Difference between constraining pre and post optimization

Question

What's the implication of constraining the optimized portfolio weights obtained using no constraints vs obtaining the weights with the constraints in the objective?

Let the asset returns be distributed with mean $\mu$ and covariance $C$.

$$r\sim(\mu,C)$$

Unconstrained portfolio optimization:

$$\min_w\frac{\gamma}{2}w^TCw-w^T\mu$$

Optimal weights-

$$w^*=\frac{1}{\gamma}C^{-1}\mu$$

Constraining post optimization-

Define $e$ as a vector of 1s.

$$e^Tw^*=1\implies\gamma=e^TC^{-1}\mu$$

$$w^*=\frac{C^{-1}\mu}{e^TC^{-1}\mu}$$

Constrained portfolio optimization:

$$\min_w\frac{\gamma}{2}w^TCw-w^T\mu-\lambda(e^Tw-1)$$

Optimal weights-

$$w^*=\frac{C^{-1}(\mu+\lambda e)}{e^TC^{-1}(\mu+\lambda e)}$$

Answer 1

Consider the equation of two variables (basically your obj func):

$$f(x,y) = x^2 + y^2$$

The unconstrained minimisation is $x=y=0$. If you now constrain this sum to be equal to one, post optimisation, well it doesn't quite work since you multiply by infinity. But, even if the obj func was slightly different and it was finite it wouldn't return the minimum of the actual constrained minimisation which in this case is $x=y=0.5$, because they are two different calculations.

Analagously its like saying: from a class of twenty students find the 3 who are the tallest, and then (post) select those who are girls, versus (pre) constrained to the girls in the class find 3 who are the tallest.