We propose a new algorithm for the stochastic unit commitment problem which is based on the Lagrangian relaxation and the column generation approach. This problem is formulated as a multi-stage nonlinear integer programming problem because the fuel cost function is assumed to be a convex quadratic function. The algorithm consists of two phases. After solving the problem by Lagrangian relaxation, the algorithm continues adding schedules from the dual solution of the restricted linear master program until the algorithm cannot generate new schedules. The schedule generation problem is solved by the calculation of dynamic programming on the scenario tree. We applied the Lagrangian relaxation-column generation approach to a test problem based on the system of a certain Japanese electric power company. Numerical results indicate a significant improvement in the quality of the solution.