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Hans Mittelmann's Benchmarks for Optimization Software. For rigorous definitions and theory, which are beyond the scope of this document, the interested reader is referred to the many LP textbooks in print, a few of which are listed in the references section.

All these entities must have consistent dimensions, of course, and you can add "transpose" symbols to taste. The matrix A is generally not square, hence you don't solve an LP by just inverting A.

The word "Programming" is used here in the sense of "planning"; the necessary relationship to computer programming was incidental to the choice of name. Hence the phrase "LP program" to refer to a piece of software is not a redundancy, although I tend to use the term "code" instead of "program" to avoid the possible ambiguity.

Although all linear programs can be put into the Standard Form, in practice it may not be necessary to do so. This allows a variable to be without an explicit upper or lower bound, although of course the constraints in the A-matrix will need to put implied limits on the variable or else the problem may have no finite solution.

Also, LP software can handle maximization problems just as easily as minimization in effect, the vector c is just multiplied college essay header format on a mac The importance of linear programming derives in part from its many applications see further below and in part from the existence of good general-purpose techniques for finding optimal solutions.

These techniques take as input only an LP in the above Standard Form, and determine a solution without reference to any information concerning the LP's origins or special structure.

They are fast and reliable over a substantial range of problem sizes and applications. Two families of solution techniques are in wide use today. Both visit a progressively improving series of trial solutions, until a solution is reached that satisfies the conditions for an optimum.

Barrier or interior-point methods, by contrast, visit points within the interior of the feasible region.

These methods derive from techniques for non-linear programming that were developed and popularized in the s by Fiacco and McCormick, but their application to linear programming dates back only to Karmarkar's innovative analysis in The related problem of integer programming or integer linear programming, strictly speaking requires some or all of the variables to take integer whole number values.

Integer programs IPs often have the advantage of being more realistic than LPs, but the disadvantage of being much harder to solve. The most widely used general-purpose techniques for solving IPs use the solutions to a series of LPs to manage the search for integer solutions and to prove optimality.

Linear and integer programming have proved valuable for modelling many and diverse types of problems in planning, routing, scheduling, assignment, and design. Industries that make use of LP and its extensions include transportation, energy, telecommunications, and manufacturing of many kinds.

A sampling of applications can be found in many LP textbooksin books on LP modelling systemsand among the application cases in the journal Interfaces.

Thanks to the advances in computing of the past decade, linear programs in a few thousand variables and constraints are nowadays viewed as "small". Problems having tens or hundreds of thousands of continuous variables are regularly solved; tractable integer programs are necessarily smaller, but are still commonly in the hundreds or thousands of variables and constraints.

The computers of choice for linear and integer programming applications are Pentium-based PCs and the several varieties of Unix workstations. There is more to linear programming than optimal solutions and number-crunching, however.

This can be appreciated by observing that modern LP software comes in two related but very different kinds of packages: Algorithmic codes are devoted to finding optimal solutions to specific linear programs. A code takes as input a compact listing of the LP constraint coefficients the A, b, c and related values in the standard form and produces as output a similarly compact listing of optimal solution values and related information.

An LP modelling system takes as input a description of a linear program in a form that people find reasonably natural and convenient, and allows the solution output to be viewed in similar terms; conversion to the forms requried by algorithmic codes is done automatically.

The collection of statement forms for the input is often called a modelling language. Most modelling systems support a variety of algorithmic codes, while the more popular codes can be used with many different modelling systems. Because packages of the two kinds are often bundled for convenience of marketing or operation, the distinction between them is sometimes obscured, but it is important to keep in mind when attempting to sort through the many alternatives available.

Large-scale LP algorithmic codes rely on general-structure sparse matrix techniques and numerous other refinements developed through years of experience. The fastest and most reliable codes thus represent considerable development effort, and tend to be expensive except in very limited demonstration or "student" versions.

Those codes that are free -- to all, or at least for research and teaching -- tend to be somewhat less robust, though they are still useful for many problems. The ability of a code to solve any particular class of problems cannot easily be predicted from problem size alone; some experimentation is usually necessary to establish difficulty.

Large-scale LP modelling systems are commercial products virtually without exception, and tend to be as expensive as the commercial algorithmic codes again with the exception of small demo versions.

They vary so greatly in design and capability that a description in words is adequate only to make a preliminary decision among them; your ultimate choice is best guided by using each candidate to formulate a model of interest.

Listed below are summary descriptions of available free codesand a tabulation of many commercial codes and modelling systems for linear and integer programming. A list of free demos of commercial software appears at the end of this section.

It contains references to about 75 available software packages not all of them just LPand goes into more detail than is possible in this FAQ; see in particular the sections on "linear programming" and on "modelling languages and optimization systems. Much information can also be obtained through the websites of optimization software developers, many of which are identified in the writeup and tables below.AFAM Intro to African American Studies This course provides an overview of African American history and culture.

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Free demos of commercial codes An increasing number of commercial LP software developers are making demo or academic versions available for downloading through websites or .

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