## terça-feira, 18 de julho de 2017

### Mathematical optimization

https://en.wikipedia.org/wiki/Mathematical_optimization

http://labs.strava.com/slide/

http://labs.strava.com/projects/

### Minimum and maximum value of a function

Consider the following notation:
${\displaystyle \min _{x\in \mathbb {R} }\;(x^{2}+1)}$
This denotes the minimum value of the objective function ${\displaystyle x^{2}+1}$, when choosing x from the set of real numbers ${\displaystyle \mathbb {R} }$. The minimum value in this case is ${\displaystyle 1}$, occurring at ${\displaystyle x=0}$.
Similarly, the notation
${\displaystyle \max _{x\in \mathbb {R} }\;2x}$
asks for the maximum value of the objective function 2x, where x may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "infinity" or "undefined".

### Optimal input arguments

Consider the following notation:
${\displaystyle {\underset {x\in (-\infty ,-1]}{\operatorname {arg\,min} }}\;x^{2}+1,}$
or equivalently
${\displaystyle {\underset {x}{\operatorname {arg\,min} }}\;x^{2}+1,\;{\text{subject to:}}\;x\in (-\infty ,-1].}$
This represents the value (or values) of the argument x in the interval ${\displaystyle (-\infty ,-1]}$ that minimizes (or minimize) the objective function x2 + 1 (the actual minimum value of that function is not what the problem asks for). In this case, the answer is x = –1, since x = 0 is infeasible, i.e. does not belong to the feasible set.
Similarly,
${\displaystyle {\underset {x\in [-5,5],\;y\in \mathbb {R} }{\operatorname {arg\,max} }}\;x\cos(y),}$
or equivalently
${\displaystyle {\underset {x,\;y}{\operatorname {arg\,max} }}\;x\cos(y),\;{\text{subject to:}}\;x\in [-5,5],\;y\in \mathbb {R} ,}$
represents the ${\displaystyle (x,y)}$ pair (or pairs) that maximizes (or maximize) the value of the objective function ${\displaystyle x\cos(y)}$, with the added constraint that x lie in the interval ${\displaystyle [-5,5]}$ (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form (5, 2kπ) and (−5,(2k+1)π), where k ranges over all integers.
arg min and arg max are sometimes also written argmin and argmax, and stand for argument of the minimum and argument of the maximum.

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