Chapter 4 Duality Stanford University（第四章二元性斯坦福大学）.pdf

Chapter 4 Duality Given any linear program, there is another related linear program called the dual. In this chapter, we will develop an understanding of the dual linear program. This understanding translates to important insights about many optimization problems and algorithms. 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 Section 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 constraints. There are two decision variables: the number of cars x1 in thousands and the number of trucks x2 in thousands. The linear program is given by maximize 3x1 + 2.5x2 (profit in thousands of dollars) subject to 4.44x1 ≤ 100 (car assembly capacity) 6.67x2 ≤ 100 (truck assembly capacity) 4x1 + 2.86x2 ≤ 100 (metal stamping capacity) 3x1 + 6x2 ≤ 100 (engine assembly capacity) x ≥ 0 (nonnegative production). The optimal solution is given approximately by x1 = 20.4 and x2 = 6.5, generating a profit of about $77.3 million. The constraints, feasible region, and optimal solution are illustrated in Figure 4.1. 83 84 40 truck assembly engine assembly ) s d 30 optimal n car assembly