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}


1 Answer 1

  1. Yes, it is ok to split the necessary KKT as you do. It is identical to if you write the complete KKT (with all multipliers) and then apply the (stanadard) switching assumptions on the corresponding constraints being in turn active.
  2. To check the Hessian is not enough. To begin with, for constrained optimization problems positive/negative-definiteness of Hessian is replaced with more involved condition, which means basically that it should be only partially definite - only along all "almost" feasible directions (pointing inside the set or tangent). Secondly, even this more involved condition is not going to ensure the global optimum, only a local one, which is not really very interesting. To make the necessary condition in 1 work, you have to prove that the optimal point does exist. It is commonly done via convexity (if the cost function is convex/concave) or by using the Weierstrass theorem. The latter is tricky if the set is not bounded (as in your case). Then you need to verify that the cost function is getting larger for minimization problem (in your case, smaller) when you approach infinity.

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