1. Linear Programming - Wikipedia, The Free Encyclopedia In mathematics, linear programming (LP) problems involve the optimization of a linear objective function, subject to linear equality and inequality http://en.wikipedia.org/wiki/Linear_programming

Linear programming From Wikipedia, the free encyclopedia Jump to: navigation search In mathematics linear programming (LP) problems involve the optimization of a linear objective function , subject to linear equality and inequality constraints Put very informally, LP problems determine the way to achieve the best outcome (such as maximum profit or lowest cost) given some list of requirements represented as linear equation. More formally, given a polytope (for example, a polygon or a polyhedron ), and a real -valued affine function defined on this polytope, the goal is to find a point in the polytope where this function has the smallest (or largest) value. Such points may not exist, but if they do, searching through the polytope vertices is guaranteed to find at least one of them. Linear programs are problems that can be expressed in canonical form: Maximize Subject to Where represents the vector of variables, while and are vectors of coefficients and is a matrix of coefficients. The expression to be maximized or minimized is called the objective function ( in this case). The equations

2. Linear Programming FAQ From the newsgroup sci.opresearch, maintained at the Optimization Technology Center of Northwestern University and Argonne National Laboratory. http://www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html

Linear Programming Frequently Asked Questions Optimization Technology Center of Northwestern University and Argonne National Laboratory Posted at http://www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html Changes posted to Usenet newsgroup sci.op-research Date of this version: September 1, 2005 What is Linear Programming Where is there good software to solve LP problems?" "Free" codes Commercial codes and modeling systems Free demos of commercial codes Q3. "Oh, and we also want to solve it as an integer program Q4. "I wrote an optimization code. Where are some test models Q5. "What is MPS format Topics briefly covered: : "What is a modeling language?" : "How do I diagnose an infeasible LP model?" : "I want to know the specific constraints that contradict each other." : "I just want to know whether or not a feasible solution exists : "I have an LP, except it's got several objective functions." : "I have an LP that has large almost-independent matrix blocks that are linked by a few constraints. Can I take advantage of this?" : "I am looking for an algorithm to compute the convex hull of a finite number of points in n-dimensional space."

3. Summary Linear Programming linear programming (LP) Problem Sketching the Solution Set of a Linear Inequality Feasible Region Graphical Method Standard Maximization Problem http://people.hofstra.edu/Stefan_Waner/Realworld/Summary4.html

Summary of Chapter 4 in Finite Mathematics Topic: Linear Programming Student Home True/False Quiz On-Line Tutorial Review Exercises ... Everything for Finite Math Chapter 3 Summary Chapter 5 Summary Tools: Linear Programming Grapher Pivot and Gauss-Jordan Tool Excel Pivot and Gauss-Jordan Tool Simplex Method Tool ... Solving a Matrix Game Using the Simplex Method Linear Programming (LP) Problem A linear programming problem is one in which we are to find the maximum or minimum value of a linear expression ax + by + cz + . . . (called the objective function ), subject to a number of linear constraints of the form or The largest or smallest value of the objective function is called the optimal value , and a collection of values of x, y, z, . . . that gives the optimal value constitutes an optimal solution . The variables x, y, z, . . . are called the decision variables Example Here is an example of an LP problem: Find the maximum value of p = 3x subject to The objective function is p = 3x 2y + 4z. The constraints are Q Wait a minute! Why can't I simply choose, say, z to be really large (z = 1,000,000 say) and thereby make p as large as I want? A You can't because Select one that would make p too large.

4. Linear Programming: Foundations And Extensions This book is an introductory graduate textbook on linear programming although upperlevel graduate students and researchers will find plenty of material http://www.princeton.edu/~rvdb/LPbook/

Linear Programming: Foundations and Extensions Order Table of Contents Lecture Notes-Undergraduate level (pdf format) Lecture Notes-Graduate level (pdf format) Online version of 2nd Edition (note that the hyperlinks in the text really work) Associated software (C/C++ source code) Additional Exercises Online Exercises Errata Simple pivot tool (a java applet to help solve exercises from Chapters 1 to 5) Advanced pivot tool (a java applet to help solve exercises from Chapters 5 to 7) Online pivot exam (a java applet to evaluate your knowledge of the simplex method) The Network-Simplex Pivot Tool Intended Audience This book is an introductory graduate textbook on linear programming although upper-level graduate students and researchers will find plenty of material here that cannot be found in other books. It has also been used successfully to teach undergraduates majoring in Operations Research. Features Balanced treatment of the simplex method and interior-point methods.

5. An Introduction To Linear Programming And The Simplex Algorithm This text is intended to function as an introduction to linear programming (LP) and the Simplex algorithm. The specific topics covered and the structure of http://www.isye.gatech.edu/~spyros/LP/LP.html

Next: The LP formulation and An Introduction to Linear Programming and the Simplex Algorithm by Spyros Reveliotis Acknowledgements There are a number of people who have contributed during the inceptional and implementational phases of this project through constructive comments and discussions, provision of information and technical assistance. At this point, I would like to acknowledge their valuable aid. Hence, I would like to say a "big" THANKS to: Evangelia Dimaraki, Amy Pritchett, Bhaskar Manda, Darren Hunt, Victoria Burse, Randy Riegsecker, Balu Vandor, Javier Ruiz and Fangjun Zhou . Many thanks go also to my students of the past IE3231 classes , since it was my interaction with them that motivated and inspired this whole effort. I would also like to acknowledge the financial (and not only) support of the School of Industrial and Systems Engineering and the CETL group at the Georgia Institute of Technology The integrated software supporting the execution of interactive examples "re-uses" the very nice software modules developed by (i) Drs Ken Goldberg and Ilan Adler at the Dept. of Industrial Engineering and Operations Research

6. Linear Programming -- From Wolfram MathWorld linear programming, sometimes known as linear optimization, is the problem of maximizing or minimizing a linear function over a convex polyhedron specified http://mathworld.wolfram.com/LinearProgramming.html

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Less... Linear Programming Linear programming, sometimes known as linear optimization, is the problem of maximizing or minimizing a linear function over a convex polyhedron specified by linear and non-negativity constraints. Simplistically, linear programming is the optimization of an outcome based on some set of constraints using a linear mathematical model. Linear programming is implemented in Mathematica as LinearProgramming c m b ], which finds a vector which minimizes the quantity subject to the constraints and for Linear programming theory falls within convex optimization theory and is also considered to be an important part of operations research . Linear programming is extensively used in business and economics, but may also be used to solve certain engineering problems. Examples from economics include Leontief's input-output model, the determination of shadow prices, etc., an example of a business application would be maximizing profit in a factory that manufactures a number of different products from the same raw material using the same resources, and example engineering applications include Chebyshev approximation and the design of structures (e.g., limit analysis of a planar truss). Linear programming can be solved using the simplex method (Wood and Dantzig 1949, Dantzig 1949) which runs along

7. Linear Programming Software for linear programming (including network linear programming) consumes more computer cycles than software for all other kinds of optimization http://www-fp.mcs.anl.gov/otc/Guide/OptWeb/continuous/constrained/linearprog/

Software for linear programming (including network linear programming) consumes more computer cycles than software for all other kinds of optimization problems combined. There is a proliferation of linear programming software with widely varying capabilities and user interfaces. The most recent survey of linear programming software for desktop computers carried out by OR/MS Today (19 (1992), pp. 44-59) gave details on 49 packages! The basic problem of linear programming is to minimize a linear objective function of continuous real variables, subject to linear constraints. For purposes of describing and analyzing algorithms, the problem is often stated in the standard form where is the vector of unknowns, is the cost vector, and is the constraint matrix. The feasible region described by the constraints is a polytope, or simplex , and at least one member of the solution set lies at a vertex of this polytope. The simplex algorithm , so named because of the geometry of the feasible set, underlies the vast majority of available software packages for linear programming. However, this situation may change in the future, as more software for interior-point algorithms becomes available.

8. EE236A: Linear Programming Introduction; Basic definitions and geometry of linear programming; Engineering applications; Duality; The simplex method; Interiorpoint methods http://www.ee.ucla.edu/ee236a/

EE236A: Linear Programming UCLA Electrical Engineering Department Prof. Lieven Vandenberghe Fall Quarter 2007-2008 Lecture notes Homework Course information Software Lecture notes Introduction and overview 4/page Linear inequalities 4/page ... 4/page Homework Homework 1 (due 10/11): Exercises 2, 6, 8 (a,c,d,e,g), 9, 11. Homework 2 (due 10/18): Exercises 12, 14 (b), 15, 20, 36. Homework 3 (due 10/25): Exercises 22, 23, 26, 30, 31 (a). Homework 4 (due 11/1): Exercises 28, 29, 33, 40, 41. Homework 5 (due 11/8): Exercises 42, 44, 50, 53, 55. Homework 6 (due 11/15): Exercises 45, 54, 58, 63, 66. Homework 7 (due 11/29): Exercises 46, 47, 69, 71, 72, 74. Homework 8 (due 12/7): Exercises 75, 76, 86. Submit your code for problem 86 by email to vandenbe@ee.ucla.edu. The homework assignments are from the EE236A Exercises . Some of the problems require Matlab files: ex9data.m ex15data.m ex17data.m ex18data.m ... ex35data.m Course information Lectures: Instructor: Prof. Lieven Vandenberghe Office: 68-119 Engineering IV Tel: (310) 206-1259 Email: vandenbe@ee.ucla.edu

9. RIOT: Interactive Linear Programming: Introduction Objective To create an interface between the WWW and a linear programming solver allow anyone with access to the Web to submit a linear program and have it http://ford.ieor.berkeley.edu/riot/Tools/InteractLP/

Interactive Linear Programming RIOT HOME Linear Programming What is LP? ... Linear Programming This portion of the RIOT site is still under construction. The navigational are "uneven" across the pages, and the editorial content is still being polished. Please bear with us during the consruction. Project Overview Objective: To create an interface between the WWW and a linear programming solver allow anyone with access to the Web to submit a linear program and have it solved. There has been a proliferation of linear programming solver software since 1980 (ex : CPLEX, LINGO, MINOS, etc). Each solver implements different algorithms (Simplex method, Interior-Point method, etc) and offers different options (sensititvity analysis, basic certificates, etc). Depending on the problem to be solved, some solvers can be more or less efficient than other in terms of speed, accuracy, number of iterations and available options. The first step of our project will be to create an interface between a linear programming solver and the web so that a user can submit his problem through the internet in order to have it solved.

10. Linear Programming Faq This page has moved to. http//wwwunix.mcs.anl. gov/otc/Guide/faq/linear-programming-faq.html. Please update your links. http://www.mcs.anl.gov/home/otc/Guide/faq/linear-programming-faq.html

This page has moved to http://www-unix.mcs.anl.gov/otc/Guide/faq/linear-programming-faq.html Please update your links.

11. Linear Programming linear programming is a particular case of constrained optimization problems. What sets the linear programming aside is that optimal values are sought for a http://www.cut-the-knot.org/do_you_know/lin_pr.shtml

var MyPageLoc = document.location; var MyPageTitle = document.title; G o o g ... e Web CTK Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Linear Programming Linear Programming is a particular case of constrained optimization problems . What sets the linear programming aside is that optimal values are sought for a linear function subject to linear constraints. To have a general formulation let's assume that we are given matrix A column ) vector b row ) vector c The role of the unknown x which is required to satisfy two constraints: A x b , and x the latter means that all the components of x are nonnegative. Vectors that satisfy constraints 1 and 2 are called feasible . We are interested in the situations where the set of feasible vectors is not empty. Finally, the cost or objective function f is given by f( x x , where c x is the scalar product of two vectors c and x c x = c x + ... + c n x n . In linear programming, one is requested to maximize (or minimize) the cost function f subject to constraints 1 and 2: (P) Maximize f( x c x subject to A x b and x Example Assume that we have decided to position defenders of a square castle according to the following plan: p q p q q p q p so that the total number of defenders is 4(p+q) while (2p+q) fighters face the enemy on every side. Let's denote p = x

12. GLPK - GNU Project - Free Software Foundation (FSF) The GLPK (GNU linear programming Kit) package is intended for solving largescale linear programming (LP), mixed integer programming (MIP), http://www.gnu.org/software/glpk/

GLPK (GNU Linear Programming Kit) Introduction Get the Software Documentation Mailing Lists/Newsgroups ... Introduction to GLPK The GLPK ( G NU L inear P rogramming K it) package is intended for solving large-scale linear programming (LP), mixed integer programming (MIP), and other related problems. It is a set of routines written in ANSI C and organized in the form of a callable library. GLPK supports the GNU MathProg language, which is a subset of the AMPL language. The GLPK package includes the following main components: Revised simplex method. Primal-dual interior point method. Branch-and-bound method. Translator for GNU MathProg. Application program interface (API). Stand-alone LP/MIP solver. Downloading GLPK The GLPK distribution can be found in the subdirectory /gnu/glpk/ on your favorite GNU mirror . For other ways to obtain GLPK, please read How to get GNU Software Documentation The GLPK documentation consists of the Reference Manual and the description of the GNU MathProg language. Both these documents are included in the distribution (in LaTeX, DVI, and PostScript formats). Mailing Lists/Newsgroups GLPK has two mailing lists: and The main discussion list is , and is used to discuss all aspects of GLPK, including its development and porting.

13. Mathematics Archives - Topics In Mathematics - Linear/Nonlinear Programming KEYWORDS Article, linear programming model, stochastic programming KEYWORDS Software, Constrained and Unconstrained Optimization, linear programming http://archives.math.utk.edu/topics/linearProg.html

Topics in Mathematics Linear / Nonlinear Programming Optimization Advanced Linear Programming ADD. KEYWORDS: Sensitivity Analysis, Model Simplification, Graphs of Linear Programs, Embedded and Hidden Structures, Sparse Matrix Techniques for Simplex and Interior Methods, Course Notes AMS's Materials Organized by Mathematical Subject Classification - Economics, Operations Research, Programming, Games ADD. KEYWORDS: Electronic Journals, Preprints, Web Sites, Databases AMS's Materials Organized by Mathematical Subject Classification ADD. KEYWORDS: Electronic Journals, Preprints, Web Sites, Databases AMS's Materials Organized by Mathematical Subject Classification ADD. KEYWORDS: Electronic Journals, Preprints, Web Sites, Databases Anima-LP: A Tool for Teaching Linear Programming Another Copy Annals of Operations Research ADD. KEYWORDS: Published Journal Bibliography on Interior Point Methods for Mathematical Programming CCC - Constraints Archive ADD. KEYWORDS: People, Bibliographies, Papers, Software, Benchmarks

14. Solver Technology - Linear Programming And Quadratic Programming Solver Technology Frontline Systems optimizers solve linear programming (LP) and quadratic programming (QP) problems using the Primal and Dual Simplex http://www.solver.com/technology2.htm

solver.com Frontline Systems, Inc. Developers of Your Spreadsheet's Solver Products Solutions Support Pricing ... Login Solver Technology - Linear Programming and Quadratic Programming Frontline Systems' optimizers solve linear programming (LP) and quadratic programming (QP) problems using these methods: Primal and Dual Simplex Method Active Set Method Interior Point or Newton-Barrier Method Quadratic Programming Methods ... More Solver Technology For an explanation of these types of problems, please see Optimization Problem Types: Linear Programming and Quadratic Programming Primal and Dual Simplex Method The standard Microsoft Excel Solver uses a basic implementation of the primal Simplex method to solve LP problems. It is limited to 200 decision variables. The Premium Solver uses an improved primal Simplex method with two-sided bounds on the variables. It handles up to 1,000 decision variables. The Premium Solver Platform uses an extended LP/Quadratic version of this Simplex Solver to handle problems of up to 2,000 decision variables. It optionally uses a dual Simplex method to solve LP subproblems in a mixed-integer (MIP) problem. However, this Simplex algorithm does not exploit sparsity in the model.

15. Linear Programming FAQ What is linear programming? * Q2. Where is there good software to solve LP problems? o Free codes o Commercial codes and modeling systems o Free demos http://faqs.org/faqs/linear-programming-faq/

Usenet FAQs Search Web FAQs Documents ... RFC Index Linear Programming FAQ There are reader questions on this topic! Help others by sharing your knowledge From: 4er@iems.nwu.edu (Robert Fourer) Newsgroups: sci.op-research 4er@iems.nwu.edu (Robert Fourer) Summary: A List of Frequently Asked Questions about Linear Programming Keywords: FAQ, LP, Linear Programming Posted-By: auto-faq 2.4 Archive-name: linear-programming-faq Last-modified: November 1, 1997 [ ] Linear Programming Frequently Asked Questions Optimization Technology Center of Northwestern University and Argonne National Laboratory [ ] Posted monthly to Usenet newsgroup sci.op-research World Wide Web version: http://www.mcs.anl.gov/home/otc/Guide/faq/linear-programming-faq.html michel@es.ele.tue.nl ) says has solved models with up to 30,000 variables and 50,000 constraints. The author requests that people retrieve it from ftp://ftp.es.ele.tue.nl/pub/lp_solve (numerical address at last check: 131.155.20.126). There is an older version to be found in the Usenet archives, but it contains bugs that have been fixed in the meantime, and hence is unsupported. The author also made available a program that converts data files from MPS-format into lp_solve's own input format; it's in the same directory, in file mps2eq_0.2.tar.Z. The documentation states that it is not public domain, and the author wants to discuss it with would-be commercial users. As an editorial opinion, I must state that difficult models will give lp_solve trouble; it's not as good as a commercial code. But for someone who isn't sure what kind of LP code is needed, it represents a reasonable first try. LP-Optimizer is a simplex-based code for linear and integer programs, written by Markus Weidenauer (

16. Operations Research Management Science Today - June 2005 Survey of modern software for linear programming (by R.Fourer, 2005). http://www.lionhrtpub.com/orms/orms-6-05/frsurvey.html

17. Linear Programming Ratfor code for the primaldual log barrier form of the interior point LP solver of Lustig, Marsten and Shanno, ORSA J Opt 1992. http://www.econ.uiuc.edu/~roger/research/rqn/rqn.b

<'End of fnc.r' #Outer wrapper for the new rqn functioncalls a frisch-newton LP solver #Does preprocessing using the functions globit and checkit to reduce initial n subroutine rqm(n2,p,a,y,rhs,d,wn,wp,beta,eps,tau,s,aa,hist) integer n,p,s(n2),hist(3,32),kit,nit,mit,m,mm,n2,maxnit,maxmit,mlim double precision a(p,n2),y(n2),rhs(p),d(n2),wn(1),wp(p,p+3),aa(p,p) double precision beta,eps,tau,omega,sparsity #real ut,time,udt data zero/0.0d0/ data one/1.0d0/ data two/2.0d0/ maxmit=8 maxnit=4 n=n2-2 m=2*nint(n**(2./3.)) mlim = 5*m mit=0 kit=0 #outer loop on the initial sample size while(mit <=x <0)deltap=dmin1(deltap,-x(i)/dx(i)) if(ds(i) <0)deltap=dmin1(deltap,-s(i)/ds(i)) if(dz(i) <0)deltad=dmin1(deltad,-z(i)/dz(i)) if(dw(i) <0)deltap=dmin1(deltap,-x(i)/dx(i)) else deltap=dmin1(deltap,-s(i)/ds(i)) if(dz(i) <0)deltad=dmin1(deltad,-z(i)/dz(i)) if(dw(i) <'End of sparsity.r' #This is a Siddiqui sparsity function estimate based on residuals double precision function sparsity(n,u,tau) integer n,nd,enuf double precision u(n),tau,h,qhi,qlo,half data half/0.5d0/ data enuf/600/ #bandwidth: approximate Hall-Sheather method - quadratic approx max error 1% nd=nint((.05 + 3.65*tau - 3.65*tau**2)*(n**(2./3.))) h=dfloat(nd)/dfloat(n) #compute Siddiqui estimator call kuantile(n,u,tau+h,qhi,enuf,half,half) call kuantile(n,u,tau-h,qlo,enuf,half,half) sparsity=(qhi-qlo)/(h+h) return end #function to compute pth quantile of a sample of n observations subroutine kuantile(n,x,p,q,mmax,cs,cd) integer n,k,l,r,mmax double precision x(n),p,q,cs,cd if(p

18. Linear Programming - Simplex Method First Choice Menu With this options you can chose clear the field results or clear the field linear program. It is also, available the options of no clear http://algos.inesc.pt/lp/

Linear Programming - Simplex Applet By Pedro Miguel Silva and Tiago Castro Guise Version 1.0 - Lisbon, July 1998, updated on October 1999 You are visitor number to this page since last reboot. Simplex Applet: The available LP algorithms are: Simplex Method, Revised Method, Primal Dual and Simplex Dual. Enter Your Linear Program: ... Applet Controls here, if you had java in your browser. How the Applet Works: Buttons: Solve - Solve your linear program. Abort Abort the execution of the algorithm. Clear - Allows you to clear fields. About - Brings up an about window. Choice Menus: First Choice Menu - With this options you can chose clear the field results or clear the field linear program. It is also, available the options of no clear fields and clear all fields. Second Choice Menu - Chose the algorithm you want Simplex, Revised Simplex, Primal Dual or Simplex Dual. . Third Choice Menu - Chose output options. Linear Programming: A linear program is a problem a problem that can be expressed as follows: min cx (Standard Form) subject to Ax = b x >= Where "x" is the vector of variables to be solved, "A" is the matrix of known coefficients and "c" and "b" are vectors of known coefficients. The Expression "cx" is called the objective function and the equations "Ax = b" are called the constraints.

19. Linear And Nonlinear Programming FAQs The FAQs for linear programming and Nonlinear programming have moved to a different site http//www.mcs.anl.gov/home/otc/Guide/faq/ http://www.skypoint.com/~ashbury/nonlinear-programming-faq.html

20. Dash Optimization - Leading Optimization Software The XpressOptimizer features optimization algorithms which enable you to solve linear programming problems (LP), mixed integer programming problems (MIP), http://www.dashoptimization.com/home/bd/linear_programming.html

Javascript Menu by Deluxe-Menu.com var dmWorkPath = "home/menu/deluxe/"; [Home] [Client Area] Xpress-MP Optimization Software - Linear Programming Dash's Xpress-MP Suite is the leading optimization software product for linear programming, modeling and optimization. The Xpress-Optimizer features optimization algorithms which enable you to solve linear programming problems (LP), mixed integer programming problems (MIP), quadratic programming problems (QP) and mixed integer quadratic programming problems (MIQP). The simplex optimizer, which includes primal and dual methods, solves linear programming problems. The Barrier optimizer is uses interior point methods to solve linear programming and quadratic programming problems. Xpress-MP is noted for its ability to solve numerically hard or unstable linear programming problems, which is one of the reasons it is the clear market leader in the process industries. Read more about our products , our success stories and our support services Related Topics Home Products What is Optimization?

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