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Linear programming 1 Basics - MIT Mathematics

Lecture notesMarch 17, 2015 Linear programmingLecturer: Michel Goemans1 BasicsLinear Programmingdeals with the problem of optimizing a linearobjective functionsubject tolinear equality and inequalityconstraintson thedecision variables. Linear programming has manypractical applications (in transportation, production planning, ..). It is also the building block forcombinatorial optimization. One aspect of Linear programming which is often forgotten is the factthat it is also a useful proof technique. In this first chapter, we describe some Linear programmingformulationsfor some classical problems. We also show that Linear programs can be expressed in avariety of equivalent The Diet ProblemIn the diet model, a list of available foods is given together with the nutrient content and the costper unit weight of each food.

In some cases, another form of linear program is used. A linear program is in canonical form if it is of the form: Max z= cTx subject to: Ax b x 0: A linear program in canonical form can be replaced by a linear program in standard form by just replacing Ax bby Ax+ Is= b, s 0 where sis a vector of slack variables and Iis the m m identity matrix.

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