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# Linear programming

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 relationships. [1]

## Affine transformation

In geometry, an affine transformation, affine mapBerger, Marcel (1987), p. 38.

## AIMMS

AIMMS (an acronym for "Advanced Interactive Multidimensional Modeling System") is a software system designed for modeling and solving large-scale optimization and scheduling-type problems.

## Albert W. Tucker

Albert William Tucker (28 November 1905 – 25 January 1995) was a Canadian mathematician who made important contributions in topology, game theory, and non-linear programming.

## Algebraic modeling language

Algebraic Modeling Languages (AML) are high-level computer programming languages for describing and solving high complexity problems for large scale mathematical computation (i.e. large scale optimization type problems).

## Algorithm

In mathematics and computer science, an algorithm is a self-contained step-by-step set of operations to be performed.

## AMPL

AMPL, an acronym for "A Mathematical Programming Language", is an algebraic modeling language for describing and solving high-complexity problems for large-scale mathematical computation (i.e. large-scale optimization and scheduling-type problems).

The Apache License is a free software license written by the Apache Software Foundation (ASF).

## APMonitor

Advanced process monitor (APMonitor), is a modeling language for differential algebraic (DAE) equations.

## Application programming interface

In computer programming, an application programming interface (API) is a set of routines, protocols, and tools for building software applications.

## Approximation algorithm

In computer science and operations research, approximation algorithms are algorithms used to find approximate solutions to optimization problems.

## Assignment problem

The assignment problem is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics.

## Barrier function

In constrained optimization, a field of mathematics, a barrier function is a continuous function whose value on a point increases to infinity as the point approaches the boundary of the feasible region (Nocedal and Wright 1999).

## Benders' decomposition

Benders' decomposition (or Benders's decomposition) is a technique in mathematical programming that allows the solution of very large linear programming problems that have a special block structure.

## Block diagram

A block diagram is a diagram of a system in which the principal parts or functions are represented by blocks connected by lines that show the relationships of the blocks.

## Block matrix

In mathematics, a block matrix or a partitioned matrix is a matrix which is interpreted as having been broken into sections called blocks or submatrices.

## Branch and bound

Branch and bound (BB or B&B) is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as general real valued problems.

## 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 price

In applied mathematics, branch and price is a method of combinatorial optimization for solving integer linear programming (ILP) and mixed integer linear programming (MILP) problems with many variables.

BSD licenses are a family of permissive free software licenses, imposing minimal restrictions on the redistribution of covered software.

## Canonical form

In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression.

## Cassowary (software)

Cassowary is an incremental constraint solving toolkit that efficiently solves systems of linear equalities and inequalities.

Christos Harilaos Papadimitriou (Greek: Χρήστος Χαρίλαος Παπαδημητρίου; born August 16, 1949, Athens) is a Greek engineer, computer scientist, and professor of Computer Science at the University of California, Berkeley.

## Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but in any case does not involve any variables of the expression.

## COIN-OR

COIN-OR, which stands for Computational Infrastructure for Operations Research, is a project that aims to "create for mathematical software what the open literature is for mathematical theory." The open literature (e.g., a research journal) provides the OR community with a peer-review process and an archive.

## Column generation

Column generation or delayed column generation is an efficient algorithm for solving larger linear programs.

## Combinatorial optimization

In applied mathematics and theoretical computer science, combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects.

## Comparability

In mathematics, any two elements x and y of a set P that is partially ordered by a binary relation ≤ are comparable when either x ≤ y or y ≤ x. If it is not the case that x and y are comparable, then they are called incomparable.

## Computers and Intractability

In computer science, more specifically computational complexity theory, Computers and Intractability: A Guide to the Theory of NP-Completeness is an influential textbook by Michael Garey and David S. Johnson.

## Concave function

In mathematics, a concave function is the negative of a convex function.

## Cone

A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.

## Constraint (mathematics)

In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy.

## Convex function

In mathematics, a real-valued function defined on an interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph, in a Euclidean space (or more generally a vector space) of at least two dimensions.

## Convex lattice polytope

A convex lattice polytope (also called Z-polyhedron or Z-polytope) is a geometric object playing an important role in discrete geometry and combinatorial commutative algebra.

## Convex optimization

Convex minimization, a subfield of optimization, studies the problem of minimizing convex functions over convex sets.

## Convex polytope

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn.

## Convex set

In Euclidean space, a convex set is the region such that, for every pair of points within the region, every point on the straight line segment that joins the pair of points is also within the region.

## Coopr

Coopr is a collection of Python software packages that supports a diverse set of optimization capabilities for formulating and analyzing optimization models.

## Copyleft

Copyleft (a play on the word copyright) is the practice of offering people the right to freely distribute copies and modified versions of a work with the stipulation that the same rights be preserved in derivative works down the line.

## Covering problems

In combinatorics and computer science, covering problems are computational problems that ask whether a certain combinatorial structure 'covers' another, or how large the structure has to be to do that.

## CPLEX

IBM ILOG CPLEX Optimization Studio (often informally referred to simply as CPLEX) is an optimization software package.

## Criss-cross algorithm

In mathematical optimization, the criss-cross algorithm denotes a family of algorithms for linear programming.

## Cutting-plane method

In mathematical optimization, the cutting-plane method is an umbrella term for optimization methods which iteratively refine a feasible set or objective function by means of linear inequalities, termed cuts.

## Dantzig–Wolfe decomposition

Dantzig–Wolfe decomposition is an algorithm for solving linear programming problems with special structure.

## David S. Johnson

David Stifler Johnson (born December 9, 1945, Washington, D.C.) is an American computer scientist specializing in algorithms and optimization.

## Distance (graph theory)

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them.

## Dominating set

In graph theory, a dominating set for a graph G.

## Duality (optimization)

In mathematical optimization theory, duality means that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem (the duality principle).

## Dynamic programming

In mathematics, computer science, economics, and bioinformatics, dynamic programming is a method for solving a complex problem by breaking it down into a collection of simpler subproblems.

## Dynamical system

In mathematics, a dynamical system is a set of relationships among two or more measurable quantities, in which a fixed rule describes how the quantities evolve over time in response to their own values.

## Economics

Economics is the social science that seeks to describe the factors which determine the production, distribution and consumption of goods and services.

## Ellipsoid method

In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions.

## Feasible region

In mathematical optimization, a feasible region, feasible set, search space, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potentially including inequalities, equalities, and integer constraints.

## Flow network

In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow.

## FortMP

FortMP is a software package for solving large-scale optimization problems.

## Fourier–Motzkin elimination

Fourier–Motzkin elimination, also known as the FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities.

## Fractional coloring

Fractional coloring is a topic in a young branch of graph theory known as fractional graph theory.

## Frank Lauren Hitchcock

Frank Lauren Hitchcock (1875&ndash;1957) was an American mathematician and physicist known for his formulation of the transportation problem in 1941.

## FreeMat

FreeMat is a free open source numerical computing environment and programming language, similar to MATLAB and GNU Octave.

## Game theory

Game theory is the study of strategic decision-making.

## Günter M. Ziegler

Günter Matthias Ziegler (born 19 May 1963) is a German mathematician.

## General Algebraic Modeling System

The General Algebraic Modeling System (GAMS) is a high-level modeling system for mathematical optimization.

## George Dantzig

George Bernard Dantzig (November 8, 1914 – May 13, 2005) was an American mathematical scientist who made important contributions to operations research, computer science, economics, and statistics.

## GNU Linear Programming Kit

The GNU Linear Programming Kit (GLPK) is a software package intended for solving large-scale linear programming (LP), mixed integer programming (MIP), and other related problems.

## Graph (mathematics)

In mathematics, and more specifically in graph theory, a graph is a representation of a set of objects where some pairs of objects are connected by links.

## Gurobi

The Gurobi Optimizer is a commercial optimization solver for linear programming (LP), quadratic programming (QP), quadratically constrained programming (QCP), mixed integer linear programming (MILP), mixed-integer quadratic programming (MIQP), and mixed-integer quadratically constrained programming (MIQCP).

## Half-space (geometry)

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional Euclidean space.

## Hirsch conjecture

In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n − d.

## IMSL Numerical Libraries

IMSL (International Mathematics and Statistics Library) is a commercial collection of software libraries of numerical analysis functionality that are implemented in the computer programming languages of C, Java, C#.NET, and Fortran.

## Independent set (graph theory)

In graph theory, an independent set or stable set is a set of vertices in a graph, no two of which are adjacent.

## Institute for Operations Research and the Management Sciences

The Institute for Operations Research and the Management Sciences (INFORMS) is an international society for practitioners in the fields of operations research (OR) and management science.

## Integer programming

An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers.

## Interior point method

Interior point methods (also referred to as barrier methods) are a certain class of algorithms that solves linear and nonlinear convex optimization problems.

## Intersection

In mathematics, the intersection of two or more objects is another, usually "smaller" object.

## Iterative method

In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems.

## Java (programming language)

Java is a general-purpose computer programming language that is concurrent, class-based, object-oriented, and specifically designed to have as few implementation dependencies as possible.

## Jiří Matoušek (mathematician)

Jiří (Jirka) Matoušek (10 March 1963 – 9 March 2015) was a Czech mathematician working in computational geometry and algebraic topology.

## Job shop scheduling

Job shop scheduling (or job-shop problem) is an optimization problem in computer science and operations research in which ideal jobs are assigned to resources at particular times.

## John von Neumann

John von Neumann (Hungarian: Neumann János,; December 28, 1903 – February 8, 1957) was a Hungarian-American pure and applied mathematician, physicist, inventor, polymath, and polyglot.

## Joseph Fourier

Jean-Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations.

## Karmarkar's algorithm

Karmarkar's algorithm is an algorithm introduced by Narendra Karmarkar in 1984 for solving linear programming problems.

## Karp's 21 NP-complete problems

In computational complexity theory, Karp's 21 NP-complete problems are a set of computational problems which are NP-complete.

## Klee–Minty cube

The Klee–Minty cube (named after Victor Klee and George J. Minty) is a unit cube whose corners have been slightly perturbed.

## Leonhard Euler

Leonhard Euler (17071783) was a pioneering Swiss mathematician and physicist.

## Leonid Kantorovich

Leonid Vitaliyevich Kantorovich (a) (19 January 19127 April 1986) was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources.

## Leonid Khachiyan

Leonid Genrikhovich Khachiyan (Լեոնիդ Գենրիխովիչ Խաչիյան; Леонид Генрихович Хачиян; May 3, 1952 – April 29, 2005) was a Soviet mathematician of Armenian descent who taught Computer Science at Rutgers University.

## LINDO

LINDO is a software package for linear programming, integer programming, nonlinear programming, stochastic programming and global optimization.

## Linear equation

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.

## Linear form

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars.

## Linear inequality

In mathematics a linear inequality is an inequality which involves a linear function.

## Linear programming relaxation

In mathematics, the linear programming relaxation of a 0-1 integer program is the problem that arises by replacing the constraint that each variable must be 0 or 1 by a weaker constraint, that each variable belong to the interval.

## Linear-fractional programming

In mathematical optimization, linear-fractional programming (LFP) is a generalization of linear programming (LP).

## Linearity

In common usage, linearity refers to a mathematical relationship or function that can be graphically represented as a straight line, as in two quantities that are directly proportional to each other, such as voltage and current in an RLC circuit, or the mass and weight of an object.

## Lingo (programming language)

Lingo is a verbose object-oriented scripting language developed by John H. Thompson for use in Adobe Director (formerly Macromedia Director).

## Loss function

In mathematical optimization, statistics, decision theory and machine learning, a loss function or cost function is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event.

## LP-type problem

In the study of algorithms, an LP-type problem (also called a generalized linear program) is an optimization problem that shares certain properties with low-dimensional linear programs and that may be solved by similar algorithms.

## Maple (software)

Maple is a commercial computer algebra system developed and sold commercially by Maplesoft, a software company based in Waterloo, Ontario, Canada.

## Mark Overmars

Markus Hendrik "Mark" Overmars (born 29 September 1958 in Zeist, Netherlands) is a Dutch computer scientist and teacher of game programming known for his game development application Game Maker.

## Matching (graph theory)

In the mathematical discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices.

Mathcad is computer software primarily intended for the verification, validation, documentation and re-use of engineering calculations.

## Mathematica

Mathematica is a computational software program used in many scientific, engineering, mathematical and computing fields, based on symbolic mathematics.

## Mathematical model

A mathematical model is a description of a system using mathematical concepts and language.

## Mathematical optimization

In mathematics, computer science and operations research, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

## MATLAB

MATLAB (matrix laboratory) is a multi-paradigm numerical computing environment and fourth-generation programming language.

## Matrix (mathematics)

In mathematics, a matrix (plural matrices) is a rectangular array—of numbers, symbols, or expressions, arranged in rows and columns—that is interpreted and manipulated in certain prescribed ways.

## Maxima and minima

In mathematical analysis, the maxima and minima (the plural of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).

## Maximum principle

In mathematics, the maximum principle is a property of solutions to certain partial differential equations, of the elliptic and parabolic types.

## Michael Garey

Michael Randolph Garey is a computer science researcher, and co-author (with David S. Johnson) of Computers and Intractability: A Guide to the Theory of NP-completeness.

## Microeconomics

Microeconomics (from Greek prefix mikro- meaning "small") is a branch of economics that studies the behavior of individuals and firms in making decisions regarding the allocation of limited resources.

## Microsoft Excel

Microsoft Excel is a spreadsheet application developed by Microsoft for Microsoft Windows,, and iOS.

In geometry, the Minkowski sum (also known as dilation) of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B, i.e., the set Analogously, the Minkowski difference is defined as The concept is named for Hermann Minkowski.

## MINTO

MINTO is an integer programming solver which uses branch and bound algorithm.

## MOSEK

MOSEK is a software package for the solution of linear, mixed-integer linear, quadratic, mixed-integer quadratic, quadratically constraint, conic and convex nonlinear mathematical optimization problems.

## NAG Numerical Library

The NAG Numerical Library is a software product developed and sold by The Numerical Algorithms Group.

## Narendra Karmarkar

Narendra Krishna Karmarkar (born 1957) is an Indian mathematician, who developed Karmarkar's algorithm.

## Naum Z. Shor

Naum Zuselevich Shor (Шор Наум Зуселевич) (1 January 1937 – 26 February 2006) was a Soviet and Ukrainian Jewish mathematician specializing in optimization.

## NMath Stats

NMath Stats is a statistical package for the Microsoft.NET Framework.

## Nobel Memorial Prize in Economic Sciences

The Nobel Memorial Prize in Economic Sciences (officially Sveriges riksbanks pris i ekonomisk vetenskap till Alfred Nobels minne, or the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel), commonly referred to as the Nobel Prize in Economics, and as a "category of the Nobel Prize" by the Nobel Foundation itself, which owns the name Nobel Prize, though the Foundation itself does not refer to it as such.

## Nonlinear programming

In mathematics, nonlinear programming is the process of solving an optimization problem defined by a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are nonlinear.

## NP-hardness

NP-hardness (''n''on-deterministic ''p''olynomial-time hard), in computational complexity theory, is a class of problems that are, informally, "at least as hard as the hardest problems in NP".

## Numerical Algorithms Group

The Numerical Algorithms Group (NAG) is a software company which provides methods for the solution of mathematical and statistical problems, and offers services to users of High performance computing (HPC) systems.

## O-Matrix

O-Matrix is a matrix programming language for mathematics, engineering, science, and financial analysis, marketed by Harmonic Software.

## Octave

In music, an octave (octavus: eighth) or perfect octave is the interval between one musical pitch and another with half or double its frequency.

## Odysseus

Odysseus (Ὀδυσσεύς), also known by the Latin name Ulysses (Ulyssēs, Ulixēs), was a legendary Greek king of Ithaca and the hero of Homer's epic poem the Odyssey.

## Operations research

Operations research, or operational research in British usage, is a discipline that deals with the application of advanced analytical methods to help make better decisions.

## Optimization Toolbox

Optimization Toolbox is an optimization software package developed by MathWorks.

## OptimJ

OptimJ is an extension of the Java with language support for writing optimization models and abstractions for bulk data processing.

## Oriented matroid

An oriented matroid is a mathematical structure that abstracts the properties of directed graphs and of arrangements of vectors in a vector space over an ordered field (particularly for partially ordered vector spaces).

## P (complexity)

In computational complexity theory, P, also known as PTIME or DTIME(nO(1)), is one of the most fundamental complexity classes.

## Packing problems

Packing problems are a class of optimization problems in mathematics that involve attempting to pack objects together into containers.

## Permissive free software licence

A permissive free software licence is a class of free software licence with minimal requirements about how the software can be redistributed.

## PHP

PHP is a server-side scripting language designed for web development but also used as a general-purpose programming language.

## Polyhedron

In elementary geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.

## Polytope

In elementary geometry, a polytope is a geometric object with flat sides, and may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope.

## Proceedings of the USSR Academy of Sciences

The Proceedings of the USSR Academy of Sciences (Доклады Академии Наук СССР, Doklady Akademii Nauk SSSR (DAN SSSR), Comptes rendus de l'Académie des sciences de l'URSS) was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biology.

## Profit maximization

In economics, profit maximization is the short run or long run process by which a firm determines the price and output level that returns the greatest profit.

## Proprietary software

Proprietary software, non-free software (in the sense of missing freedoms), or closed-source software is software, where the developers or distributors reserve all freedoms and rights.

## Python (programming language)

Python is a widely used general-purpose, high-level programming language.

## Qoca

Qoca is a GPL library for incrementally solving systems of linear equations with various goal functions.

Quadratic programming (QP) is a special type of mathematical optimization problem.

## R

R (named ar/or) is the 18th letter of the modern English alphabet and the ISO basic Latin alphabet.

## R (programming language)

R is a programming language and software environment for statistical computing and graphics.

## Real number

In mathematics, a real number is a value that represents a quantity along a continuous line.

## Robert J. Vanderbei

Robert J. Vanderbei (born 1955) is an American mathematician and Professor in the Department of Operations Research and Financial Engineering at Princeton University.

## Routing

Routing is the process of selecting best paths in a network.

## SageMath

SageMath (previously Sage or SAGE, System for Algebra and Geometry Experimentation) is mathematical software with features covering many aspects of mathematics, including algebra, combinatorics, numerical mathematics, number theory, and calculus.

## SAS (software)

SAS (Statistical Analysis System) is a software suite developed by SAS Institute for advanced analytics, multivariate analyses, business intelligence, data management, and predictive analytics.

## Scheduling (production processes)

Scheduling is the process of arranging, controlling and optimizing work and workloads in a production process or manufacturing process.

## Scilab

Scilab is an open source, cross-platform numerical computational package and a high-level, numerically oriented programming language.

## SCIP (optimization software)

SCIP (Solving Constraint Integer Programs) is a mixed integer programming solver and a framework for Branch and cut and Branch and price, developed at Zuse Institute Berlin.

## Semidefinite programming

Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function (that is, a function to be maximized or minimized) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron.

## Set cover problem

The set cover problem is a classical question in combinatorics, computer science and complexity theory.

## Set packing

Set packing is a classical NP-complete problem in computational complexity theory and combinatorics, and was one of Karp's 21 NP-complete problems.

The term "Shadow Price" or "Shadow Pricing" is used to refer to monetary values assigned to currently unknowable or difficult to calculate costs.

## Simplex algorithm

In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.

## Slack variable

In an optimization problem, a slack variable is a variable that is added to an inequality constraint to transform it to an equality.

## Smale's problems

Smale's problems are a list of eighteen unsolved problems in mathematics that was proposed by Steve Smale in 1998, republished in 1999.

Springer Science+Business Media or Springer is a global publishing company that publishes books, e-books and peer-reviewed journals in science, technical and medical (STM) publishing.

## Stephen Smale

Stephen Smale (born July 15, 1930) is an American mathematician from Flint, Michigan.

## Stochastic programming

In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty.

## Strong duality

Strong duality is a concept in optimization such that the primal and dual solutions are equivalent.

## Submodular set function

In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function, that a single element makes when added to an input set, decreases as the size of the input set increases.

## Sysquake

Sysquake is a numerical computing environment based on a programming language mostly-compatible with MATLAB.

## The Mathematical Intelligencer

The Mathematical Intelligencer is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common amongst such journals.

## Time complexity

In computer science, the time complexity of an algorithm quantifies the amount of time taken by an algorithm to run as a function of the length of the string representing the input.

## Tjalling Koopmans

Tjalling Charles Koopmans (August 28, 1910 – February 26, 1985) was a Dutch American mathematician and economist, the joint winner with Leonid Kantorovich of the 1975 Nobel Memorial Prize in Economic Sciences.

## TOMLAB

The TOMLAB Optimization Environment is a modeling platform for solving applied optimization problems in MATLAB.

## Total dual integrality

In mathematical optimization, total dual integrality is a sufficient condition for the integrality of a polyhedron.

## Transpose

In linear algebra, the transpose of a matrix A is another matrix AT (also written A′, Atr, tA or At) created by any one of the following equivalent actions.

## Travelling salesman problem

The travelling salesman problem (TSP) asks the following question: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science.

## Unimodular matrix

In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or &minus;1.

## Unit cube

A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long.

## Variable (computer science)

In computer programming, a variable or scalar is a storage location paired with an associated symbolic name (an identifier), which contains some known or unknown quantity of information referred to as a value.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context.

## Vertex cover

In the mathematical discipline of graph theory, a vertex cover (sometimes node cover) of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex of the set.

## VisSim

VisSim is a visual block diagram language for simulation of dynamical systems and model based design of embedded systems.

## Weak duality

In applied mathematics, weak duality is a concept in optimization which states that the duality gap is always greater than or equal to 0.

## World War II

World War II (WWII or WW2), also known as the Second World War, was a global war that lasted from 1939 to 1945, though related conflicts began earlier.

## Worst-case complexity

In computer science, the worst-case complexity (usually denoted in asymptotic notation) measures the resources (e.g. running time, memory) an algorithm requires in the worst-case.

## Yinyu Ye

Yinyu Ye (born 1948) is a Chinese American theoretical computer scientist working on mathematical optimization.

## .NET Framework

.NET Framework (pronounced dot net) is a software framework developed by Microsoft that runs primarily on Microsoft Windows.

## References

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