0

Lets say we have $4$ variables: $$ x_1, x_2, x_3, x_4 $$ with coefficients: $a,b,c,d$ respectively, and output $y$

With different combinations of $a,b,c,d$, we have a blackbox/unknown function, that returns the numeric output $(y)$.

Constraints here are:

$a + b + c + d = 1 $ and $ a,b,c,d \geq 0$

How can we find the coefficients $(a,b,c,d) $ for maximum $ y?$ Or, to get started, what assumptions could we start with.

user26857
  • 53,190
skadoosh
  • 105

2 Answers2

1

If the "unknown function" is really unknown, you cannot find the best output. You need some further assumptions about $f$ before you can start to calculate anything.

5xum
  • 126,227
  • 6
  • 135
  • 211
  • Thank you. But, to get started, if we assume it to be linear function, how can we proceed? – skadoosh Jun 13 '16 at 06:15
  • @skadoosh If it's a linear function, then it's of the form $\alpha a + \beta b + \gamma c + \delta d$, and it's very very very easy to maximize this. You just have to evaluate the factors. – 5xum Jun 13 '16 at 06:17
  • Could you point me to the right resources if possible, to achieve this? As I am not sure how to evaluate the factors, with the constraints given. Thank you. – skadoosh Jun 13 '16 at 06:23
  • @skadoosh $\alpha=f(1,0,0,0)$... no resource needed. – 5xum Jun 13 '16 at 06:29
  • @skadoosh To expand further on what 5xum said, if you have $f(a,b,c,d)=x_0+ax_1+bx_2+cx_3+dx_4$ then the maximum must occur (see here) at one of the four points $(a,b,c,d)=(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$, and you can just evaluate these to see which is greatest. – Oscar Cunningham Jun 13 '16 at 10:32
0

If the function really is unknown (no linearity assumption), there are free user friendly softwares available to solve the problem, such as NOMAD.

Kuifje
  • 9,802