I want give some extra homework about Lagrange multipliers and constrained optimization. I am interested in examples that are complex (for example, those having more than two variables and more than one restriction). Where I can find sufficient examples for students' practice?
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$\begingroup$ I need any webpage address that contains these examples. $\endgroup$– HuseyinMar 18, 2014 at 7:00
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$\begingroup$ Just curious, is this for a calculus class? I'm tutoring multivariable calculus and this could be of help. $\endgroup$– Mark FantiniMar 18, 2014 at 10:54
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$\begingroup$ Yes @Fantini . I need more examples for Economics students in Calculus class (Math II). $\endgroup$– HuseyinMar 18, 2014 at 15:16
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$\begingroup$ Ask the Economics faculty for relevant, simple to explain, examples? You might substitute "realistic" functions by simpler ones with roughly the right behaviour. $\endgroup$– vonbrandMar 19, 2014 at 1:56
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$\begingroup$ @vonbrand I need exapmles. $\endgroup$– HuseyinMar 19, 2014 at 6:45
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1 Answer
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Some simple examples you can try:
1) Maximize the product of $n$ positive numbers, given their sum is one.
2)Maximize entropy of $p_1,\dots, p_n$ (all $p_i>0$, summing to one) that is maximizing $-\sum p_i \log p_i$.
3) More conditions: Maximizing entropy, but with the extra condition that $ \sum a_i p_i = \mu$. You can also try introducing more conditions (variance?). The $a_i$s are known constants.
4) You can find a few more good problems here: https://www.math.ucdavis.edu/~thomases/W11_16C1_lec_2_4_11.pdf
5) A few more good problems here: http://homepages.math.uic.edu/~dcabrera/practice_exams/topics/lagrangemultipliers.html
answered Mar 19, 2014 at 19:22
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$\begingroup$ Second link was exactly which I had needed. $\endgroup$– HuseyinMar 19, 2014 at 19:57