41. ILOG CPLEX: High-performance Software For Mathematical Programming And Optimizat ILOG CPLEX provides flexible, highperformance optimizers used when solving linear programming, quadratic programming, quadratically constrained programming http://www.ilog.com/products/cplex/

Sign In BRMS Optimization Visualization ... ILOG CPLEX Product Info Latest version CPLEX interfaces CPLEX algorithms ILOG Parallel CPLEX ... Datasheet Solutions Customers Manufacturing Transportation and travel Newsletters ... Contact info ILOG CPLEX The World's Leading Mathematical Programming Optimizers MM_preloadImages('../../images/imgSmall_CPLEX.jpg','../../images/imgSmall_ODM.jpg','../../images/imgSmall_OPL.jpg','../../images/imgSmall_CP.jpg'); Improve efficiency, quickly implement strategies and increase profitability. ILOG CPLEX's mathematical optimization technology enables better decision-making for efficient resource utilization. With ILOG CPLEX, complex business problems can be represented as mathematical programming models. Advanced optimization algorithms allow you to rapidly find solutions to these models. Today, over 1,000 corporations and government agencies use ILOG CPLEX, along with researchers at over 1,000 universities. ILOG CPLEX helps solve planning and scheduling problems in virtually every industry. More than 100 of the world's leading software companies are also ILOG CPLEX customers, including market leaders like SAP, Oracle and Sabre. Robust algorithms for demanding problems ILOG pioneered development of the algorithms needed to solve today's most demanding mathematical optimization problems. ILOG CPLEX has solved problems with millions of constraints and variables, and consistently sets new standards for mathematical programming software performance.

42. Linear Programming - Formulation linear programming formulation. You will recall from the Two Mines example that the conditions for a mathematical model to be a linear program (LP) were http://people.brunel.ac.uk/~mastjjb/jeb/or/lp.html

OR-Notes J E Beasley OR-Notes are a series of introductory notes on topics that fall under the broad heading of the field of operations research (OR). They were originally used by me in an introductory OR course I give at Imperial College. They are now available for use by any students and teachers interested in OR subject to the following conditions A full list of the topics available in OR-Notes can be found here Linear programming - formulation You will recall from the Two Mines example that the conditions for a mathematical model to be a linear program (LP) were: all variables continuous (i.e. can take fractional values) a single objective (minimise or maximise) the objective and constraints are linear i.e. any term is either a constant or a constant multiplied by an unknown. LP's are important - this is because: many practical problems can be formulated as LP's there exists an algorithm (called the simplex algorithm) which enables us to solve LP's numerically relatively easily. We will return later to the simplex algorithm for solving LP's but for the moment we will concentrate upon formulating LP's.

43. Virtual Optima Information on LPL, a mathematical modeling language, related product, projects and links. http://www.virtual-optima.com/

44. The Diet Problem An Application of linear programming. cute icon The problem is formulated as a linear program where the objective is to minimize cost and meet http://www-neos.mcs.anl.gov/CaseStudies/dietpy/WebForms/index.html

The Diet Problem: An Application of Linear Programming CLICK HERE for the demo! What's New Edit the Constraints! Helpful Suggestions for Infeasible Diets!! Description of the diet problem The goal of the diet problem is to find the cheapest combination of foods that will satisfy all the daily nutritional requirements of a person. The problem is formulated as a linear program where the objective is to minimize cost and meet constraints which require that nutritional needs be satisfied. We include constraints that regulate the number of calories and amounts of vitamins, minerals, fats, sodium and cholesterol in the diet. The mathematical formulation is simple, but you will find out by running the model that people do not actually choose their menus by solving this model. Our nutritional requirements can be met yet our concerns for taste and variety go unheeded. We would never drink gallons of vinegar nor include a few boullion cubes in our meals; however, such "optimal" menus have been created using this model. Read more about the history of the diet problem for more interesting facts.

45. Dantzig, G.: Linear Programming And Extensions. of the book linear programming and Extensions by Dantzig, G., published by Princeton University Press....... http://pup.princeton.edu/titles/413.html

Book Search: Keywords Author Title or ISBN More Options Power Search Search Hints Google contents of this website: Google full text of our books: Linear Programming and Extensions George B. Dantzig One of Princeton University Press's Notable Centenary Titles. Shopping Cart Endorsements Table of Contents Google full text of this book: In real-world problems related to finance, business, and management, mathematicians and economists frequently encounter optimization problems. In this classic book, George Dantzig looks at a wealth of examples and develops linear programming methods for their solutions. He begins by introducing the basic theory of linear inequalities and describes the powerful simplex method used to solve them. Treatments of the price concept, the transportation problem, and matrix methods are also given, and key mathematical concepts such as the properties of convex sets and linear vector spaces are covered. George Dantzig is properly acclaimed as the "father of linear programming." Linear programming is a mathematical technique used to optimize a situation. It can be used to minimize traffic congestion or to maximize the scheduling of airline flights. He formulated its basic theoretical model and discovered its underlying computational algorithm, the "simplex method," in a pathbreaking memorandum published by the United States Air Force in early 1948. Linear Programming and Extensions provides an extraordinary account of the subsequent development of his subject, including research in mathematical theory, computation, economic analysis, and applications to industrial problems.

46. Problem 8: Linear Programming: Strongly Polynomial? Problem 8 linear programming Strongly Polynomial? http://maven.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 Nimrod Megiddo [ 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), established in dimensions 2 and 3 in [ ] and for all dimensions in [ Appearances Categories linear programming Entry Revision History J. O'Rourke, 2 Aug 2001, 16 Jul 2007. 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.

47. Zuse Institute Berlin - Mathprog - Linear Programming Software HOPDM, Package for solving large scale linear programming problems, LIPSOL, linear programming Interior Point Solvers package by Yin Zhang. http://elib.zib.de/pub/Packages/mathprog/linprog/

48. Mixed Integer Programming If all the variables can be rational (the set D is empty), this is a linear programming problem, which can be solved in polynomial time. http://www.cs.sandia.gov/opt/survey/mip.html

Mixed Integer Programming Overview A mixed-integer program is the minimization or maximization of a linear function subject to linear constraints. More explicitly, a mixed-integer program with n variables and m constraints has the form: T minimize c x subject to A x = b 1 1 A x <= b 2 2 l <= x If all the variables can be rational (the set D is empty), this is a linear programming problem, which can be solved in polynomial time. In practice linear programs can be solved efficiently for reasonable-sized problems, or even for big problems with special structure. However when some or all of the variables must be integer, corresponding to pure integer and mixed integer programming respectively, the problem becomes NP-complete (formally intractible). For the sake of this discussion, assume that all variables must be integral. The general methods can be easily extended to the mixed-integer case. Branch and Bound The most widely used method for solving integer programs is branch and bound. Subproblems are created by restricting the range of the integer variables. For binary variables, there are only two possible restrictions: setting the variable to 0, or setting the variable to 1. More generally, a variable with lower bound l and upper bound u will be divided into two problems with ranges l to q and q+1 to u respectively. Lower bounds are provided by the linear-programming relaxation to the problem: keep the objective function and all constraints, but relax the integrality restrictions to derive a linear program. If the optimal solution to a relaxed problem is (coincidentally) integral, it is an optimal solution to the subproblem, and the value can be used to terminate searches of subproblems whose lower bound is higher. See the

49. Linear Programming Selected topics in linear programming, including problem formulation checklist, sensitivity analysis, binary variables, simulation, useful functions, http://www.quickmba.com/ops/lp/

QuickMBA Operations Linear Programming Linear Programming Topics 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. Problem Formulation Checklist The objective function and constraints are formulated from information extracted from the problem statement. The following checklist is useful for minimizing the risk of errors in problem formulation. Every number in problem statement should be either implemented in the formulation or rejected as irrelevant, e.g. sunk costs. Don't forget any initial conditions, e.g. initial staff on hand at beginning of first staffing period. Ensure every variable in the objective function is listed somewhere in the constraints. Ensure that any non-negativity constraints are listed. Ensure that binary integer variables are restricted to 0,1. For good form, move all variables to left hand side of equation, writing them in the order of their subscripts. Sensitivity Analysis While problems may be modeled using deterministic objective functions, in the real world there is variation. A

50. Linear Programming: Introduction Explains the terminology and demonstrates the basic techniques for linear programming ; that is, for maximizing or minimizing a linear relation subject to http://www.purplemath.com/modules/linprog.htm

powered by FreeFind Print-friendly page Linear Programming: Introduction (page 1 of 5) Sections: Optimizing linear systems, Setting up word problems Linear programming is the process of taking various linear inequalities relating to some situation, and finding the "best" value obtainable under those conditions. A typical example would be taking the limitations of materials and labor, the determining the "best" production levels for maximal profits under those conditions. In "real life", linear programming is part of a very important area of mathematics called "optimization techniques". This field of study (or at least the applied results of it) are used every day in the organization and allocation of resources. These "real life" systems can have dozens or hundreds of variables, or more. In algebra, though, you'll only work with the simple (and graphable) two-variable linear case. The general process is to graph the inequalities (which, in this context, are called "constraints") to form a walled-off area on the

51. Two Dimensional Linear Programming 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, http://valis.cs.uiuc.edu/~sariel/research/CG/applets/linear_prog/main.html

52. BPMPD Home Page I m collecting linear programming problems (in MPS format, formulated as STOCHLP, deterministic equvivalent of stochastic linear programming problems http://www.sztaki.hu/~meszaros/bpmpd/

BPMPD Home Page This page is devoted to the interior point solver called BPMPD. The solver is developed by at the MTA SZTAKI, Computer and Automation Research Institute, Hungarian Academy of Sciences Budapest Hungary ABOUT BPMPD Newest version: 2.21 (June, 1998) New features: New sparsity heuristics More efficient presolve Convex QP extension Warm-start BPMPD is a state-of-the-art implementation of a primal-dual interior point algorithm, written in C programming language. Recent version is 2.21. You can find a performance comparison of differnent IPM packages (including BPMPD) on the pages of Decision Tree for Optimization Software . Press here to view the BPMPD readme file There is a downloadable Windows95/NT executable and DLL version of the recent release. The standard BPMPD package has the following features: Standard MPS input Advanced presolve, which: removes empty rows/columns removes singleton rows eliminates free (or implied free) singleton columns removes redundant constraints and fixes variables based on minimum/maximum constraints activity does the above on the dual problem substitutes duplicated variables removes redundant bounds on variables eliminates free (or implied free) variables makes the constraint matrix sparser Sophisticated heuristic to make decision between normal equation and augmented system approach Advanced sparsity analyzer for sparse quadratic problems Scaling the problem for better numerical properties Advanced symbolic ordering for normal equations approach Left-looking supernodal factorization algorithm with loop unrolling

53. Practical Optimization: A Gentle Introduction Chapter 2 Introduction to linear programming. The basic notions of linear programming and the simplex method. The simplex method is the easiest way to http://www.sce.carleton.ca/faculty/chinneck/po.html

Practical Optimization: A Gentle Introduction John W. Chinneck Systems and Computer Engineering Carleton University Ottawa, Ontario K1S 5B6 Canada http://www.sce.carleton.ca/faculty/chinneck/po.html The chapters appearing below are draft chapters of an introductory textbook on optimization. The ultimate objective is the creation of a complete, yet compact, introductory survey text on the major topics in optimization. The material is derived from the lecture notes used by the author in engineering courses at Carleton University, and reflects the design considerations for those courses: Students need to have a solid intuitive understanding of how and why optimization methods work. This enables them to recognize when things have gone wrong, and to diagnose the source of the difficulty and take appropriate action. It also permits students to see how methods can be combined or modified to solve non-standard problems. Explanation and diagrams are more effective in transmitting a real understanding than a total reliance on mathematics. There should be some exposure to how things are done in practice, which can be significantly different than how they are usually done in textbooks.

54. Annotated Bibliography On Linear Programming Models This bibliography consists of the early papers on linear programming model formulations. It includes some papers that are not about linear programming http://itorms.iris.okstate.edu/index-10.html

55. GNU Linear Programming Kit - Summary [Savannah] GLPK is a callable library in ANSI C intended for solving large scale linear programming (LP), mixed integer programming (MIP), and other related problems. http://savannah.gnu.org/projects/glpk/

Login Status: Not Logged In Login New User This Page Clean Reload Printer Version Search in Projects People Cookbook Support Bugs Tasks Patches Hosted Projects Register New Project Full List Contributors Wanted Statistics Site Help User Docs: Cookbook User Docs: In Depth Guide Get Support Contact Us ... FAQ Developer Info Cryptographic software legal notice Anonymous CVS GNU Webmasters Documentation Organization Projects map www project GNU Project GNU GPL v3 Coming Events Free Software Directory Help GNU ... GNU Mirrors GNU Linear Programming Kit - Summary Group Main Main View Members Search ... Manage Show feedback again Membership Info Project Admins: Andrew Makhorin Andrew Makhorin members View Members Group identification Id: System Name: glpk Name: GNU Linear Programming Kit Group Type: Official GNU software Search in this Group in Cookbook Support Tasks Patches This project is part of the GNU Project. GLPK is a callable library in ANSI C intended for solving large scale linear programming (LP), mixed integer programming (MIP), and other related problems. Registration Date: Thursday 05/10/01 at 15:56 UTC License: GNU General Public License v3 or later Development Status: - Undefined Latest News No news items found [Submit News] [0 news in archive] Quick Overview Project Homepage Download Area Docs Project Memberlist members) Communication Tools Tech Support Manager open items

56. Egwald Mathematics: Operations Research - Linear Programming Operations research entry page, online linear programming solver, online solver of two person games, solve lpproblems online. http://www.egwald.com/operationsresearch/orpage.php3

Egwald Web Services Domain Names Web Site Design Egwald Mathematics: Operations Research Linear Programming by Elmer G. Wiens A. Linear Programming. 1. Step through a linear programming graphical example . Then solve a problem by hand using the simplex algorithm , or the dual simplex algorithm 2. Use either the primal simplex tableaux generator , or the dual simplex tableaux generator , to set up and solve your own linear programming problem, with a user-friendly interface. 3. Perform a sensitivity and/or postoptimal analysis of the solution to the data of an economics application of linear programming, with a user-friendly interface. a. The profit maximization decision for a firm. b. The least cost formula decision for a firm. 4. Step through a linear programming example using the online linear programming solver. 5. Use the online linear programming solver to solve your own L.P. problem. 6. As an application of linear programming, solve a zero sum two person game 7. Try your skill on these

57. Optimization Toolbox - Quadratic, Linear, And Binary Integer Programming linear programming problems consist of a linear expression for the objective function, and linear equality and inequality constraints. http://www.mathworks.com/products/optimization/description5.html

var host_pre = "http://www"; Home Select Country Contact Us Store Search Create Account Log In Industries Academia ... Documentation Other Resources Technical Literature User Stories Related Books Product Description Introduction and Key Features Defining, Solving, and Assessing Optimization Problems Nonlinear Optimization and Multi-Objective Optimization Nonlinear Least-Squares, Data Fitting, and Nonlinear Equations Quadratic, Linear, and Binary Integer Programming Solving Optimization Problems Using Parallel Computing Quadratic, Linear, and Binary Integer Programming Quadratic Programming Quadratic programming problems involve minimizing a multivariate quadratic function subject to linear equality and inequality constraints. The toolbox implements three methods for solving these problems: trust-region, preconditioned conjugate gradient, and active set. Trust-region is used for bound constrained problems. Preconditioned conjugate gradient is used for problems subject to equality constraints. Active set minimizes the objective at each iteration over the active set (a subset of the constraints that are locally active) until it reaches a solution. Linear Programming Linear programming problems consist of a linear expression for the objective function and linear equality and inequality constraints. Two methods are used to solve this type of problem: simplex and interior point.

58. Linear Programming With Equality Constraints linear programming with equality constraints. linear programming with equality constraints. Consider the following classical optimization problem http://www.scilab.org/doc/lmidoc/node11.html

Next: Sylvester Equation Up: Examples Previous: Descriptor Riccati equations Linear programming with equality constraints Consider the following classical optimization problem where A and C are matrices and e b and d are vectors with appropriate dimensions. Here the sign is to be understood elementwise. This problem can be formulated in LMITOOL as follows: and solved in Scilab by (assuming A C e b and d and an initial guess x0 exist in the environment): Scilab Group

59. Mixed Integer Linear Programming (MILP) For Control Mixedinteger linear programming (MILP) is a very general framework for capturing problems with both discrete decisions and continuous variables. http://acl.mit.edu/milp/

Mixed-Integer Linear Programming for Control Department of Aeronautics and Astronautics Aerospace Controls Laboratory Massachusetts Institute of Technology Slides for 2005 ACC tutorial on Mixed-integer Programming for Control RHTA Movie by Arthur Richards and Jonathan P. How Mixed-integer linear programming (MILP) is a very general framework for capturing problems with both discrete decisions and continuous variables. This includes: Assignment problems Control of hybrid systems Piecewise-affine (PWA) systems (including approximations of nonlinear systems) Problems with non-convex constraints (e.g., collision avoidance) You can obtain some sample code for trajectory design and assignment problem for UAV guidance. It requires Matlab and AMPL/CPLEX system to run the code. The MILP problems are sized down so that AMPL CPLEX student version can solve them. Trajectory optimization for an autonomous UAV Developed a new approach to trajectory optimization for autonomous aerial vehicles. The planner can design a trajectory that reaches targets, while satisfying constraints such as no-fly zones and aircraft dynamics.

60. Linear Programming - Randomization And Abstract Frameworks Recent years have brought some progress in the knowledge of the complexity of linear programming in the unit cost model, and the best result known at this http://citeseer.ist.psu.edu/grtner96linear.html

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