introduction-lp-duality1

xn satisfying the constraints of a linear programme

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Unformatted text preview: the standard form (See Exercices 9.1 and 9.2.). A n-tuple (x1 , . . . , xn ) satisfying the constraints of a linear programme is a feasible solution of this problem. A solution that maximizes the objective function of the problem is called an optimal solution. Beware that a linear programme does not necessarily admits a unique optimal solution. Some problems have several optimal solutions while others have none. The later case may occur for two opposite reasons: either there exist no feasible solutions, or, in a sense, there are too many. The ﬁrst case is illustrated by the following problem. Maximize Subject to: 3x1 − x2 x1 + x2 ≤ 2 −2x1 − 2x2 ≤ −10 x1 , x2 ≥ 0 (9.2) Maximize x1 − x2 Subject to: −2x1 + x2 ≤ −1 −x1 − 2x2 ≤ −2 x1 , x2 ≥ 0 (9.3) which has no feasible solution (See Exercise 9.3). Problems of this kind are referred to as unfeasible. At the opposite, the problem has feasible solutions. But none of them is optimal (See Exercise 9.3). As a matter of fact, for every number M , there exists a feasible solution x1 , x2 such that x1 − x2 > M . The problems verifying this property are referred to as unbounded. Every linear programme satisﬁes exactly one the following assertions: either it admits an optimal solution, or it is unfeasible, or it is unbounded. Geometric interpretation. The set of points in IRn at which any single constraint holds with equality is a hyperplane in n IR . Thus each constraint is satisﬁed by the points of a closed half-space of IRn , and the set of feasible solutions is the intersection of all these half-spaces, a convex polyhedron P. Because the objective function is linear, its level sets are hyperplanes. Thus, if the maximum value of cx over P is z∗ , the hyperplane cx = z∗ is a supporting hyperplane of P. Hence cx = z∗ contains an extreme point (a corner) of P. It follows that the objective function attains its maximum at one of the extreme points of P. 9.2. THE SIMPLEX METHOD 9.2 131 The Simplex Method ´ The authors advise you, in a humanist elan, to skip this section if you are not ready to suffer. In this section, we present the principle of the Simplex Method. We consider here only the most general case and voluntarily omit here the degenerate cases to focus only on the basic principle. A more complete presentation can be found for example in [2]. 9.2.1 A ﬁrst example We illustrate the Simplex Method on the following example: Maximize 5x1 Subject to: 2x1 4x1 3x1 + 4x2 + 3x3 + 3x2 + x3 + x2 + 2x3 + 4x2 + 2x3 x1 , x2 , x3 ≤5 ≤ 11 ≤8 ≥ 0. (9.4) The ﬁrst step of the Simplex Method is to introduce new variables called slack variables. To justify this approach, let us look at the ﬁrst constraint, 2x1 + 3x2 + x3 ≤ 5. (9.5) For all feasible solution x1 , x2 , x3 , the value of the left member of (9.5) is at most the value of the right member. But, there often is a gap between these two values. We note this gap x4 . In other words, we deﬁne x4 = 5 − 2x1 − 3x2 − x3 . W...
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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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