# Constrained optimization with inequalities

I want to solve a constrained maximization problem: $F =\log(\theta(1+x)(1-\phi \times y)- \omega\times y\times z - \delta)+\gamma \log (\Psi^x\times y\times (1+z)^\alpha)$

$\max F$ such that $0\leq x \leq 1$, $y>0$, $z \geq 0$.

I can characterize the solutions in terms of the parameters quite easily by solving several maximizations involving only equality constraints (similar to Kuhn-Tucker, but without incorporating the multipliers):

1 - The fully interior solution.

2 - $\max F$ such that $x=0$, $y>0$, $z > 0$.

3 - $\max F$ such that $x= 1$, $y>0$, $z > 0$.

4 - $\max F$ such that $0\leq x \leq 1$, $y>0$, $z = 0$.

5 - $\max F$ such that $x = 0$, $y>0$, $z = 0$.

6 - $\max F$ such that $x = 1$, $y>0$, $z = 0$.

Is this approach right and is there any theorem I can invoke to sustain it? I just want to characterize the solutions for each regime in terms of the parameters. This is, I am OK saying that if for a given set of parameters the solution consists of $0 \leq x \leq 1,\, y>0,\, z=0$ the first order conditions found when solving problem number 4 give the maximum.

Next, I would like to show that solutions are indeed maxima using the Hessian. My second question is: can I check for negative semi-definiteness of each of the Hessian matrices I obtain by plugging the constraints into the objective and, when necessary, assuming that, for instance, $0 \leq x \leq 1$? So, treating them as unconstrained problems. For example, for the second case, I would have:

$\log (\theta(1+0)(1 -\phi \times y)-\omega \times y \times x-\delta)+\gamma \log (\Psi^0 \times y \times z^\alpha)$

And its Hessian:

\begin{bmatrix} F_{y,y} & F_{y,z} \\ F_{z,y} & F_{z,z} \end{bmatrix}