Let $x_j$ denote the ratio of assets to allocate to investment option $j$, where $j=1,...,n$
Let $c_j$ denote the annual expected rate of return on investment option $j$
How do we write the following constraints?
- The difference between the ratios of your assets allocated among any two different investment options should not be over a given threshold value δ, where δ ∈ (0, 1).
- Financial experts suggest a portfolio allocation vector given by $(r_1, . . . , r_n)$, where $r_j$ denotes the ratio allocated to investment option $j$, $j = 1, . . . , n,$ and $\sum_{j=1}^n r_j=1$. You wish to ensure that the total deviation between your allocation vector and the suggested allocation vector does not exceed a given threshold value β.
The first one is $\left|x_i-x_j\right| \leq \delta$ for all $i,j$. Every such inequality can be rewritten using the linear constraints $-\delta \leq x_i - x_j \leq\delta$.
In the second one, it is not entirely clear to me what "total deviation" is, but it sounds like $\sum_i \left|x_i-r_i\right| \leq \beta$, which can be rewritten using additional decision variables $w_i$ with $\sum_i w_i \leq \beta$ and $-w_i \leq x_i-r_i \leq w_i$.