The quadratic optimization (min variance) $$ w^{T} \Sigma w \rightarrow \text{min}, $$ where $w$ is the vector of portfolio weights and $\Sigma$ is the covariance matrix of asset returns, is a well studied problem. In practice we have to define certain constraints. These are easy linear, continuous ones (like $\sum_{i=1}^n w_i = 1$ or constraints on turn over) or difficult binary constraints (if an asset is bought than with at least $x\%$, a cardinality constraint, etc.).
Setting $\hat{w} = w_{Pf} - w_{BM}$ as the vector of active weights of the portfolio relative to a benchmark, the above formula describes minimizing the (squared) tracking error.
The difficult binary constraints lead to mixed integer programming (MIP). Commercial solvers solve this by methods such as branch and bound which are very time consuming for large problems (e.g. 1000+ assets).
I heard of an approach to formulate such problems as second order cone programs (SOCP) and that such problems are usually solved more efficiently.
I have plenty of experience with branche and bound and I would like to switch to SOCP. Do you know of any good refernce of SOCP and portfolio/TE optimization with hard real world constraints (especially with binary variables)? Do you have any experiences whether switching is worth the efforts?
PS: Let us assume that $\Sigma$ is well estimated and that variance makes sense. I know that this is debatable ...
EDIT: This paper Applications of second-order cone programming describes the formulation of a quadratically constrainted quadratic program as SOCP. Will also have a look here.
EDIT 2: To formulate every detail. I have the mixed-inter quadratic program (I formulate the TE case): $$ w^{T} \Sigma w + w^T c \rightarrow \text{min}, $$ with $0 \le w_i \le u$ for $i = 1,\ldots, N$ with $N \approx 1000$ and a constant vecor $c$. The constraints are of the form $$ w_i \ge b_i*l \text{ for } i = 1,\ldots,N \\ \sum b_i \le K \\ b_i \in \{0,1\} \text{ for } i = 1,\ldots,N $$ with a small real $l \in (0,1)$ and some large integer $K$. Moreover I have lots of continuous constraints $$ A w \le b. $$ If I am totally precise then there are additional continuous variables that appear in the constraints only (these model turn over).
If I follow the link above then this can be formulated as mixed SOCP with $$ t \rightarrow \text{min}. $$ and all the above constraints and one additional constraint: $$ \| \Sigma^{1/2} w + \Sigma^{-1/2}c \| \le t. $$
Is there a chance that the mixed-integer SOCP problem as formulated above can be solved more efficiently than the mixed-integer quadratic program (with linear and binary but no quadratic constraints)?