Can somebody either explain how to show the equivalence of the three alternative optimization problems in MPT (or point me to some literature)?
I am looking for the necessary "algebraic steps" if at all possible rather than the intuitive explanation so that I can apply the same principle to other optimization problems.
The three equivalent optimization problems I am thinking of are:
$ \renewcommand{\vec}[1]{\boldsymbol{#1}} \newcommand{\mat}[1]{\boldsymbol{#1}} \newcommand{\code}[1]{\texttt{#1}} $
$ \begin{aligned} &\underset{\vec x}{\text{minimize}} && \vec x^T \mat\Sigma \vec x \\ &\text{subject to} && \vec1 \cdot \vec x = 1, \\ & && \vec\mu_{\vec x} \cdot \vec x \ge r_{min} \\\\ \end{aligned} $
$ \begin{aligned} &\underset{\vec x}{\text{maximize}} && \vec\mu_{\vec x} \cdot \vec x \\ &\text{subject to} && \vec1 \cdot \vec x = 1, \\ & && \vec x^T \mat\Sigma \vec x \le \sigma_{max}^2\\\\ \end{aligned} $
$ \begin{aligned} &\underset{\vec x}{\text{minimize}} && \vec x^T \mat\Sigma \vec x -\gamma \vec\mu_{\vec x} \cdot \vec x \\ &\text{subject to} && \vec1 \cdot \vec x = 1, \\ \end{aligned} $
