated with certain properties, called axioms. A mathematical proof consists of a succession of applications of some deductive rules to already known results, including previously proved theorems, axioms and some basic properties that are considered as true starting points of the theory under consideration.
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.
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing in a given space, a kissing number can also be defined for each individual sphere as the number of spheres it touches. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another.
AMPL is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing . It was developed by Robert Fourer, David Gay, and Brian Kernighan at Bell Laboratories. AMPL supports dozens of solvers, both open source and commercial software, including CBC, CPLEX, FortMP, MINOS, IPOPT, SNOPT, KNITRO, and LGO. Problems are passed to solvers as nl files. AMPL is used by more than 100 corporate clients, and by government agencies and academic institutions.
The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at each (triennial) International Symposium of the MOS. Originally, the prizes were paid out of a memorial fund administered by the AMS that was established by friends of the late Delbert Ray Fulkerson to encourage mathematical excellence in the fields of research exemplified by his work. The prizes are now funded by an endowment administered by MPS.
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.
Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron.
The Mathematical Optimization Society (MOS), known as the Mathematical Programming Society until 2010, is an international association of researchers active in optimization. The MOS encourages the research, development, and use of optimization—including mathematical theory, software implementation, and practical applications.
In mathematical optimization, a quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and the constraints are quadratic functions. It has the form
Ellis Lane Johnson is the Professor Emeritus and the Coca-Cola Chaired Professor in the H. Milton Stewart School of Industrial and Systems Engineering at Georgia Institute of Technology in Atlanta, Georgia.
Yinyu Ye is a Chinese American theoretical computer scientist working on mathematical optimization. He is a specialist in interior point methods, especially in convex minimization and linear programming. He is a professor of Management Science and Engineering and Kwoh-Ting Li Chair Professor of Engineering at Stanford University. He also holds a courtesy appointment in the Department of Electrical Engineering. Ye also is a co-founder of minMax Optimization Inc.
Robert J. Vanderbei is an American mathematician and Professor in the Department of Operations Research and Financial Engineering at Princeton University.
Arkadi Nemirovski is a professor at the H. Milton Stewart School of Industrial and Systems Engineering at the Georgia Institute of Technology. He has been a leader in continuous optimization and is best known for his work on the ellipsoid method, modern interior-point methods and robust optimization.
Michel Xavier Goemans is a Belgian-American professor of applied mathematics and the RSA Professor of Mathematics at MIT working in discrete mathematics and combinatorial optimization at CSAIL and MIT Operations Research Center.
Alexander (Lex) Schrijver is a Dutch mathematician and computer scientist, a professor of discrete mathematics and optimization at the University of Amsterdam and a fellow at the Centrum Wiskunde & Informatica in Amsterdam. Since 1993 he has been co-editor in chief of the journal Combinatorica.
Yurii Nesterov is a Russian mathematician, an internationally recognized expert in convex optimization, especially in the development of efficient algorithms and numerical optimization analysis. He is currently a professor at the University of Louvain (UCLouvain).
Martin Grötschel is a German mathematician known for his research on combinatorial optimization, polyhedral combinatorics, and operations research. From 1991 to 2012 he was Vice President of the Zuse Institute Berlin (ZIB) and served from 2012 to 2015 as ZIB's President. From 2015 to 2020 he was President of the Berlin-Brandenburg Academy of Sciences and Humanities (BBAW).
Katya Scheinberg is a Russian-American applied mathematician known for her research in continuous optimization and particularly in derivative-free optimization. She works at Cornell University and is a professor in the School of Operations Research and Information Engineering.
Kim-Chuan Toh is a Singaporean mathematician, and Leo Tan Professor in Science at the National University of Singapore (NUS). He is known for his contributions to the theory, practice, and application of convex optimization, especially semidefinite programming and conic programming.
In convex geometry and polyhedral combinatorics, the extension complexity is a convex polytope is the smallest number of facets among convex polytopes that have as a projection. In this context, is called an extended formulation of ; it may have much higher dimension than .
