Suppose you currently own a portfolio of eight stocks. Using the Markowitz model, you computed the optimal mean/variance portfolio. The weights of these two portfolios are shown in the following table:
[![enter image description here][1]][1]

You would like to rebalance your portfolio in order to be closer to the M/V portfolio. To avoid excessively high transaction costs, you decide to rebalance only three stocks from your portfolio. Let $x_i$ denote the weight of stock $i$ in your rebalanced portfolio. The objective is to minimize the quantity:

$|x_1-0.02|+|x_2-0.05|+|x_3-0.25|+...+|x_8-0.12|$

which measures how closely the rebalanced portfolio matches the M/V portfolio.
Now I want to formulate this problem as a mixed integer linear program but I am only familiar with index tracking problems based on characteristic similarities and when transaction costs are neglected, hence this problem is pretty confusing for me.

**Progress:** I have figured how to write the objective function in a linear form.
$$min\sum_{i=1}^8 t_i$$
$$st \quad t_1\geq x_1-0.02$$
$$t_1\geq 0.02-x_1$$
$$...$$
$$t_8\geq x_8-0.12$$
$$t_8\geq 0.12-x_8$$
$$\sum_{i=1}^8x_i=1$$

What are the further constraints?
I would appreciate any helps!


  [1]: https://i.sstatic.net/49Re7.png