Read e-book online Linear Programming: Foundations and Extensions PDF

By Robert J Vanderbei

ISBN-10: 1475756623

ISBN-13: 9781475756623

ISBN-10: 147575664X

ISBN-13: 9781475756647

This Fourth variation introduces the newest concept and purposes in optimization. It emphasizes restricted optimization, starting with a considerable remedy of linear programming after which continuing to convex research, community flows, integer programming, quadratic programming, and convex optimization. Readers will find a host of sensible company functions in addition to non-business applications.

Topics are essentially constructed with many numerical examples labored out intimately. particular examples and urban algorithms precede extra summary subject matters. With its specialise in fixing useful difficulties, the publication good points loose C courses to enforce the key algorithms lined, together with the two-phase simplex approach, primal-dual simplex process, path-following interior-point strategy, and homogeneous self-dual equipment. additionally, the writer presents on-line JAVA applets that illustrate numerous pivot principles and variations of the simplex technique, either for linear programming and for community flows. those C courses and JAVA instruments are available at the book's site. the web site additionally contains new on-line educational instruments and exercises.

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Extra info for Linear Programming: Foundations and Extensions

Example text

We start by adding so-called slack variables. 1) we introduce a new variable that represents the difference between the right-hand side and the left-hand side. For example, for the first inequality, we introduce the slack variable w 1 defined by It is clear then that this definition of WI, together with the stipulation that WI be nonnegative, is equivalent to the original constraint. 2) maximize ( subject to WI = = 5- 2xi - 3x2 - w2 = 11 - 4xl - X3 x2 - 2x3 w3 = 8 - 3xi - 4x2 - 2x3 14 2. THE SIMPLEX METHOD Note that we have included a notation,(, for the value of the objective function, 5x 1 + 4x2 + 3x3.

Within each iteration of the simplex method, exactly one variable goes from nonbasic to basic and exactly one variable goes from basic to nonbasic. We saw this process in our example, but let us now describe it in general. The variable that goes from nonbasic to basic is called the entering variable. It is chosen with the aim of increasing (;that is, one whose coefficient is positive: pick k from {j E N : Cj > 0}. Note that if this set is empty, then the current solution is optimal. If the set consists of more than one element (as is normally the case), then we have a choice of which element to pick.

We avoided the issue before, but now we must face what to do if a denominator in one of these ratios vanishes. If the numerator is nonzero, then it is easy to see that the ratio should be interpreted as plus or minus infinity depending on the sign of the numerator. For the case of0/0, the correct convention (as we'll see momentarily) is to take this as a zero. What if all of the ratios, aik /bi, are nonpositive? In that case, none of the basic variables will become zero as the entering variable increases.

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Linear Programming: Foundations and Extensions by Robert J Vanderbei

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