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I have a cost function $f(x,y)=ax^2+bx+cy^2+dy+e$ with a constraint $x+y\leq m$ that must be optimized. I can solve the problem using Lagrange multipliers method when constraint written as $x+y=m$ but how the solution can be extended to the new constraint?

By the way I write the solution for $x+y=m$ as follow:

$$[L]: \max_{x,y,\lambda} ax^2+bx+cy^2+dy+e+\lambda(m-x-y) $$ $$[x]: ax+b-\lambda^*=0$$ $$[y]: cy+d-\lambda^*=0$$ $$[\lambda]: m-x^*-y^*=0$$ Optimality condition is obtained by eliminating $\lambda$ from $[x]$ and $[y]$: $$ax+b=cy+d \to x=\frac{cy+d-b}{a} $$ Now we have two unknown and two equations which results in: $$x=\frac{cm+b-d}{c+a}$$ $$y=\frac{am+b-d}{c+a}$$

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    Hint: There are two cases: either the extreme value is found on the border line $x+y=m$ or in the interior of the half-plane $x+y$\frac d{dx} f(x^*,y^*) = \frac d{dy} f(x^*,y^*) = 0$. – 2017-01-25

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