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In mathematics and computer science, an algorithm is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions and use mathematical and logical tests to divert the code execution through various routes. Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus".
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming.
Mathematical optimization 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 subfields: discrete optimization and continuous optimization. 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 mathematical optimization, Dantzig's simplex algorithm is a popular algorithm for linear programming.
Johan Torkel Håstad is a Swedish theoretical computer scientist most known for his work on computational complexity theory. He was the recipient of the Gödel Prize in 1994 and 2011 and the ACM Doctoral Dissertation Award in 1986, among other prizes. He has been a professor in theoretical computer science at KTH Royal Institute of Technology in Stockholm, Sweden since 1988, becoming a full professor in 1992. He is a member of the Royal Swedish Academy of Sciences since 2001.
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.
Russell Graham Impagliazzo is a professor of computer science at the University of California, San Diego specializing in computational complexity theory, having joined the faculty of UCSD in 1989. He received a BA in mathematics from Wesleyan University. He obtained a doctorate from the University of California, Berkeley in 1992. His advisor was Manuel Blum. He is a 2004 Guggenheim fellow.
The Frank–Wolfe algorithm is an iterative first-order optimization algorithm for constrained convex optimization. Also known as the conditional gradient method, reduced gradient algorithm and the convex combination algorithm, the method was originally proposed by Marguerite Frank and Philip Wolfe in 1956. In each iteration, the Frank–Wolfe algorithm considers a linear approximation of the objective function, and moves towards a minimizer of this linear function.
In mathematical optimization, Bland's rule is an algorithmic refinement of the simplex method for linear optimization.
Dantzig–Wolfe decomposition is an algorithm for solving linear programming problems with special structure. It was originally developed by George Dantzig and Philip Wolfe and initially published in 1960. Many texts on linear programming have sections dedicated to discussing this decomposition algorithm.
In theoretical computer science, smoothed analysis is a way of measuring the complexity of an algorithm. Since its introduction in 2001, smoothed analysis has been used as a basis for considerable research, for problems ranging from mathematical programming, numerical analysis, machine learning, and data mining. It can give a more realistic analysis of the practical performance of the algorithm compared to analysis that uses worst-case or average-case scenarios.
Dimitri Panteli Bertsekas is an applied mathematician, electrical engineer, and computer scientist, a McAfee Professor at the Department of Electrical Engineering and Computer Science in School of Engineering at the Massachusetts Institute of Technology (MIT), Cambridge, Massachusetts, and also a Fulton Professor of Computational Decision Making at Arizona State University, Tempe.
Michael Ezra Saks is an American mathematician. He is currently the Department Chair of the Mathematics Department at Rutgers University (2017–) and from 2006 until 2010 was director of the Mathematics Graduate Program at Rutgers University. Saks received his Ph.D. from the Massachusetts Institute of Technology in 1980 after completing his dissertation titled Duality Properties of Finite Set Systems under his advisor Daniel J. Kleitman.
In mathematical optimization, the criss-cross algorithm is any of a family of algorithms for linear programming. Variants of the criss-cross algorithm also solve more general problems with linear inequality constraints and nonlinear objective functions; there are criss-cross algorithms for linear-fractional programming problems, quadratic-programming problems, and linear complementarity problems.
The Klee–Minty cube or Klee–Minty polytope is a unit hypercube of variable dimension whose corners have been perturbed. Klee and Minty demonstrated that George Dantzig's simplex algorithm has poor worst-case performance when initialized at one corner of their "squashed cube". On the three-dimensional version, the simplex algorithm and the criss-cross algorithm visit all 8 corners in the worst case.
In the study of algorithms, an LP-type problem is an optimization problem that shares certain properties with low-dimensional linear programs and that may be solved by similar algorithms. LP-type problems include many important optimization problems that are not themselves linear programs, such as the problem of finding the smallest circle containing a given set of planar points. They may be solved by a combination of randomized algorithms in an amount of time that is linear in the number of elements defining the problem, and subexponential in the dimension of the problem.
MINOS is a Fortran software package for solving linear and nonlinear mathematical optimization problems. MINOS may be used for linear programming, quadratic programming, and more general objective functions and constraints, and for finding a feasible point for a set of linear or nonlinear equalities and inequalities.
The Tucker Prize for outstanding theses in the area of optimization is sponsored by the Mathematical Optimization Society (MOS). Up to three finalists are presented at each (triennial) International Symposium of the MOS. The winner will receive an award of $1000 and a certificate. The Albert W. Tucker Prize was established by the Society in 1985, and was first awarded at the Thirteenth International Symposium on Mathematical Programming in 1988.
In mathematical optimization, Zadeh's rule is an algorithmic refinement of the simplex method for linear optimization.
In mathematical optimization, Cunningham's rule is an algorithmic refinement of the simplex method for linear optimization.
References
- ↑ "Heinz Schwärtzel Dissertation Award" (in German). Archived from the original on 2018-08-16. Retrieved 2018-03-14.
- ↑ Oliver Friedmann at the Mathematics Genealogy Project
- ↑ "Kleene Award Winners" . Retrieved 2018-03-14.
- ↑ "An Exponential Lower Bound for the Parity Game Strategy Improvement Algorithm as We Know it" . Retrieved 2018-03-14.
- ↑ "STOC Best Paper Award". Archived from the original on 2017-12-22. Retrieved 2018-03-14.
- ↑ "Günter Ziegler: 1000$ from Beverly Hills for a Math Problem". 20 January 2011. Retrieved 2018-03-14.
- ↑ "Exponential Lower Bounds for Solving Infinitary Payoff Games and Linear Programs" (Mathematical Optimization Society)
