<div> We develop an efficient method for solving non-convex constrained optimization problems that are pervasive in economics. The optimal solution to these problems often involves randomization. We employ a Lagrangian framework and prove that the value of the saddle point characterizing the optimal random solution equals the value of the deterministic dual problem. Our algorithm solves this dual via subgradient descent and recovers the optimal random solution directly from deterministic optima computed along the iterations. For many non-convex economic problems, our method is orders of magnitude faster than linear programming, making previously intractable lottery problems feasible. As an application, we solve for optimal Mirrleesian income taxation with multi-dimensional types. We show that heterogeneity in productivity and Frisch elasticity can make randomization welfare-improving over the optimal deterministic schedule. </div>