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We describe an optimization-based tax-aware portfolio construction method that adds tax liability to standard Markowitz-based portfolio construction. Our method produces a trade list that specifies the number of shares to buy of each asset and the number of shares to sell from each tax lot held. To avoid wash sales (in which some realized capital losses are disallowed), we assume that we trade monthly, and cannot simultaneously buy and sell the same asset. The tax-aware portfolio construction problem is not convex, but it becomes convex when we specify, for each asset, whether we buy or sell it. It can be solved using standard mixed-integer convex optimization methods at the cost of very long solve times for some problem instances. We present a custom convex relaxation of the problem that borrows curvature from the risk model. This relaxation can provide a good approximation of the true tax liability, while greatly enhancing computational tractability. This method requires the solution of only two convex optimization problems: the first determines whether we buy or sell each asset, and the second generates the final trade list. In our numerical experiments, our method almost always solves the nonconvex problem to optimality, and when it does not, it produces a trade list very close to optimal. Backtests show that the performance of our method is indistinguishable from that obtained using a globally optimal solution, but with significantly reduced computational effort.