10

The graph below shows the distance from the origin (black dot) to the (blue) convex polygonal curve, which alternately has vertices (red) on the unit-radius semicircle (brown), and slightly inside that semicircle. The distance $d$ is plotted with respect to $\theta$, the angle w.r.t. the $x$-axis. You can see it is multimodal.         &...


10

How about this? Say the two curves are $A$ and $B$, with the min distance achieved at point $a \in A$ and $b \in B$. Let me assume both curves are smooth, or at least have 1st derivatives everywhere. Claim: the segment $ab$ is orthogonal to $A$ at $a$ and to $B$ at $b$. Suppose to the contrary that $ab$ is not orthogonal to $A$ at $a$. Then one can move $a$ ...


5

Four things come to mind. 1) You could focus on how the linearity of the objective function and the convexity of the feasible region yields that local optima are global, and how the linear nature of the constraints allow you to restrict your attention to corner points. This would accent the fundamental differences between linear and nonlinear optimization. ...


4

Memorization of multiplication tables is a skill which used to be valuable, but is much less so nowadays. I do not actually see a valid reason to force students to learn multiplication tables, other than it is a minor convenience to be able to multiply numbers in one's head. Bear in mind there are always a couple of iPhone's sitting around that can do much ...


3

I would be more careful with the phrasing of the question. Here are some points to explain why. The first point is an answer under certain conditions. When looking for the distance between a point and a curve (in two dimensions), do you prove that for a point $P$ not on the curve when $B$ is the closest point on the curve, then $\overleftrightarrow{PB}$ is ...


3

I have not tried anything similar myself, but if I had to, I would start by looking at Gilbert Strang's book "Introduction to applied mathematics" in chapter 8 "Optimization" (8.1 is "Introduction to linear programming"). My version is probably not the latest one and misses some recent developments (it seems to be written when interior point methods were ...


3

Maybe consider a sinusoid(ish) curve superimposed on a general trend. A place where you can see this is in business (e.g. volumes of production) where there is some general trend over time but there is SEASONALITY of the business. Some examples: ND oil production from 2009-2014 (low during winters because of weather issues but growing over time with ...


2

One of my classes actually has linear optimization on their corriculum (Switzerland, Berufsmatur). I teach according to the book "Mathematik in der Berufsschule". It's focused on students who want to go into economics. The way it's taught is very visual and seems well suited for your class if you have but 2 variables. Every linear condition is drawn on a ...


2

Just an idea: If your students know some statistics, including least squares regression, you could introduce optimization via some machine learnings methods, such as the lasso. The lasso can be solved as a quadratic programming problem, but simpler methods like coordinate descent work perfectly well. One book where this is detailed is "Statistical Learning ...


2

One thing that has been very helpful for me is reinforcing how valuable quick mental arithmetic can be. I recently taught about some great mental math shortcuts for squaring any reasonably sized number ending in 5, and using the difference of squares to quickly multiply any two numbers that are centered around an "easy" square (and now 5s are in that list as ...


2

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 ...


2

[T]o justify that it should occur when the slopes of the two curves are parallel: If $ab$ is a minimum, then the curve through $b$ has no points in the interior of the circle centered at $a$ with radius $ab$. (Any point inside is closer to $a$ than $b$.) Likewise, the curve through $a$ has no points in the interior of the circle centered at $b$ with ...


1

So, you want to find the minimum distance between two curves in the most general form using calculus and optimization. The curves you are interested at this particular example are: \begin{equation} xy=1 ~~and~~ y=-x \tag{01} \end{equation} The very first step is to write them in parametric form. i.e, \begin{equation} xy=1 \Rightarrow \begin{cases} x_1 &...


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