Nonlinear Programs with Complicating Variables: Theoretical Analysis and Numerical Experience

Nonlinear programming problems with complicating variables (those which, when fixed, render the remaining problem simpler to solve) are analyzed and solved. The concept of support functions, which allows us to develop and interpret geometrically the generalized benders decomposition, is introduced. After a brief differentiability study of the related perturbation functions, a primal method is proposed, which can handle problems for which the Geoffrion´s property P (or ¿) is not present. The property P states that the optimal solution of the associated Lagrangian function is independent of any feasible value of the complicating variables, and it is essential to build up efficiently the cuts that define the master problem. A large-scale problem the unit commitment of thermal plants with start-up costs with ten boolean variables, 240 continuous variables, and 528 constraints is solved. The computational experience included shows the great numerical efficiency of the decomposition technique.