We develop and implement a portfolio optimization method for building investment portfolios that dominate a given benchmark index in terms of third-degree stochastic dominance. Our approach relies on the properties of the semi-variance function, a refinement of an existing 'super-convex' dominance condition and quadratic constrained programming. To reduce the computational burden in large-scale applications, we propose a problem reduction method based on vertex enumeration. We apply our method to historical stock market data using an industry momentum strategy. Our enhanced portfolio generates important performance improvements compared with alternatives based on mean-variance dominance and second-degree stochastic dominance. Relative to the benchmark, our portfolio increases average out-of-sample return by almost seven percentage points per annum without incurring more downside risk, using quarterly rebalancing and without short selling.