An important practical problem is that of determining a nonparametric estimate of the conditional quantile of y given x. If we balance fidelity to the data with a smoothness requirement, the resulting quantile function is a smoothing spline. We reformulate this estimation procedure as a quadratic programming problem, with associated optimality conditions. A recently developed interior point algorithm with proven convergence is extended to solve the quadratic program. This solution characterizes the desired nonparametric conditional quantile function. These methods are illustrated in a study of audiologic performance following cochlear implants.