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Integer Linear Program
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear. Integer programming is NP-complete. In particular, the special case of 0-1 integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's 21 NP-complete problems. If some decision variables are not discrete, the problem is known as a mixed-integer programming problem. Canonical and standard form for ILPs In integer linear programming, the ''canonical form'' is distinct from the ''standard form''. An integer linear program in canonical form is expressed thus (note that it is the \mathbf vector which is to be decided): : \begin & \text && \mathbf^\mathrm \mathbf\\ & \text && A \mathbf \le \mathbf, \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 the more general approach, an optimization problem consists of maxima and minima, maximizing or minimizing a Function of a real variable, real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming Relaxation
In mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constraints are of the form :x_i\in\. The relaxation of the original integer program instead uses a collection of linear constraints :0 \le x_i \le 1. The resulting relaxation is a linear program, hence the name. This relaxation technique transforms an NP-hard optimization problem (integer programming) into a related problem that is solvable in polynomial time (linear programming); the solution to the relaxed linear program can be used to gain information about the solution to the original integer program. Example Consider the set cover problem, the linear programming relaxation of which was first considered by . In this problem, one is given as input a family of sets ''F'' = ; the task is to find a subfamily, with as few sets as possible, having the same union as ''F''. To formulate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hill Climbing
numerical analysis, hill climbing is a mathematical optimization technique which belongs to the family of local search. It is an iterative algorithm that starts with an arbitrary solution to a problem, then attempts to find a better solution by making an incremental change to the solution. If the change produces a better solution, another incremental change is made to the new solution, and so on until no further improvements can be found. For example, hill climbing can be applied to the travelling salesman problem. It is easy to find an initial solution that visits all the cities but will likely be very poor compared to the optimal solution. The algorithm starts with such a solution and makes small improvements to it, such as switching the order in which two cities are visited. Eventually, a much shorter route is likely to be obtained. Hill climbing finds optimal solutions for convex problems – for other problems it will find only local optima (solutions that cannot be imp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tabu Search
Tabu search is a metaheuristic search method employing local search methods used for mathematical optimization. It was created by Fred W. Glover in 1986 and formalized in 1989. Local (neighborhood) searches take a potential solution to a problem and check its immediate neighbors (that is, solutions that are similar except for very few minor details) in the hope of finding an improved solution. Local search methods have a tendency to become stuck in suboptimal regions or on plateaus where many solutions are equally fit. Tabu search enhances the performance of local search by relaxing its basic rule. First, at each step ''worsening'' moves can be accepted if no improving move is available (like when the search is stuck at a strict local minimum). In addition, ''prohibitions'' (henceforth the term ''tabu'') are introduced to discourage the search from coming back to previously-visited solutions. The implementation of tabu search uses memory structures that describe the visited ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-hard
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard problem is the subset sum problem. A more precise specification is: a problem ''H'' is NP-hard when every problem ''L'' in NP can be reduced in polynomial time to ''H''; that is, assuming a solution for ''H'' takes 1 unit time, ''H''s solution can be used to solve ''L'' in polynomial time. As a consequence, finding a polynomial time algorithm to solve any NP-hard problem would give polynomial time algorithms for all the problems in NP. As it is suspected that P≠NP, it is unlikely that such an algorithm exists. It is suspected that there are no polynomial-time algorithms for NP-hard problems, but that has not been proven. Moreover, the class P, in which all problems can be solved in polynomial time, is contained in the NP class. Defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry Of Numbers
Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental information on algebraic numbers. The geometry of numbers was initiated by . The geometry of numbers has a close relationship with other fields of mathematics, especially functional analysis and Diophantine approximation, the problem of finding rational numbers that approximate an irrational quantity. Minkowski's results Suppose that \Gamma is a lattice in n-dimensional Euclidean space \mathbb^n and K is a convex centrally symmetric body. Minkowski's theorem, sometimes called Minkowski's first theorem, states that if \operatorname (K)>2^n \operatorname(\mathbb^n/\Gamma), then K contains a nonzero vector in \Gamma. The successive minimum \lambda_k is defined to be the inf of the numbers \lambda such that \lambda K contains k linearly independ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Peter Van Emde Boas
Peter van Emde Boas (born 3 April 1945, Amsterdam) is a Dutch computer scientist and professor at the University of Amsterdam. He gained his doctorate in 1974 under Adriaan van Wijngaarden. The Van Emde Boas tree A van Emde Boas tree (), also known as a vEB tree or van Emde Boas priority queue, is a tree data structure which implements an associative array with -bit integer keys. It was invented by a team led by Dutch computer scientist Peter van Emde Boa ... is named after him.Peter van Emde Boas '' Preserving order in a forest in less than logarithmic time'', Proceedings of the 16th Annual Symposium on Foundations of Computer Science, 1975, S. 75-84 - doi:10.1109/SFCS.1975.26 References External links Homepage 1945 births Living people Dutch computer scientists Theoretical computer scientists University of Amsterdam faculty University of Amsterdam alumni Scientists from Amsterdam {{Netherlands-scientist-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
László Lovász
László Lovász (; born March 9, 1948) is a Hungarian mathematician and professor emeritus at Eötvös Loránd University, best known for his work in combinatorics, for which he was awarded the 2021 Abel Prize jointly with Avi Wigderson. He was the president of the International Mathematical Union from 2007 to 2010 and the president of the Hungarian Academy of Sciences from 2014 to 2020. In graph theory, Lovász's notable contributions include the proofs of Kneser's conjecture and the Lovász local lemma, as well as the formulation of the Erdős–Faber–Lovász conjecture. He is also one of the eponymous authors of the LLL lattice reduction algorithm. Early life and education Lovász was born on March 9, 1948, in Budapest, Hungary. Lovász attended the Fazekas Mihály Gimnázium in Budapest. He won three gold medals (1964–1966) and one silver medal (1963) at the International Mathematical Olympiad. He also participated in a Hungarian game show about math prodigies. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hendrik Lenstra
Hendrik Willem Lenstra Jr. (born 16 April 1949, Zaandam) is a Dutch mathematician. Biography Lenstra received his doctorate from the University of Amsterdam in 1977 and became a professor there in 1978. In 1987 he was appointed to the faculty of the University of California, Berkeley; starting in 1998, he divided his time between Berkeley and the University of Leiden, until 2003, when he retired from Berkeley to take a full-time position at Leiden. Three of his brothers, Arjen Lenstra, Andries Lenstra, and Jan Karel Lenstra, are also mathematicians. Jan Karel Lenstra is the former director of the Netherlands Centrum Wiskunde & Informatica (CWI). Hendrik Lenstra was the Chairman of the Program Committee of the International Congress of Mathematicians in 2010. Scientific contributions Lenstra has worked principally in computational number theory. He is well known for: * Co-discovering of the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (in 1982); * Developing a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Scarf
Herbert Eli "Herb" Scarf (July 25, 1930 – November 15, 2015) was an American mathematical economist and Sterling Professor of Economics at Yale University. Education and career Scarf was born in Philadelphia, the son of Jewish emigrants from Ukraine and Russia, Lene (Elkman) and Louis Scarf. During his undergraduate work he finished in the top 10 of the 1950 William Lowell Putnam Mathematical Competition, the major mathematics competition between universities across the United States and Canada. He received his PhD from Princeton in 1954, supervised by Salomon Bochner. Contributions Among his notable works is a seminal paper in cooperative game in which he showed sufficiency for a core (economics) in general balanced games. Sufficiency and necessity had been previously shown by Lloyd Shapley for games where players were allowed to transfer utility between themselves freely. Necessity is shown to be lost in the generalization. Recognition Scarf received the 1973 Frederick ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branch And Cut
Branch and cut is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some or all the unknowns are restricted to integer values. Branch and cut involves running a branch and bound algorithm and using cutting planes to tighten the linear programming relaxations. Note that if cuts are only used to tighten the initial LP relaxation, the algorithm is called branch and cut. Algorithm This description assumes the ILP is a maximization problem. The method solves the linear program without the integer constraint using the regular simplex algorithm. When an optimal solution is obtained, and this solution has a non-integer value for a variable that is supposed to be integer, a cutting plane algorithm may be used to find further linear constraints which are satisfied by all feasible integer points but violated by the current fractional solution. These inequalities may be added to the linear program, such that r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Branch And Bound
Branch and bound (BB, B&B, or BnB) is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm consists of a systematic enumeration of candidate solutions by means of state space search: the set of candidate solutions is thought of as forming a rooted tree with the full set at the root. The algorithm explores ''branches'' of this tree, which represent subsets of the solution set. Before enumerating the candidate solutions of a branch, the branch is checked against upper and lower estimated ''bounds'' on the optimal solution, and is discarded if it cannot produce a better solution than the best one found so far by the algorithm. The algorithm depends on efficient estimation of the lower and upper bounds of regions/branches of the search space. If no bounds are available, the algorithm degenerates to an exhaustive search. The method was first proposed by Ailsa Land and Alison Doig whilst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
