One period market with n securities with return R~ $N(\mu,\Sigma)$. An expected return on each security is $\mu_1,\dots,\mu_n$
Investor has 1 unit of wealth and invests into n assets, with weights $w_1,\dots,w_n$ so that the sum of the $w_i$=1.
To solve the mean-variance problem, given the expected return on this portfolio is $p$, and allow the short-sale.
The minimum variance portfolio problem can be solved by Lagrange Multiplier: $$min\quad w^T\Sigma w$$ subject to:$$w_1+\dots+w_n=1$$ $$\mu_1w_1+\dots+\mu_nw_n=p$$ The Lagrangian is: $$\mathcal{L}=w^T\Sigma w-\lambda_1(a^Tw-p)-\lambda_2(w^Te-1)$$ Where $e$ is the all-ones matrix and $a$ is $[\mu_1,\mu_2,\dots,\mu_n]$. After first order conditions,$$w=\lambda_1\Sigma^{-1}\mu+\lambda_2\Sigma^{-1}e$$ So that $$1=\lambda_1e^T\Sigma^{-1}\mu+\lambda_2e^T\Sigma^{-1}e$$ $$p=\lambda_1\mu^T\Sigma^{-1}\mu+\lambda_2\mu^T\Sigma^{-1}e$$
and in a matrix form: $$\begin{bmatrix} \lambda_1 \\ \lambda_2 \end{bmatrix}= \begin{bmatrix} e^T\Sigma^{-1}\mu & e^T\Sigma^{-1}e \\\mu^T\Sigma^{-1}\mu&\mu^T\Sigma^{-1}e \end{bmatrix}^{-1} \begin{bmatrix} 1\\p\end{bmatrix}$$
and the solution for $w$ is: $$w=\begin{bmatrix}\Sigma^{-1}\mu&\Sigma^{-1}e\end{bmatrix}\begin{bmatrix} e^T\Sigma^{-1}\mu & e^T\Sigma^{-1}e \\\mu^T\Sigma^{-1}\mu&\mu^T\Sigma^{-1}e \end{bmatrix}^{-1} \begin{bmatrix} 1\\p\end{bmatrix}$$
Follow this logic, if I want to solve the maximum expected return mean-variance problem, I have an objective function: $$max\quad w^Ta$$ subject to: $$w^T\Sigma w=\sigma^2,\quad where \ \sigma^2\text{is the given variance of the portfolio}$$ $$w^Te=1$$
But when I applied equivalent Lagrange multiplier approach, I can't get the correct results which should be similar as the minimum variance context. Is this because the quadratic programming cannot be used when we have non-linear constraints? And I hope to get a similar result so I can put it into the MatLab to do an empirical study project.