Hi everyone, I'm new here. I have a problem and want to know if my general approach is correct:
\(\displaystyle \max_{x_1,x_2,x_3 \ge 0} \min \left\{ f(x_1,x_2,x_3), g(x_1,x_2,x_3), h(x_1,x_2,x_3) \right\}\)
I actually know the functions \(\displaystyle f,g\) and \(\displaystyle h\) so I can use some extra informations (see below).
I want to know if my approach is right:
Since \(\displaystyle f\) is continue and monotonically increasing and \(\displaystyle g\) is continue and monotonically decreasing w.r.t \(\displaystyle x_1\) (for each fixed value of \(\displaystyle x_2,x_3\)) I find the "equivalence point" betwenn the two:
\(\displaystyle f(x_1,x_2,x_3) = g(x_1,x_2,x_3)\)
and then solve for \(\displaystyle x_1\) to get \(\displaystyle x_1 = l(x_2,x_3)\) (Here \(\displaystyle l\) stands for something depending on \(\displaystyle x_1,x_2\)).
Now if \(\displaystyle x_1 \ge l(x_2,x_3)\) the problem becomes: \(\displaystyle \displaystyle \max_{x_1,x_2,x_3 \ge 0} \min \left\{ f(x_1,x_2,x_3), h(x_1,x_2,x_3) \right\}\) else it becomes \(\displaystyle \displaystyle \max_{x_1,x_2,x_3 \ge 0} \min \left\{ g(x_1,x_2,x_3), h(x_1,x_2,x_3) \right\}\).
Each of the two new problems can be resolved with the same method using the fact that one function is monotonically increasing and the other is monotonically decreasing w.r.t. one of their variables.
Does this make sense?
\(\displaystyle \max_{x_1,x_2,x_3 \ge 0} \min \left\{ f(x_1,x_2,x_3), g(x_1,x_2,x_3), h(x_1,x_2,x_3) \right\}\)
I actually know the functions \(\displaystyle f,g\) and \(\displaystyle h\) so I can use some extra informations (see below).
I want to know if my approach is right:
Since \(\displaystyle f\) is continue and monotonically increasing and \(\displaystyle g\) is continue and monotonically decreasing w.r.t \(\displaystyle x_1\) (for each fixed value of \(\displaystyle x_2,x_3\)) I find the "equivalence point" betwenn the two:
\(\displaystyle f(x_1,x_2,x_3) = g(x_1,x_2,x_3)\)
and then solve for \(\displaystyle x_1\) to get \(\displaystyle x_1 = l(x_2,x_3)\) (Here \(\displaystyle l\) stands for something depending on \(\displaystyle x_1,x_2\)).
Now if \(\displaystyle x_1 \ge l(x_2,x_3)\) the problem becomes: \(\displaystyle \displaystyle \max_{x_1,x_2,x_3 \ge 0} \min \left\{ f(x_1,x_2,x_3), h(x_1,x_2,x_3) \right\}\) else it becomes \(\displaystyle \displaystyle \max_{x_1,x_2,x_3 \ge 0} \min \left\{ g(x_1,x_2,x_3), h(x_1,x_2,x_3) \right\}\).
Each of the two new problems can be resolved with the same method using the fact that one function is monotonically increasing and the other is monotonically decreasing w.r.t. one of their variables.
Does this make sense?