New Lagrangian Relaxation Based Algorithm for Resource Scheduling with Homogeneous Subproblems

Solution oscillations, often caused by identical solutions to the homogeneous subproblems, constitute a severe and inherent disadvantage in applying Lagrangian relaxation based methods to resource scheduling problems with discrete decision variables. In this paper, the solution oscillations caused by homogeneous subproblems in the Lagrangian relaxation framework are identified and analyzed. Based on this analysis, the key idea to alleviate the homogeneous oscillations is to differentiate the homogeneous subproblems. A new algorithm is developed to solve the problem under the Lagrangian relaxation framework. The basic idea is to introduce a second-order penalty term in the Lagrangian. Since the dual cost function is no longer decomposable, a surrogate subgradient is used to update the multiplier at the high level. The homogeneous subproblems are not solved simultaneously, and the oscillations can be avoided or at least alleviated. Convergence proofs and properties of the new dual cost function are presented in the paper. Numerical testing for a short-term generation scheduling problem with two groups of identical units demonstrates that solution oscillations are greatly reduced and thus the generation schedule is significantly improved.