  Mean-variance portfolio optimization problems often involve separable
nonconvex terms, including penalties on capital gains, integer share
constraints, and minimum position and trade sizes. We propose a heuristic
algorithm for this problem based on the alternating direction method of
multipliers (ADMM). This method allows for solve times in tens to hundreds of
milliseconds with around 1000 securities and 100 risk factors. We also obtain a
bound on the achievable performance. Our heuristic and bound are both derived
from similar results for other optimization problems with a separable objective
and affine equality constraints. We discuss a concrete implementation in the
case where the separable terms in the objective are piecewise-quadratic, and we
demonstrate their effectiveness empirically in realistic tax-aware portfolio
construction problems.
