OPTIMIZATION Course

Constrained polygon

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It's most likely the easiest method, but it won't work if you got more than 2 parameters. Rewrite all of your constraints to have constraints like these

\[ \begin{split} x \le 10\\ y \le 15\\ y \le 3/5 * x + 7\\ \end{split} \]

You may use greater, greater equals, or lesser/lesser equals, but you can't have something like x + y < 5 since we want only one variable on the left side.


constrained space

On you have all of your inequalities, simply trace the lines corresponding to each one. Remember to add a small arrow to visualize the constrained space.

If you have a constraint y < 3 and x > 2 it would look like this

lines

You should remember that something like that y <= 3/5 * x + 7 means that

  • you got a point at x=0, y=7
  • if x increase by 5, then y increase by 3
  • so you can guess the next point (x=5, y=10)

Minimum / Maximum

The critical points are the edges of our constrained space like C in the screenshot.

You had a function f(x,y)=..., and since each point is got (x,y), then replace each critical point in the function. Take the highest (maximum) or the lowest (minimum) value, and you're done.


Translation

If you don't want to test all the points, trace your function then translate it, the last point touching it is the point you are looking for.