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.
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