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# lec9 - 15.053 March 8 2007 Duality 2 The Dual Problem 1...

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15.053 March 8, 2007 Duality 2. The Dual Problem 1

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Quote of the Day Every word or concept, clear as it may seem to be, has only a limited range of applicability. -- Werner Heisenberg Just as we have two eyes and two feet, duality is a part of life. -- Carlos Santana 2
3 Prices Review of Last Lecture A set of prices for a linear program is a collection of real numbers associated with each constraint, other than the nonnegativity constraints. 17.2 -.33 max z = -3x 1 + 2x 2 s.t. -3x 1 + 3x 2 6 red -4x 1 + 2x 2 2 gray x 1 0, x 2 0

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4 Pricing out to get reduced costs z For a maximization problem, treat the prices as though they really are prices on the RHS. Prices 2 1 max z = -3x 1 + 2x 2 s.t. -3x 1 + 3x 2 6 red -4x 1 + 2x 2 2 gray x 1 0, x 2 0 Reduced costs = original costs minus column coefficients time prices. x 1 x 2 - 3 - (1 × -3) - (2 × -4) 4 2 - (1 × 3) - (2 × 2) -3 i ij i j a p c
5 Pricing out a new variable If we introduce a new variable, we price it out just as we priced out all other variables. max z = -3x 1 + 2x 2 + 3y s.t. -3x 1 + 3x 2 + 4y + x 3 = 6 -4x 1 + 2x 2 + 3y + x 4 = 2 x 1 0, x 2 0, x 3 0, x 4 0, y 0 Shadow prices 1/2 1/3 y 3 - (1/3 × 4) - (1/2 × 3) 1/6 If we price out a new variable, and it is its reduced cost is positive, then the opt objective value will increase if the variable can be positive.

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Simplex Multipliers Start with an original problem Obtain a modified problem and modified tableau after several pivots. The simplex multipliers are those prices such that the reduced costs wrt to those prices are exactly the same as the reduced costs in the modified tableau. The basic variables have a reduced cost of 0. 6
7 3 1 0 1 2/3 1/2 1 0 1/3 -1/2 x 1 = = x 2 x 4 x 3 z 0 0 -1 = = = 2 6 x 1 x 2 = x 4 x 3 0 0 z -3 +2 0 0 0 -1 Simplex multipliers for the tableau below. 1/3 1/2 The tableau after two pivots. 0 0 -3 -1/2 -1/3 -3 3 -4 2 1 0 0 1

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Important Fact z For a problem with equality constraints, optimizing wrt the reduced costs is the same as optimizing wrt the original costs. 8
Review of Results z The shadow prices are the unit change in the optimal objective value per unit change in the RHS coefficients. z The simplex multipliers are chosen so that the reduced costs of the basic variables are 0. z The simplex multipliers for the optimal tableau are equal to the shadow prices. z The reduced costs of the nonbasic variables in the final tableau are the same as the reduced costs of the optimal solution. These are also the shadow prices of the nonnegativity constraints. 9

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Nooz, this looks like where we left off last time. Are we finally going to see what duality is all about? That’s right. We will see that dual prices are great for computing upper bounds on the objective function of a max problem. And they often have interesting interpretations.
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lec9 - 15.053 March 8 2007 Duality 2 The Dual Problem 1...

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