Context: Given a vector $Y$ with $N$ elements indexed $i=1,2,...,N$, provide a vector $X^*$ that has a sample correlation of exactly $\rho$ to $Y$, i.e. $corr(Y,X^*)=\rho$.
Procedure: There are probably many and perhaps easier ways to do this, but the following occurred to me because I saw it applied to a more difficult problem in a paper years ago. The paper, however, skips over the calculations and it still keeps me up at night that I was never able to figure out that Lagrangian.
Start with a vector $X$ where every element $x_i=1$, and an initial vector of weights $\omega$ where again every element $\omega_i=1$. Then find $\omega^*$, the vector of weights that (1) is Kullback-Leibler closest to $\omega$ and (2) satisfies $corr(Y,\omega^* X)=\rho$. In other words, minimize the Kullback-Leibler distance between $\omega^*$ and $\omega$, subject to $corr(Y,\omega^* X)=\rho$:
$\displaystyle \min_{\omega^*} \quad \sum_{i=1}^{N}\omega^*_iln\left(\frac{\omega^*_i}{\omega_i} \right)$
$s.t. \quad \frac{cov(Y,\omega^* X)}{sd(Y)sd(\omega^* X)}=\frac{\frac{1}{N-1} \sum_{i=1}^{N}(y_i-\frac{1}{N}\sum_{i=1}^{N}y_i)(\omega^*_ix_i-\frac{1}{N}\sum_{i=1}^{N}\omega^*_ix_i)}{\sqrt{\frac{1}{N-1} \sum_{i=1}^{N}(y_i-\frac{1}{N}\sum_{i=1}^{N}y_i)^2}\sqrt{\frac{1}{N-1} \sum_{i=1}^{N}(\omega^*_ix_i-\frac{1}{N}\sum_{i=1}^{N}\omega^*_ix_i)^2}} = \rho$
The Lagrangian is then:
$L(\omega^*,\lambda)= \sum_{i=1}^{N}\omega^*_iln\left(\frac{\omega^*_i}{\omega_i} \right) - \lambda(cov(Y,\omega^* X)-\rho \times sd(Y)sd(\omega^* X))$
And it is not difficult to take derivatives to come up with first-order conditions:
$\frac{\partial L}{\partial \omega^*}= \sum_{i=1}^{N} \left[ ln\left(\frac{\omega^*_i}{\omega_i} \right)+\frac{1}{\omega_i} \right] - \lambda \frac{\partial}{\partial \omega^*}cov(Y,\omega^* X) + \lambda \rho\times sd(Y)\frac{\partial}{\partial \omega^*}sd(\omega^*X)$
$\quad \quad = \sum_{i=1}^{N} \left[ ln\left(\frac{\omega^*_i}{\omega_i} \right)+\frac{1}{\omega_i} \right] -\lambda cov(Y,X)+\lambda\rho \times sd(Y) \\ \quad \quad \quad \quad\times \frac{2}{\sqrt{\omega^*_i}(N-1)}\sum_{i=1}^{N} \left[ \omega^*_ix_i^2 - \frac{x_i}{N}\sum_{i=1}^{N} \omega^*_ix_i - \frac{\omega^*_ix_i}{N} + \frac{2}{N^2}(\sum_{i=1}^{N}x_i) (\sum_{i=1}^{N}\omega^*_ix_i) \right] \stackrel{!}{=} 0$
$\frac{\partial L}{\partial \lambda}=cov(Y,\omega^* X)-\rho \times sd(Y)sd(\omega^* X) \stackrel{!}{=} 0$
So now I somehow need to solve the $\frac{\partial L}{\partial \omega^*}\stackrel{!}{=} 0$ for $\omega^*_i(\lambda)$, then I can substitute that expression for $\omega^*_i$ in $\frac{\partial L}{\partial \lambda}$ and, finally, give $\frac{\partial L}{\partial \lambda(\omega^*)}\stackrel{!}{=} 0$ to Newton-Raphson or whatever optimizer to compute the optimal $\lambda$.
My question is how in the world could I ever hope to solve $\frac{\partial L}{\partial \omega^*}\stackrel{!}{=} 0$ for $\omega^*_i(\lambda)$, given that the $\omega^*_i$ appears inside sums in there? Any hints and links to similar (solved) problems will be greatly appreciated. Thanks.