Generalized benders decomposition master problem

The basic idea in Generalized Benders Decomposition GBD is the generation, at each iteration, of an upper bound and a lower bound on the sought solution of the MINLP model. The upper bound results from the primal problem, while the lower bound results from the master problem. The primal problem corresponds to problem (6.2) with fixed y-variables (i.e., it is in the jr-space only), and its solution provides information about the upper bound and the Lagrange... 

This section presents the theoretical development of the Generalized Benders Decomposition GBD. The primal problem is analyzed first for the feasible and infeasible cases. Subsequently, the theoretical analysis for the derivation of the master problem is presented. 

The derivation of the primal master problem follows the same steps as the derivation of the master problem in Generalized Benders Decomposition GBD. The final form of the primal master problem is ... 

The branch and bound method can be used for MINLP problems, but it requires solving a large number of NLP problems and is, therefore, computationally intensive. Instead, methods such as the Generalized Benders Decomposition and Outer Approximation algorithms are usually preferred. These methods solve a master MILP problem to initialize the discrete variables at each stage and then solve an NLP subproblem to optimize the continuous variables. Details of these methods are given in Biegler et al. (1997) and Diwekar (2003). 

---