A gradient-based optimization approach is employed to select design tolerances for the component dimensions of a mechanical assembly to minimize manufacturing cost while achieving a desired probability of meeting functional requirements, known as the yield. Key to the feasibility of such an approach is to be able to use Monte Carlo simulation to make estimates of the derivatives of the yield with respect to the design tolerances quickly and accurately. A new approach for making these estimates is presented and is shown to be far faster and more accurate than previous approaches. Gradient-based optimization using the new approach for estimating the derivatives is applied to example problems from the literature. The solutions are superior to all previously published solutions and are obtained with very reasonable computer run times. Additional advantages of a gradient-based approach are described.