introduction-lp-duality1

# Introduction-lp-duality1

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Unformatted text preview: ts that the values of the variables x1 , x2 and x3 can be chosen at will while the values of x4 , x5 , x6 and z are ﬁxed. In this dictionary, the decision variables x1 , x2 , x3 act as independent variables while the slack variables x4 , x5 , x6 are related to each other. In the dictionary (9.13), the independent variables are x2 , x4 , x6 and the related ones are x3 , x1 , x5 , z. Property 9.2. The equations of a dictionary have to express m variables among x1 , x2 , . . . , xn+m , z in function of the n remaining others. Properties 9.1 and 9.2 deﬁne what a dictionary is. In addition to these two properties, the dictionaries (9.6),(9.11) and (9.13) have the following property. Property 9.3. When putting the right variables to zero, one obtains a feasible solution by evaluating the left variables. The dictionaries that have this last property are called feasible dictionaries. As a matter of fact, any feasible dictionary describes a feasible solution. However, all feasible solutions cannot be described by a feasible dictionary. For example, no dictionary describe the feasible solution x1 = 1, x2 = 0, x3 = 1, x4 = 2, x5 = 5, x6 = 3 of (9.4). The feasible solutions that can be described by dictionaries are referred as basic solutions. The Simplex Method explores only basic solutions and ignores all other ones. But this is valid because if an optimal solution exists, then there is an optimal and basic solution. Indeed, if a feasible solution cannot be improved by the Simplex Method, then increasing any of the n right variables to a positive value never increases the objective function. In such case, the objective function must be written as a linear function of these variables in which all the coefﬁcient are non-positive, and thus the objective function is clearly maximum when all the right variables equal zero. For example, it was the case in (9.14). 9.2.3 Finding an initial solution In the previous examples, the initialisation of the simplex method was not a problem. As a matter of fact, we carefully chosen problems with all bi non negative. This way x1 = 0, x2 = 0, 136 CHAPTER 9. LINEAR PROGRAMMING · · · , xn = 0 was a feasible solution and the dictionary was easily built. These problems are called problems with a feasible origin. What happens when confronted with a problem with an unfeasible origin? Two difﬁculties arise. First, a feasible solution can be hard to ﬁnd. Second, even if we ﬁnd a feasible solution, a feasible dictionary has then to be built. A way to solve these difﬁculties is to use an other problem called auxiliary problem: Minimise x0 n Subject to: ∑ j=1 ai j x j − x0 ≤ bi (i = 1, 2, · · · , m) x j ≥ 0 ( j = 0, 1, · · · , n). A feasible solution of the auxiliary problem is easily available: it is enough to set x j = 0∀ j ∈ [1 . . . n] and to give to x0 a big enough value. It is now easy to see that the original problem has a feasible solution if and only if the auxiliary problem has a feasible solution with x0 = 0. In other words, the original problem has a...
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## This note was uploaded on 11/20/2013 for the course CS 101 taught by Professor Smith during the Fall '13 term at Mitchell Technical Institute.

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