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# Topic: Linear programming

###### In the News (Thu 23 May 13)

 Linear Programming - Wolfram Mathematica Linear programming problems are optimization problems where the objective function and constraints are all linear. The simplex and revised simplex algorithms solve a linear programming problem by moving along the edges of the polytope defined by the constraints, from vertices to vertices with successively smaller values of the objective function, until the minimum is reached. The simplex and revised simplex algorithms solve linear programming problems by constructing a feasible solution at a vertex of the polytope defined by the constraints, and then moving along the edges of the polytope to vertices with successively smaller values of the objective function until the minimum is reached. reference.wolfram.com /mathematica/tutorial/ConstrainedOptimizationLinearProgramming.html   (1834 words)

 Linear programming - Wikipedia, the free encyclopedia In mathematics, linear programming (LP) problems are optimization problems in which the objective function and the constraints are all linear. Linear programming is an important field of optimization for several reasons. Leonid Kantorovich was one of founders of linear programming en.wikipedia.org /wiki/Linear_programming   (1542 words)

 PlanetMath: linear programming   (Site not responding. Last check: ) A linear programming problem, or LP, is a problem of optimizing a given linear objective function over some polyhedron. The linear constraints for a linear programming problems define a convex polyhedron, called the feasible region for the problem. This is version 8 of linear programming, born on 2003-06-16, modified 2006-02-17. planetmath.org /encyclopedia/LinearProgrammingProblem.html   (257 words)

 Online Encyclopedia and Dictionary - Linear programming In fact for some time it was not known whether the linear programming problem was NP-complete or polynomial-time solvable. It was based on the ellipsoid method in nonlinear optimization by Naum Shor, which is the generalization of the ellipsoid method in convex optimization by Arkadi Nemirovski, a 2003 John von Neumann Theory Prize winner, and D. Yudin. An advanced algorithm for solving large integer linear programs is delayed column generation www.fact-archive.com /encyclopedia/Linear_programming   (1011 words)

 Good Math, Bad Math : Linear Programming Linear programming is a general technique for solving a huge family of optimization problems. Integer linear programs can be solved for example by a method of tree searching using LP relaxations (that is the same problem with removed integrality constraints) to establish bounds on the optimal solution. As an example, in the distant past I wrote a program that determined the composition of piles of scrap metal based upon the analysis of the metal that resulted when various quantities of each were mixed together and melted. scienceblogs.com /goodmath/2008/05/linear_programming.php   (2301 words)

 [No title]   (Site not responding. Last check: ) Linear programming is the process of solving a system of linear equalities and linear inequalities over a set of unknown real variables, along with a linear objective function to be maximized. Due to the geometry of these kinds of linear constraints and the linear objective function, LP problems are "convex", which means that there are no local optima (aside from the global optimum). Furthermore, in LP the solution must be at a vertex of the n-dimensional polytope defined by the constraints. wikiwhat.com /encyclopedia/l/li/linear_programming.html   (470 words)

 Linear Programming   (Site not responding. Last check: ) Linear programming Linear programming This applet lets students experiment graphically with the value of linear functions of the form f(x,y) = ax by c subject to linear constraints on x and y. Linear Programming, like many applications of math, was developed and used in defense in the beginning in the 1940s. Linear programming is a widely-used method of optimization that arises in various... www.memsoptimization.com /linearprogramming   (790 words)

 Timetabling an Academic Department with Linear Programming Linear programming involves the optimization (maximization or minimization) of a linear objective function subject to linear constraints. In a second article describing an approach based on linear programming (Tripathy, 1984), 900 course sections in 25 different areas of specialization at the university graduate level were timetabled by grouping courses into larger groups of courses in which enrolment was required of most of an identifiable group of students. Linear programming packages that can do integer programming generally solve the problem first without restricting the variables to integers and then impose the integer restriction on those variables that are not already integers. www.unb.ca /centres/nbcea/lptimtab.html   (7477 words)

 Linear Programming LINEAR PROGRAMMING, a specific class of mathematical problems, in which a linear function is maximized (or minimized) subject to given linear constraints. Linear programming was developed as a discipline in the 1940's, motivated initially by the need to solve complex planning problems in wartime operations. Linear programs in thousands or even millions of variables are routinely solved using the simplex method on modern computers. www.cs.nyu.edu /cs/faculty/overton/g22_lp/encyc/article_web.html   (1036 words)

 Management Science - HISTORY, BREADTH OF MANAGEMENT SCIENCE, MATHEMATICAL PROGRAMMING, LINEAR PROGRAMMING, SIMPLEX ... Linear programming problems are a special class of mathematical programming problems for which the objective function and all constraints are linear. Nonlinear programming is useful when the objective function or at least one of the constraints is not linear with respect to values of at least one decision variable. Stochastic programming is useful when the value of a coefficient in the objective function or one of the constraints is not know with certainty but has a known probability distribution. www.referenceforbusiness.com /management/Log-Mar/Management-Science.html   (2435 words)

 Linear Programming Linear programming is a means by which a real-valued objective function expressible as a linear equation can be optimized subject to a set of constraints. The concept of the linear program (see, e.g., Dantzig and Thapa, 1997) was introduced by George B. Dantzig in 1947. Linear programming is a subset of mathematical programming, which includes integer programming, nonlinear programming, stochastic programming, combinatorial programming, and network flow maximization (Dantzig and Thapa, 1997). www.io.com /~chuyser/Linear_Programming.htm   (1937 words)

 Linear Programming FAQ FAQ It may not be obvious that integer programming is a very much harder problem than ordinary linear programming, but that is nonetheless the case, in both theory and practice. The network linear programming problem is to minimize the (linear) total cost of flows along all arcs of a network, subject to conservation of flow at each node, and upper and/or lower bounds on the flow along each arc. Linear programs in MPS format and nonlinear problems in the form of a C or Fortran program may be submitted by e-mail, Web page, or a high speed, socket-based Unix interface. www.non.com /news.answers/linear-programming-faq.html   (10632 words)

 5.6 - Linear Programming The linear function (equal sign) representing cost, profit, or some other quantity to be maximized of minimized subject to the constraints. If there is going to be an optimal solution to a linear programming problem, it will occur at one or more corner points, or on a line segment between two corner points. If there is a solution to a linear programming problem, then it will occur at a corner point, or on a line segment between two corner points. people.richland.edu /james/lecture/m116/systems/linear.html   (821 words)

 Linear Programming Linear programming models consist of an objective function and the constraints on that function. In addition to these linearity assumptions, linear programming assumes certainty; that is, that the coefficients are known and constant. With computers able to solve linear programming problems with ease, the challenge is in problem formulation - translating the problem statement into a system of linear equations to be solved by computer. www.netmba.com /operations/lp   (625 words)

 Linear Programming In general, linear programming allows you to find the global minimum or maximum of any linear function subject to a set of constraints defined by linear inequalities. It turns out that the solution to any linear programming problem always corresponds to a point which lies at the boundary of the region defined for that problem. Sometimes, however, it is more convenient to handle linear programming problems simply by setting up vectors and matrices which represent the appropriate coefficients of the linear functions that appear. documents.wolfram.com /v4/MainBook/3.9.9.html   (556 words)

 Linear Programming FAQ These methods derive from techniques for nonlinear programming that were developed and popularized in the 1960s by Fiacco and McCormick, but their application to linear programming dates back only to Karmarkar's innovative analysis in 1984. The network linear programming problem is to minimize the (linear) total cost of flows along all arcs of a network, subject to conservation of flow at each node, and upper and/or lower bounds on the flow along each arc. Linear programs in MPS format and nonlinear problems in the form of a C or Fortran program may be submitted by sending e-mail, by submitting URLs through a Web page, or via a high-speed socket-based Unix interface. www.faqs.org /faqs/linear-programming-faq   (11971 words)

 Linear Programming While the basic simplex algorithm is not too difficult to program, there is a considerable art to producing an efficient implementation capable of solving large linear programs. Linear programming is one algorithmic problem of such economic importance that commercial implementations are far superior to free versions. Pascal implementations of the revised and dual simplex methods for linear programming, as well as cutting plane and explicit enumeration algorithms for integer programming, are provided in [SDK83]. www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK3/NODE141.HTM   (1514 words)

 Linear Programming: Introduction Linear programming is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions. In "real life", linear programming is part of a very important area of mathematics called "optimization techniques". Somebody really smart proved that, for linear systems like this, the maximum and minimum values of the optimization equation will always be on the corners of the feasibility region. www.purplemath.com /modules/linprog.htm   (356 words)

 Nonlinear Programming FAQ Semidefinite Programming is a generalization of linear programming to the space of block diagonal, symmetric, positive semidefinite matrices. Communicating with a nonlinear programming code can be particularly tedious and error-prone, especially if you have to write programs in a language like Fortran or C to compute function (and maybe gradient) values for your objective and constraints. Schittkowski, by Hock and Schittkowski, and by Torn and Zilinskas. www-unix.mcs.anl.gov /otc/Guide/faq/nonlinear-programming-faq.html   (5846 words)

 Linear Programming FAQ   (Site not responding. Last check: ) These methods derive from techniques for nonlinear programming that were developed and popularized in the 1960s by Fiacco and McCormick, but their application to linear programming dates back only to Karmarkar's innovative analysis in 1984. ABACUS is a C++ class library that "provides a framework for the implementation of branch-and-bound algorithms using linear programming relaxations that can be complemented with the dynamic generation of cutting planes or columns" (branch-and-cut and/or branch-and-price). This approach is based on a linear programming problem that decides how many rolls should be cut in each of several patterns, which means that fractional numbers of patterns may show up in the results. www.freewebtown.com /lpsolve/LinearProgrammingFAQ.htm   (13244 words)

 Linear Programming   (Site not responding. Last check: ) That's not how you "solve" linear programming problems- the point of "displacing the object function" is to show why the methods work. A linear programming problem involves a linear object function that you want to maximize or minimize, such as, say, ax+ by. THAT'S how you solve a linear programming problem (in 2 variables): determine the vertices of the polygon (where the line graphs of the equations associated with the inequalities intersect) and evaluate the object function at those points. www.physicsforums.com /showthread.php?p=905936   (741 words)

 Chapter4 Linear programming is the most widely applied of all of the optimization methods. Linear programming was developed about 1947, before the advent of the computer, when George B. Dantzig (1) recognized a generalization in the mathematics of scheduling and planning problems. In this chapter a geometric representation and solution of a simple linear programming problem will be given initially to introduce the subject and illustrate the way to capitalize on the mathematical structure of the problem. www.mpri.lsu.edu /textbook/Chapter4.htm   (466 words)

 Linear programming library First way is a linear program construction by the provided set of functions, such as specification of the variables and constraints count and assignment of the constraint matrix elements. The second way is importing linear program from a text file in MPS format. After a linear program (LP) is specified or imported from MPS file it is ready for the calculation. www.optimalon.com /product_gipals32.htm   (676 words)

 Linear Programming Linear programming is a quantitative analysis technique for optimizing an objective function given a set of constraints. As the name implies, the functions must be linear in order for linear programming techniques to be used. Linear programming techniques assume certainty and by themselves do not deal well with significant randomness. www.quickmba.com /ops/lp   (720 words)

 Linear Programming FAQ GULF is a a small-scale implementation of the simplex method for linear programs and linear-fractional programs (having a ratio of two linear functions in the objective). Interactive Linear Programming appears to be useful for solving small models that can be entered by hand. Linear programs in MPS format and nonlinear problems in the form of a C or Fortran program may be submitted by sending e-mail, by submitting URLs through a Web page, or via a high-speed socket-based Unix interface. frcatel.fri.utc.sk /linear-programming-faq.html   (12135 words)

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