Dear All,

Define

f1(x) = x * (1 - x - y)

and

f2(y) = y * (1 - x - y).

Consider now the following coupled optimization problem:

max f1(x) and max f2(y) (simultaneously).

I know that the analytical solution is

x = 1/3 and y = 1/3,

but how can one solve this numerically? My question is motivated by
the fact that I have to solve similar problems, but large-scale ones.

Thanks in advance,

Paul

Define

f1(x) = x * (1 - x - y)

and

f2(y) = y * (1 - x - y).

Consider now the following coupled optimization problem:

max f1(x) and max f2(y) (simultaneously).

I know that the analytical solution is

x = 1/3 and y = 1/3,

but how can one solve this numerically? My question is motivated by
the fact that I have to solve similar problems, but large-scale ones.

max t
s.t. t <= f1(x)
t <= f2(x)

Dear All,

Define

f1(x) = x * (1 - x - y)

and

f2(y) = y * (1 - x - y).

Consider now the following coupled optimization problem:

max f1(x) and max f2(y) (simultaneously).

I know that the analytical solution is

x = 1/3 and y = 1/3,

but how can one solve this numerically? My question is motivated by
the fact that I have to solve similar problems, but large-scale ones.

f1(x) = x * (1 - x - y)

and

f2(y) = y * (1 - x - y).

Consider now the following coupled optimization problem:

max f1(x) and max f2(y) (simultaneously).

Both f1 and f2 depend on both x and y. Do you mean that you only want
to maximize f1 wrt x and f2 wrt y, i.e. find (x,y) such that
x (1 - x - y) >= x' (1 - x' - y) for all x' and
y (1 - x - y) >= y' (1 - x - y') for all y'?

Or do you want to somehow maximize both objectives with respect to
both variables?

I know that the analytical solution is

x = 1/3 and y = 1/3,

I guess this supports the first interpretation...

but how can one solve this numerically? My question is motivated by
the fact that I have to solve similar problems, but large-scale ones.

You could use numerical methods to look for solutions to the equations
df1/dx = 0, df2/dy = 0.