81. Problem 8: Linear Programming: Strongly Polynomial? Problem 8 linear programming Strongly Polynomial? Statement Is linear programming strongly polynomial? Polynomial algorithm in linear programming. http://cs.smith.edu/~orourke/TOPP/P8.html

Next: Problem 9: Edge-Unfolding Convex Up: The Open Problems Project Previous: Problem 7: k-sets Problem 8: Linear Programming: Strongly Polynomial? Statement Is linear programming strongly polynomial? Origin Uncertain, pending investigation. Status/Conjectures Open. Partial and Related Results It is known to be weakly polynomial, exponential in the bit complexity of the input data [ ]. Subexponential time is achievable via a randomized algorithm [ ]. In any fixed dimension, linear programming can be solved in strongly polynomial linear time (linear in the input size) [ Appearances Categories linear programming Entry Revision History J. O'Rourke, 2 Aug. 2001. Bibliography M. E. Dyer. Linear time algorithms for two- and three-variable linear programs. SIAM J. Comput. N. Karmarkar. A new polynomial-time algorithm for linear programming. Combinatorica L. G. Khachiyan. Polynomial algorithm in linear programming. U.S.S.R. Comput. Math. and Math. Phys. N. Megiddo. Linear programming in linear time when the dimension is fixed. J. ACM

82. Linear Programming Section 34 linear programming. Demo linear programming (Exploremath - requires Shockwave). Try the quiz at the bottom of the page! http://home.alltel.net/okrebs/page34.html

Section 3-4:  Linear Programming Demo: Linear Programming (Exploremath - requires Shockwave) Try the quiz at the bottom of the page! go to quiz Linear programming is a method used to identify optimal maximum or minimum values.  It is used in business for practical planning, decision-making problems, and many other problems that can be done using a computer.  Each different resource can be written as a linear inequality called a constraint .  These constraints can be resources like the number of workers, amount of time on a given shift, number of machines, availability of these machines, etc., etc.  By using what we call the corner point theorem , we can find an optimal solution(s) for our problem.  When we graph these constraints, we will get a feasible region that contains our solutions.  The corner point theorem says that if a maximum or minimum value exists, it will occur at a corner point of this feasible region. Sample problem 1)  Find and graph the feasible region for the following constraints: x + y 2x + y x 0, y

83. LINEAR PROGRAMMING linear programming. Syllabus. Exercises. Some material presented in class. LINDO Links. Links. http://www.cs.bgu.ac.il/~berend/teaching/Past-Courses/Linear-Programming-Fall97/

Linear Programming Syllabus Exercises Some material presented in class LINDO Links ... Links

84. Linear Programming linear programming. Discussion linear programming is the most important problem in mathematical optimization and operations research. Applications include http://www2.toki.or.id/book/AlgDesignManual/BOOK/BOOK3/NODE141.HTM

Next: Random Number Generation Up: Numerical Problems Previous: Constrained and Unconstrained Optimization Linear Programming Input description : A set S of n linear inequalities on m variables , and a linear optimization function Problem description : Which variable assignment X ' maximizes the objective function f while satisfying all inequalities S Discussion : Linear programming is the most important problem in mathematical optimization and operations research. Applications include: Resource allocation We seek to invest a given amount of money so as to maximize our return. Our possible options, payoffs, and expenses can usually be expressed as a system of linear inequalities, such that we seek to maximize our possible profit given the various constraints. Very large linear programming problems are routinely solved by airlines and other corporations. Approximating the solution of inconsistent equations A set of m linear equations on n variables , is overdetermined if m n Such overdetermined systems are often inconsistent , meaning that no assignment of variables simultaneously solves all the equations. To find the variable assignment that best fits the equations, we can replace each variable

85. Linear Programming linear programming. linear programming Many practical problems in operations research can be expressed as linear programming problems. For instance http://www.fact-index.com/l/li/linear_programming.html

Main Page See live article Alphabetical index Linear programming 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. Many practical problems in operations research can be expressed as linear programming problems. For instance, if x1 is the number of acres planted with wheat and x2 is the number planted with corn, and a farmer has a limited number of acres A, and has a limited permissible amount F of fertilizer and P of insecticide which can be used, each of which is required in different amounts per acre (F1, F2, P1, P2) for wheat and corn respectively, and knows the selling price of wheat S1 and the selling price of corn S2, then the optimal number of acres to plant with wheat vs corn can be expressed as a linear program: Subject to the linear constraints: x x x x A (limit on total acrage) F x F x F (limit on fertilizer) P x P x P (limit on insecticide) maximize the objective function: S x S x 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

86. MATHPROG - Solvers For Linear Programming Problems MATHPROG. Codes for Solving linear programming Problems. GULF, LINEARFRACTIONAL programming (LFP) and linear programming (LP) package http://elib.zib.de/pub/Packages/mathprog/linprog/

MATHPROG Computational Geometry Computational Number Theory Linear Programming Matching Maximum Flow Minimum Cost Flow Minimum Cuts Miscellaneous network optimization codes Non-linear Optimization Polytopes and Polyhedra Semide-definite Programming Solving optimization problems via the Web Tools and Libraries References to other optimization resources MATHPROG Codes for Solving Linear Programming Problems GULF LINEAR-FRACTIONAL programming (LFP) and LINEAR programming (LP) package GULF for IBM compatible MS-DOS microcomputers by E.B. Bajalinov. Greenberg's LP Tools H. Greenberg's tools ANALYZE, MODLER, and RANDMOD for analyzing, modeling and randomizing Linear Programming problems. HOPDM Package for solving large scale Linear Programming problems, implementation of primal-dual logarithmic barrier method by J. Gondzio. ILPS Interactive Linear Programming Server provided by Remote Interactive Optimization Testbed of IEOR at the University of California at Berkeley. LIPSOL Linear Programming Interior Point Solvers package LIPSOL by Yin Zhang.

87. Linear Programming linear programming. linear programming is a particular case of constrained optimization problems. The linear programming problem (P) is then interpreted as http://www.cut-the-knot.org/do_you_know/lin_pr.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site 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 an mxn matrix A an mx1 ( column ) vector b a 1xn ( row ) vector c The role of unknown is played by a nx1 (column) vector 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 situations where the set of feasible vectors is not empty. Finally, the cost or objective function f is given by f( x ) = c. 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

88. Two Dimensional Linear Programming Two dimensional linear programming. linear programming is the problem of deciding whether there exists a solution to a system of linear ineqalities. http://valis.cs.uiuc.edu/~sariel/CG/applets/linear_prog/main.html

Home Bookmarks Search Computational Geometry ... Papers Two dimensional linear programming Applet By Evgeny Gonopolsky Linear programming is the problem of deciding whether there exists a solution to a system of linear ineqalities. Alot of work was gone into this problem, resulting in the classical SIMPLEX algorithm (which is exponential in the worst case), and the Ellipsoid method. The SIMPlEX algorithm is exponential in the worstcase, and the Ellipsoid algorithm is weakly polynomial. Coming up with a strongly-polynomial time algorithm for the linear programming problem is one of the major open problems in computer science. However, if the number of variables in the system of inequalities is small (2, 3, or even a small constant) better methods are known. In this applet we demonstrate Megiddo algorithm (independently discovered by Dyer) for the planar case. Here, each inequlity: a i x + b i y <= c i corresponds to half of the plane (because, a i x + b i y = c i is a line, and all the points that lie to one side of this line fulfill this inequality). We know that there is a solution if the intersection of the all those regions is not empty (i.e. the feasible region). In fact, usually we also wish to find a solution that minimizes (or maximizes) a linear function (i.e. objective function). First, we apply a linear transformation to the points of the plane, so that the objective function becomes equal to the negative

89. Linear Programming Worksheets linear programming worksheets. linprog.zip A zip file of all worksheets; linprog.ps A postscript file of the worksheets; ma416s97.mws http://www.ms.uky.edu/~carl/linprog.html

Linear Programming worksheets linprog.zip A zip file of all worksheets linprog.ps A postscript file of the worksheets ma416s97.mws table of contents worksheet sylls97.mws intro.mws shfeas.mws wyndor.mws ... soma.mws

90. Applications Of LP Applications of linear programming. This is to give you an indication of some of the applications of linear programming, and of optimization in general. http://www.rpi.edu/~mitchj/math1900/lp/

Next: The Diet Problem Applications of Linear Programming This is to give you an indication of some of the applications of linear programming, and of optimization in general. The Diet Problem Portfolio Optimization Crew scheduling Manufacturing and transportation ... John E Mitchell

91. MATP6640 / DSES6770 Linear Programming. linear programming. Spring 2004. MATP6640 / DSES6770. Course outline. Material on reserve in the library. Myths and counterexamples in linear programming. http://www.rpi.edu/~mitchj/matp6640/

Linear Programming Spring 2004 Course outline Material on reserve in the library. You can borrow the books for up to an hour. Further, you can borrow the books overnight if you check them out less than an hour before closing time and return them early the following day. Office hours: Wednesday , April 28, 1-4pm, or by appointment, Amos Eaton 325. Student presentations . The remaining presentations will be on Tuesday, May 4 , from 11.30am-2.30pm, in Amos Eaton 216. An example of a writeup Two SDP references: Vandenberghe and Boyd and Alizadeh Nesterov and Todd article on algorithms for self-scaled cones. Alizadeh and Goldfarb survey of second order cone programming (published in Mathematical Programming , volume 95(1), 2003, pages 3-51). Midterm Exam . Here are the solutions Homework: Homework 1 . Here are the solutions Homework 2 . Here are the solutions Homework 3 . Here are the solutions Homework 4 . Here are the solutions Homework 5 . Note: the homework is now due on April 19. Problems 2 and 4 were clarified/corrected on April 15. Presentations: Presentations will take place during the scheduled final exam time, Tuesday May 4, 11.30am-2.30pm.

92. 90: Operations Research, Mathematical Programming Depending on the options and constraints in the setting, this may involve linear programming, or quadratic, convex-, integer-, or boolean-programming. History. http://www.math.niu.edu/~rusin/known-math/index/90-XX.html

Search Subject Index MathMap Tour ... Help! ABOUT: Introduction History Related areas Subfields POINTERS: Texts Software Web links Selected topics here 90: Operations research, mathematical programming Introduction Operations research may be loosely described as the study of optimal resource allocation. Mathematically, this is the study of optimization. Depending on the options and constraints in the setting, this may involve linear programming, or quadratic-, convex-, integer-, or boolean-programming. History Some links to the history of Operations Research can be found at the Military Operations Research Society and on J. E. Beasley 's home page. Applications and related fields For numerical optimization techniques (conjugate gradient, simulated annealing, etc.) see 65, Numerical Analysis Discrete optimization problems (traveling salesman, etc.) are principally treated in Combinatorics The word "programming" in this context is essentially unrelated to computer programming; for that topic see Computer Science Subfields Operations research and management science (for discrete assignment problems see also 05-XX.)

93. From Mjs@hubcap.clemson.edu (MJ Saltzman) Newsgroups Sci.math Check out the linear programming FAQ at a href= a href= a href= http//www.skypoint.com/subscribers/ashbury/linearprogramming-faq.html http//www.skypoint http://www.math.niu.edu/~rusin/known-math/96/linprog.faq

From: mjs@hubcap.clemson.edu (M. J. Saltzman) Newsgroups: sci.math Subject: Re: >>n-dimensional Convex Hull algorithm? < Date: 6 Aug 1996 00:55:21 GMT In article , Martin C. Glanvill

94. Linear Programming linear programming. An FAQ of linear programming is here. This results. Some sample usage of the linear programming package are here . http://www.cs.wustl.edu/~javagrp/help/LinearProgramming.html

Linear Programming An FAQ of Linear Programming is here This is the source codes of Java version of written by Michel Berkelaar The Java version does not recognize lp format or mps format. It does not take a file as input. You should specify the objective function and constraints by the methods provided by class solve, such as and call to solve the problem, then call to print out results. Some sample usage of the Linear Programming package are here . This is a straight forward translation from the C code of lp_solve 2.0. Almost all variables and methods use the same name as in lp_solve 2.0. go to Network Design Tool go to Network Design Tool Help Network Design Group Washington University in St. Louis Send questions to Hongzhou Ma (hma@cs.wustl.edu) Inderjeet Singh (inder@cs.wustl.edu) Last modified on Wednesday, 26-Feb-1997 15:46:25 CST

95. Large-Scale Linear Programming Techniques For The Design Of Protein Folding Pote LargeScale linear programming Techniques for the Design of Protein Folding Potentials Michael Wagner Jaroslaw Meller Ron Elber Abstract We present large http://www.optimization-online.org/DB_HTML/2002/02/444.html

Large-Scale Linear Programming Techniques for the Design of Protein Folding Potentials Michael Wagner Jaroslaw Meller Ron Elber Abstract Keywords : protein folding potentials, linear programming, parallel processing, linear inequality systems Category 1 : Applications Science and Engineering ( Basic Sciences Applications ) Category 2 : Linear, Cone and Semidefinite Programming ( Linear Programming ) Category 3 : Optimization Software and Modeling Systems ( Parallel Algorithms ) Citation : ODU-TR-2002-02, Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529-0077. Download Compressed Postscript PDF Entry Submitted : 02/12/2002 Entry Accepted : 02/12/2002 Entry Last Modified : 02/12/2002 Modify/Update this entry Back to Optimization Online

96. On A New Collection Of Stochastic Linear Programming Test Problems On a new collection of stochastic linear programming test problems KA Ariyawansa Andrew J. Felt Abstract The purpose of this paper is to introduce a new test http://www.optimization-online.org/DB_HTML/2001/05/334.html

On a new collection of stochastic linear programming test problems K. A. Ariyawansa Andrew J. Felt Abstract Keywords : stochastic programming Category 1 : Stochastic Programming ( ) Citation : Technical Report 01-4 Department of Mathematics Washington State University Pullman, WA 99164-3113 submitted for review and publication Download Postscript PDF Entry Submitted : 05/31/2001 Entry Accepted : 05/31/2001 Entry Last Modified : 05/31/2001 Modify/Update this entry Back to Optimization Online

97. CE597S: Graphical Simplex Algorithm Graphical Simplex Algorithm (2D). Graphical method provides us valuable insights about the general nature of solutions to linear programming problems. http://gunsmoke.ecn.purdue.edu:8658/ce597s/project/

Graphical Simplex Algorithm (2D) Graphical method provides us valuable insights about the general nature of solutions to linear programming problems. We build this server as an interface between graphical solution and simplex algorithm. Since graphical mothod can only apply to small problems with two, or at most three decision variables, this server is built to deal with LP problems with two decision variables but any number of constrains. In this tutorial, we adopt the standard formulation for linear programming problem. First, you are required to specify the objective function, and the number of the constrains. Then you continue to define the constrains. After you finish your definition of the LP problem, a graphical solution is provided with the feasible region shaded and optimal optimal point highlighted. A tabular solution with the decision variables and slack variables at all extreme points is also presented. The relationship between the graphical solution and the tabular solution is specified by various colors as well as labeled points. This tutorial is built as a server-side application. For users with less fast network connection, please visit

98. The Use Of Linear Programming In The Construction Of Extremal Solutions To Linea The Use of linear programming in the Construction of Extremal Solutions to Linear Inverse Problems. Stephen P. Huestis. Abstract. http://epubs.siam.org/sam-bin/dbq/article/28895

SIAM Review Volume 38, Number 3 pp. 496-506 The Use of Linear Programming in the Construction of Extremal Solutions to Linear Inverse Problems Stephen P. Huestis Abstract. While a finite collection of data does not specify a unique solution to a linear inverse problem, it can allow bounds to be placed on certain nonlinear solution functionals. Using the Dirichlet problem for the unit disc as an example, this note demonstrates the use of linear programming in constructing extremal solutions associated with a variety of such bounds. Key words. linear inverse problems, extremal solutions, linear programming AMS Subject Classifications DOI Retrieve PostScript document ( 28895.ps : 4530698 bytes) Retrieve GNU Compressed PostScript document ( ... : 39084 bytes) For additional information contact service@siam.org

99. Infinite Linear Programming And Multichain Markov Control Processes In Uncountab Infinite linear programming and Multichain Markov Control Processes in Uncountable Spaces. Onésimo HernándezLerma, Juan González-Hernández. Abstract. http://epubs.siam.org/sam-bin/dbq/article/29223

SIAM Journal on Control and Optimization Volume 36, Number 1 pp. 313-335 Infinite Linear Programming and Multichain Markov Control Processes in Uncountable Spaces Onésimo Hernández-Lerma, Juan González-Hernández Abstract. In this paper we use infinite linear programming to study Markov control processes in Borel spaces and the average cost criterion in the "unichain" and "multichain" cases. Under appropriate assumptions we show that in both cases the associated linear programs are solvable and that there is no duality gap. Moreover, conditions are given for minimizing (respectively, maximizing) sequences for the primal (respectively, dual) programs to converge to optimal solutions. Key words. (discrete-time) Markov control processes, average cost criterion, infinite linear programming, generalized Farkas theorem AMS Subject Classifications DOI Retrieve PostScript document ( 29223.ps : 1449936 bytes) Retrieve GNU Compressed PostScript document ( ... Retrieve reference links For additional information contact service@siam.org

100. Linear Programming As Applied To Stock Market Options An Introduction to linear programming As Applied to Stock Market Options, The solution method we will use is called linear programming. http://www.durangobill.com/LP_Options.html

Durango Bill's Applied Mathematics Guaranteed Profits in Stock Market Options? An Introduction to Linear Programming As Applied to Stock Market Options An interesting secondary question can be asked. Is there any combination of buying and/or selling a mixture of these options (at the given prices) that can guarantee a profit no matter the market does? Interestingly the answer is YES. In fact, given any set of option prices (real, arbitrary, real time, closing values, this year, next year, etc.) there are always buy/sell combinations that will guarantee a profit no matter what the market does. There are a few conditions that must be met. 1) Brokerage companies will subtract commissions from the profit shown by the calculations. (Good news - these are usually small enough that they are not a problem.) 2) The securities laws restrict some of the combinations. (Eliminates some of the combinations, but other combinations are open.) 3) Once you have made your commitment, the options remain unexercised until they expire. 4) For any combination, you have to execute all of the orders at (or near) the given price - usually this must be done simultaneously. (There is a long frustrating story regarding this. If you are an outside investor, your orders will probably never be executed. However, this is an example of how brokerage companies operate in the financial derivatives markets.)

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