11-hw116

# 11-hw116 - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Assignment #11 Last modiﬁed: December 12, 2007 Due Tuesday, December 18 at class. If you are leaving town early, deliver it early to Nike. Work the last two problems before Thursday, since they are useful background for the applications that will be done in class. 1. Luenberger, Exercise 5.14.16. This is a crucial part of the Lagrange multi- plier rule in section 7.7. 2. State and prove all the theorems on pages 187-188 of Luenberger, simpliﬁed to the case n = 2. 3. Luenberger, problem 7.14.12. In lecture the problem was solved for arbi- trary boundary conditions x ( - 1) and x (1). In polar coordinates the solu- tion is simple when x ( - 1) = 0 and x (1) = 0, and you need do only that case. You may have to hunt up your old calculus text to get formulas for area and arc length in polar coordinates. It is very easy, once you have the diﬀerential equation, to show that r ( θ ) = 1 is a solution, and I think that is what Luenberger had in mind. However,

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11-hw116 - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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