This chapter addresses the solution of mixed-integer nonlinear programming (MINLP) problems. The following methods for convex MINLP optimization are described: branch and bound, outer-approximation, generalized Benders decomposition. and extended cutting plane. The last three methods rely on decomposing the MINLP problem into a master MILP model thatpredicts lower bounds and new integer values, and an NLP subproblem that is solved for fixed integer variables yielding an upper bound. It is shown that the MILP master problem of generalized Benders decomposition can be derivedfrom a linear combination of the constraints of the master MILP for outer-approximation yielding a weaker lower bound. The extension of these methods for solving nonconvex MINLP problems is discussed, as well as brief reference to software such as SBB, DICOPT, and α-ECP.

Decomposition of optimization problems is a fundamental technique to reduce the computational cost and enable efficient solution of very large-scale models.Key decomposition approaches are presented in this chapter, discussing primal and dual decomposition methods, Generalized Benders Decomposition, and related applications.