My question is related to the fundamental problem of portfolio selection, but I am posting it here since in essence the question is really of mathematical optimization. If, however, I am not correct I will move it, for example to quantitative finance.
Consider the following scenario: you have random variables $r_1, \ldots, r_n$, representing rates of return of some financial assets. Let their means be $$ E_s = (E(r_1) , \ldots, E(r_n) )^t $$ and the covariance matrix between the variables be $C$. A portfolio is any linear combination $\omega_1 r_1 + \ldots \omega_n r_n = \omega^tr$ of the assets. Now consider the main optimization problem for construction a portfolio (denoted by $(r)$ ) with a rate of return $r \in \mathbb{R}$ and a minimal variance, i.e., $$ (r)\qquad \begin{aligned} & \underset{\omega\, \in\, \mathbb{R}^n}{\text{minimize}} & & \omega^tC\omega \\ & \text{subject to} & & \omega^t 1_n = 1, \\ & & & \omega^t E_S = r. \; \end{aligned} $$ where $1_n = (1, \ldots, 1)^t$ is a $n$-tuple of ones. Note that there are no restrictions for positivity of the weights (the financial interpretation of the possible negative signs is "short selling"). The problem stated is actually about parametric optimization. In the course Portfolio Selection and Risk Management in coursera (I am linking to the particular lecture) the lecturer explains that to find the solutions for all $r \in \mathbb{R}$ it is sufficient to solve only two problems:
$$ (1) \qquad \begin{array}{cc} \begin{aligned} & \underset{\omega\, \in\, \mathbb{R}^n}{\text{minimize}} & & \omega^tC\omega \\ & \text{subject to} & & \omega^t 1_n = 1.\; \end{aligned} & \qquad \qquad (2) \qquad \begin{aligned} & \underset{\omega\, \in\, \mathbb{R}^n}{\text{maximize}} & & \dfrac{\omega^t E_s - r_f}{\sqrt{\omega^t C\omega}} \\ & \text{subject to} & & \omega^t 1_n = 1. \; \end{aligned} \end{array} $$ and then if their optimal solutions are $\omega_1^*$ and $\omega_2^*$ then all of the solutions of $(r)$ can be produced by linear combinations $$ (*)\qquad \omega_{\lambda} = \lambda\omega_1^* + (1-\lambda)\omega_2^*, \quad \lambda \in \mathbb{R}. $$ /* Here $r_f$ is some positive constant, representing the risk free rate of return */
Graphically, in the standard deviation - expected return space we have the following picture: 
Indeed, problem $(1)$ is obtained by a relaxation of the constraint $\omega^t E_s = r$ in $(r)$. About problem (2), the target function is the so-called Sharpe ratio, the minimizing of which geometrically is all about minimizing the angle between the red line and the $O\sigma$ axis. The two solutions are depicted with a black and red points.
Finally, here are my questions:
- Why can the set of optimal solutions of the parametric problem $(r)$ be obtained by linear combinations of the optimal solutions of $(1)$ and $(2)$?
*It is clear to me that the constraints are preserved under $(*)$ but why $\omega_\lambda$ is optimal for the problem $(\lambda E(r_1^*) + (1-\lambda)E(r_2^*))$, i.e.,
$$ \begin{aligned} & \underset{\omega\, \in\, \mathbb{R}^n}{\text{minimize}} & & \omega^tC\omega \\ & \text{subject to} & & \omega^t 1_n = 1, \\ & & & \omega^t E_S = \lambda E(r_1^*) + (1-\lambda)E(r_2^*)? \; \end{aligned} $$
- How is really problem $(2)$ a special case of the problem $(r)$?
Here I would like some rigour.
Credit for the graph: https://people.ucsc.edu/~ealdrich/Teaching/Econ133/LectureNotes/multiAssetOpt.html.
