# 6417c04 - DUALITY THEORY CONTENTS Defining the Dual Problem...

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1 DUALITY THEORY CONTENTS Defining the Dual Problem (Sections 5.1 and 5.2) The Weak Duality Theorem (Section 5.3) The Strong Duality Theorem (Section 5.4) Simplex Algorithm and Dual Variables Complementary Slackness Conditions (Section 5.5) The Dual Simplex Algorithm (Section 5.6)

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2 Lower and Upper Bounds Maximize 4x 1 + x 2 + 3x 3 subject to x 1 + 4x 2 1 3x 1 - x 2 + x 3 3, x 1 , x 2 , x 3 0 A feasible solution : x 1 = 0, x 2 = 0, x 3 = 3, ξ = 9 Each feasible solution gives a lower bound on the optimal objective function value. We can also obtain a lower bound easily. 2 (x 1 + 4x 2 ) (1) 2 + 3 (3x 1 - x 2 + x 3 ) (3) 3 11x 1 + 5x 2 + 3x 3 11 Observe that 4x 1 + x 2 + 3x 3 11x 1 + 5x 2 + 3x 3 11. Hence ξ
3 Lower and Upper Bounds (contd.) Maximize 4x 1 + x 2 + 3x 3 subject to x 1 + 4x 2 1 3x 1 - x 2 + x 3 3, x 1 , x 2 , x 3 0 4x 1 + x 2 + 3x 3 11x 1 + 5x 2 + 3x 3 11. Hence ξ 11. How can we get the smallest possible upper bound ? Find y 1 and y 2 such that y 1 (x 1 + 4x 2 ) (1) y 1 + y 2 (3x 1 - x 2 + x 3 ) (3) y 2 (y 1 +3y 2 )x 1 + (4y 1 -y 2 )x 2 + (y 2 )x 3 y 1 + 3y 2 y 1 + 3y 2 4, 4y 1 - y 2 1, y 2 3, y 1 0, y 2 0 and y 1 +3y 2 is minimum

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4 Defining the Dual Dual LP : Primal LP : Minimize y 1 + 3y 2 Maximize 4x 1 + x 2 + 3x 3 subject to subject to y 1 + 3y 2 4 x 1 + 4x 2 1 4y 1 - y 2 1 3x 1 - x 2 + x 3 3 y 2 3 x 1 , x 2 , x 3 0 y 1 0, y 2 0
5 The Dual Problem PRIMAL LP : Maximize j=1,n c j x j subject to j=1,n a ij x j b i for all i = 1, 2, …, m x j 0 for all j = 1, 2, … , n THE ASSOCIATED DUAL LP : Minimize i=1,m b i y i subject to i=1,m y i a ij c j for all j = 1, 2, … , n y j 0 for all i = 1, 2, … , m

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6 The Dual Problem THE DUAL LP : - maximize i=1,m (-b i )y i subject to i=1,m (-a ij ) y i (-c j ) for all j = 1, 2, …, n y j 0 for all i = 1, 2, … , m DUAL OF THE DUAL LP : -minimize j=1,n (-c j )x j subject to j=1,n (-a ij ) x j (-b j ) for all i = 1, 2, …, m x j 0 for all j = 1, 2, … , n which is the same as the primal problem.
7 A Diet Problem My diet consists of four items: chocolate cake, ice cream, soda, and cheesecake Each day, I must ingest at least 500 calories, 6 oz. of chocolate, 10 oz. of sugar, and 8 oz. of fat. The nutritional contents of each type of per unit of food are given below. Find the minimum cost diet plan. Calories Chocolate (ounces) Sugar (ounces) Fat (ounces) Brownie (cost = .20) 400 3 2 2 Chocolate ice cream (cost = .20) 200 2 2 4 Cola (cost = .30) 150 0 4 1 Pineapple Cheesecake (cost = .80) 500 0 4 5

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8 A Pill Manufacturer’s Problem A pill manufacturer is planning to manufacture pills for calories, chocolates, sugar, and fat.
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## This note was uploaded on 08/27/2011 for the course ESI 6417 taught by Professor Siriphonglawphongpanich during the Spring '07 term at University of Florida.

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6417c04 - DUALITY THEORY CONTENTS Defining the Dual Problem...

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