introduction-lp-duality1

Combining all inequalities we obtain the following

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Unformatted text preview: try to upper bound the optimal value by a linear combination of the constraints. Precisely, for all i, let us multiply the ith constraint by yi ≥ 0 and then sum the resulting constraints. In the previous two examples, we had (y1 , y2 , y3 ) = (0, 5 , 0) and 3 (y1 , y2 , y3 ) = (0, 1, 1). More generally, we obtain the following inequality: y1 (x1 − x2 − x3 + 3x4 ) + y2 (5x1 + x2 + 3x3 + 8x4 ) + y3 (−x1 + 2x2 + 3x3 − 5x4 ) = (y1 − 5y2 − y3 )x1 + (−y1 + y2 + 2y3 )x2 + (−y1 + 3y2 + 3y3 )x3 + (3y1 + 8y2 − 5y3 )x4 ≤ y1 + 55y2 + 3y3 For this inequality to provide an upper bound of 4x1 + x2 + 5x3 + 3x4 , we need to ensure that, for all x1 , x2 , x3 , x4 ≥ 0, 4x1 + x2 + 5x3 + 3x4 ≤ (y1 − 5y2 − y3 )x1 + (−y1 + y2 + 2y3 )x2 + (−y1 + 3y2 + 3y3 )x3 + (3y1 + 8y2 − 5y3 )x4 . That is, y1 − 5y2 − y3 ≥ 4, −y1 + y2 + 2y3 ≥ 1, −y1 + 3y2 + 3y3 ≥ 5, and 3y1 + 8y2 − 5y3 ≥ 3. Combining all inequalities, we obtain the following minimization linear programme: Minimize y1 + 55y2 + 3y3 Subject to: y1 − 5y2 − y3 −y1 + y2 + 2y3 −y1 + 3y2 + 3y3 3y1 + 8y2 − 5y3 y1 , y2 , y3 ≥ ≥ ≥ ≥ ≥ 4 1 5 3 0 This problem is called the dual of the initial maximization problem. 140 9.3.2 CHAPTER 9. LINEAR PROGRAMMING Dual problem We generalize the example given in Subsection 9.3.1. Consider the following general maximization linear programme: Problem 9.5. Maximize ∑n=1 c j x j j Subject to: ∑n=1 ai j x j ≤ bi j xj ≥ 0 for all 1 ≤ i ≤ m for all 1 ≤ j ≤ n Problem 9.5 is called the primal. The matricial formulation of this problem is Maximize cT x Subject to: Ax ≤ b x≥0 where xT = [x1 , . . . , xn ] and cT = [c1 , . . . , cn ] are vectors in Rn , and bT = [b1 , . . . , bm ] ∈ Rm , and A = [ai j ] is a matrix in Rm×n . To find an upper bound on cT x, we aim at finding a vector yT = [y1 , . . . , ym ] ≥ 0 such that, for all feasible solutions x ≥ 0 of the initial problem, cT x ≤ yT Ax ≤ yT b = bT y, that is: Minimize bT y Subject to: AT y ≥ c y≥0 In other words, the dual of Problem 9.5 is defined by: Problem 9.6. Minimize ∑m 1 bi yi i= Subject to: ∑m 1 ai j yi ≥ c j i= yi ≥ 0 for all 1 ≤ j ≤ n for all 1 ≤ i ≤ m Notice that the dual of a maximization problem is a minimization problem. Moreover, there is a one-to-one correspondence between the m constraints of the primal ∑ j=1...n ai j x j ≤ bi and the m variables yi of the dual. Similarly, the n constraints ∑m 1 ai j yi ≥ c j of the dual correspond i= one-to-one to the n variables xi of the primal. Problem 9.6, which is the dual of Problem 9.5, can be equivalently formulated under the standard form as follows. Maximize ∑m 1 (−bi )yi i= Subject to: ∑m (−ai j )yi ≤ −c j i= yi ≥ 0 for all 1 ≤ j ≤ n for all 1 ≤ i ≤ m (9.22) Minimize ∑n=1 (−c j )x j j Subject to: ∑n=1 (−ai j )x j ≥ −bi j xj ≥ 0 for all 1 ≤ i ≤ m for all 1 ≤ j ≤ n (9.23) Then, the dual of Problem 9.22 has the following formulation which is equivalent to Problem 9.5. 9.3. DUALITY OF LINEAR PROGRAMMING 141 We deduce the following lemma. Lemma 9.7. If D is t...
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