Applied Dynamic Programming by Richard Bellman

By Richard Bellman

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43). 43) when the nonbasic variables x1 and x2 have the specified values (x01 , x02 ). 45) for i = 1, . . 45) 1 a2i1 + a¯2i2 )− 2 . If the point (x01 , x02 ) satisfies the inequality, then the where ki = (¯ geometric picture of the distance of a point from the boundary is shown in Figure 113. If the slack variables xi+2 are replaced by yi = ki xi+2 for i = 1, . . , m, and the coordinates of a point P are the values of the independent variables, then the value of the ith basic variable is just the distance from the point P to the corresponding ith constraint.

3 ........ ........ . . . . . . . .. 1 2 ........ ................................................................. A ◦ G • Starting solution plane. u • G∗ ◦ A ◦A New solution plane ◦ (b , b , 0) u1 Figure 1-18: Simplex Associated with an Iteration of the Simplex Algorithm (m = 3) difference c¯j = cj − (π0 + π1 a1j + π2 a2j ) < 0 is the vertical distance that Aj is below the plane. In this case, a three-dimensional simplex with vertices Aj , A1 , A2 , and A3 can be formed and a point G∗ found where the requirement line pierces the simplex at its lowest point.

The redundancy can be seen in the diagram in that we could remove one of the constraints without changing the feasible region. Give an example of degeneracy with a nonredundant system of inequalities. Draw a picture to demonstrate this. 4 Dantzig [1963]. (a) Show that the set of possible values of any variable xk of a linear program forms a convex set, in this case, a straight line segment a ≤ xk ≤ b. (b) Show that the set of possible values of two variables, say (x1 , x2 ) or (x1 , z) satisfying a linear program, forms a convex set in two dimensions.

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