Optimization: constructive solution

Asked Jul 04 '16 at 15:58

Active Jul 04 '16 at 16:05

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Consider the following program \begin{align} \max_{x,y \geq 0} f(x,y)\tag{1} \end{align}

I wanna construct a solution with $y^* = 0$ and $x^* > 0$. Suppose FOCs satisfy \begin{align} f_x(x^*,y^*) = 0\\ f_y(x^*,y^*) < 0 \end{align}

and SOCs also satisfy \begin{align} f_{xx}(x^*,y^*) < 0 \quad \text{and} \quad \det(H(f)(x^*,y^*)) > 0 \end{align}

where $H(\cdot)$ is the hessian.

edited Jul 04 '16 at 16:05

asked Jul 04 '16 at 15:58

clueless

It seems to me we need to know a bit more about $f(.,.)$. Is it continuous? Strictly concave (not just at the point but elsewhere)? – Nameless Jul 04 '16 at 19:40
Oh yeah, it's continuous and quasi-concave. – clueless Jul 04 '16 at 19:48
I was able to verify the constructed analytcal solution by a numerical approximation. Is this somehow proof? – clueless Jul 05 '16 at 08:42

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