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Chapter 4 Duality - Stanford University

Chapter 4 DualityGiven any linear program, there is another related linear program called thedual. In this Chapter , we will develop an understanding of the dual linearprogram. This understanding translates to important insights about manyoptimization problems and algorithms. We begin in the next section byexploring the main concepts of Duality through the simple graphical exampleof building cars and trucks that was introduced in Section Then, wewill develop the theory of Duality in greater generality and explore moresophisticated A Graphical ExampleRecall the linear program from Section , which determines the optimalnumbers of cars and trucks to build in light of capacity constraints. There aretwo decision variables: the number of carsx1in thousands and the numberof trucksx2in thousands. The linear program is given bymaximize3x1+ (profit in thousands of dollars)subject to 100(car assembly capacity) 100(truck assembly capacity)4x1+ 100 (metal stamping capacity)3x1+ 6x2 100(engine assembly capacity)x 0(nonnegative production).

4.44y 1 +4y 3 +3y 4 ≥ 3 and 6.67y 2 +2.86y 3 +6y 4 ≥ 2.5. The first constraint ensures that selling a bundle of capacity that could be used to produce a car is at least as profitable as producing the car. The second constraint is the analog associated with production of trucks. Given our requirements, Company X solves a linear program to ...

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