Chapter 4, ENGR 62, Stanford
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Chapter 4 Duality Given any linear program, there is another related linear program called the dual . In this chapter, we will develop an unde rsta nding of the dual linea r program. This understan ding translates to important insig hts about many optimization probl ems and al gor it hms. We begin in the next section by exploring the main concepts of duality through the simple graphical example of building cars and trucks that was introduced in Sect ion 3.1.1. Then, we will develop the theory of duality in greater generality and explore more sophisticated applications. 4.1 A Graphical Example Recall the linear program from Section 3.1.1, which determines the optimal numbers of cars and trucks to build in light of capacity constrai nts. There are two decision variables: the number of cars x 1 in thousands and the number of trucks x 2 in thousands. The linear program is given by maximize 3x 1 + 2 .5x 2 (profit in thousands of dollar s) sub ject to 4.44x 1 ≤ 100 (car assembly capacity) 6.67x 2 ≤ 100 (truck assembly capacity) 4x 1 + 2 .86x 2 ≤ 100 (metal stamping capa ci ty) 3x 1 + 6 x 2 ≤ 100 (e ngine as se mbl y capaci ty) x ≥ 0 (nonnegative production) . The optimal solution is given approximately by x 1 = 20.4 and x 2 = 6.5, generating a profit of about $77 .3 million. The constraints, feasible region, and optimal solution are illustrated in Figure 4.1. 83
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