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Unformatted text preview: Fall 2007 ARE211 Problem Set #11 Answer key First NPP Problem Set (1) Consider the following maximization problem (solve it graphically): max x 1 ,x 2 f ( x 1 ,x 2 ) with f ( x 1 ,x 2 ) = x 1 subject to g 1 : x 3 1 + x 2 ≤ 0 and g 2 : x 3 1 x 2 ≤ 0. a) What is the solution to the maximization problem? Ans: The feasible set is displayed in the left graph of figure 1. Since it is the task to maximize x 1 , one has to pick the point with the lowest x 1 value in the feasible set, which is the point (0,0). b) Is the Mantra satisfied for the solution to part a). If yes, write the gradient of the objective function as a postive linear combination of the gradients of the contstraints that are satisfied with equality. If not, explain why? Ans: The gradient of the objective function and the gradient of the constraints that are satisfied with equality are displayed in the right side of figure 1. Note that the gradient of the objective function does not lie in the nonnegative cone defined by the gradients of the constraints that are satified with equality. The reason is that the constraint qualification is not satisfied, i.e., the two gradients of the constraints satified with equality are colinear. c) Now, slightly change the problem and let the second constraint be g 2 : x 3 1 epsilon1x 1 x 2 ≤ 0 for epsilon1 > 0 Again, what is the solution to your problem? Ans: The feasible set is displayed in the upper graph of figure 1 for epsilon1 = 0 . 1 . Since it is the task to maximize x 1 , one has to pick the point with the lowest x 1 value in the feasible set, which is again the point (0,0). d) For the revised problem in part c), is the Mantra satisfied. If yes, write the gradi ent of the objective function as a postive linear combination of the gradients of the contstraints that are satisfied with equality. If not, explain why? 2 1 0 .8 0 .6 0 .4 0 .2 0 .2 0 .4 0 .6 0 .8 1 1 0 .8 0 .6 0 .4 0 .2 0 .2 0 .4 0 .6 0 .8 1 x 1  a x is x2axis c o n s tra in t g 2 c o n s tra in t g 1 fe a s ib le s e t F e a s ib le s e t 2 1 .5 1 0 .5 0 .5 1 1 .5 2 2 1 .5 1 0 .5 0 .5 1 1 .5 2 g ra d (g 1 ) g ra d (g 2 ) g ra d (f) x 1  a x is x2axis c o n s tra in t g 2 c o n s tra in t g 1 M a n tra a t p o in t (0 ,0 )21.510.5 0.5 1 1.5 221.510.5 0.5 1 1.5 2 grad(g1) grad(g2) grad(f) x1axis x2axis constraint g2 constraint g1 Mantra at point (0,0) Figure 1. Feasible set (left) and Mantra at point (0,0) (right) Ans: The gradient of the objective function and the gradient of the constraints that are0.50.40.30.20.1 0.1 0.2 0.3 0.4 0.50.50.40.30.20.1 0.1 0.2 0.3 0.4 0.5 x1axis x2axis constraint g2 constraint g1 feasible set Feasible set1.510.5 0.5 1 1.51.510.5 0.5 1 1.5 grad(g1) grad(g2) grad(f) x1axis x2axis constraint g2 constraint g1 Mantra at point (0,0) Figure 2. Feasible set (top) and Mantra at point (0,0) (bottom) satisfied with equality are displayed in the lower graph of figure 1. Now, the gradient of the objective function does lie in the nonnegative cone defined by the gradients of the...
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.
 Fall '07
 Simon

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