Help with defining constraints in Lagrangian

savecancel

We write $$\begin{align} x=75\;000+X\ ,\\ y=200\;000+Y\ ,\\ \end{align}$$ and the new function to maximize on $X,Y\ge 0$ is the following function $Q(X,Y)$:

sage: var( 'x,y,X,Y' );
sage: P(x,y) = 5/100*x*(1-x/150000-8/10^7*x*y) + 8/100*y*(1-y/200000-8/10^7*x*y)
sage: Q(X,Y) = P(X+75000, Y+200000).expand()
sage: Q(X,Y)
-1/25000000*X^2*Y - 1/15625000*X*Y^2 - 24001/3000000*X^2 - 79/2500*X*Y - 12001/2500000*Y^2 - 3760*X - 53627/25*Y - 236998125

The maximal value for Q is taken at the "corner" $(X,Y)=(0,0)$, because all monomials appearing in total degree $\ge 1$ have negative coefficients, so the maximal value for both P and Q is - 236998125, taken at "the one or the other corner".

Note: The theory tells us to search for the absolute extremal values

• in the inner of the closed definition domain, by searching all local extremal values, obtained as a solution of $P'_x=P'_y=0$, and

• at the (finite and/or infinite) boundary of it, by setting $x=75\;000$, $x=\infty$, $y=200\;000$, $y=\infty$, and computing here, in each of the four cases the supremum.

In our case there are no inner extremal points:

sage: for sol in solve( [diff(P,x) == 0, diff(P,y) == 0], [x,y] ):
....:     print sol
....:     
[x == 816.8214285714286, y == 506.0324074074074]
[x == -816.2645502645503, y == -514.5402298850574]
[x == (13.33265402759081 + 1414.015105182754*I), y == (3.906674551362281 - 883.7900965450355*I)]
[x == (13.33265402759081 - 1414.015105182754*I), y == (3.906674551362281 + 883.7900965450355*I)]

and no solution is in the inner domain.

Also there are no maximal values at the infinite boundaries, since $P\to -\infty$ for either $x$ or $y$ going to $+\infty$.

For the remained boundaries, we can set $x=75\;000$, respectivel $y=200\;000$, getting two new maximization problems in the simpler lower dimension...