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1-Generalizing

# 1-Generalizing - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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MATHEMATICS 116, FALL 2007 CONVEXITY AND OPTIMIZATION WITH APPLICATIONS Outline #1 (Generalizing from two dimensions) Last modiﬁed: September 10, 2007 Reading. Luenberger, Chapter 1 Lecture topics. 1. Why such a long introduction? The general approach in Luenberger, which has made the book a classic, is this. Identify techniques from algebra, elementary single-variable calculus, or elementary multivariable calculus that can be used to solve opti- mization problems. Reformulate the solution geometrically. Using geometry for inspiration, generalize the solution, typically to inﬁnite-dimension vector spaces and noneuclidean norms, and prove (algebraically) that it is still valid. All the ﬁnite-dimensional problems in this outline should be familiar, though they may be valuable review for some students. The inﬁnite-dimensional problems are just stated, not solved, and we will take quite a while to get to them. Important concepts are in CAPITAL LETTERS. These will appear in Chapters 2, 3, and 5, generally in a context where there is no mention of optimization, just some challenging mathematics. My hope for these introductory lectures is to convince you that optimization problems are fun and relevant, that some of the best ones can only be formulated in inﬁnite-dimensional vector spaces, and that it is worth your while to learn quite a few new deﬁnitions and theorems in order to be able to solve them. 1

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You are entering a student competition to draw up a business plan for a company with m scientists and n other employees. Entries with n 2 > 2 m 2 get rejected. You want to have the highest possible ratio of scientists to other employees. Does this optimization problem have a solution? As director of the state lottery, you are designing scratch tickets by assigning probabilities to the possible payoﬀs from 0 through \$4. Since a ticket sells for \$5, you want to be as generous as possible. Does this optimization problem in R 5 have a solution? Change the problem so that any integer payoﬀ is allowed. Does this opti- mization problem in an inﬁnite-dimensional vector space have a solution? Can you remember the rule for when a real-valued function f is guaranteed to have a maximum in a subset X R 2 ? 3. An easy allocation problem Your small bakery can produce only two products. A batch of frosted cookies uses up 1 pound of ﬂour and 3 pounds of sugar. A batch of cake uses up 2 pounds of ﬂour and 1 pound of sugar. Each day your suppliers bring you 14 pounds of ﬂour and 17 pounds of sugar. Your optimization problem is to look at the market price of cookies and cakes and decide what to produce. Let
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1-Generalizing - MATHEMATICS 116 FALL 2007 CONVEXITY AND...

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