Mixed integer nonlinear program (MINLP) problems arise in many fields, including engineering design, the process industry, communications, and finance. In this book, the author discusses various methods to solve nonconvex structured MINLPs. The book demonstrates a generic branch-cut-and-price framework for MINLPs, which is the basic concept in almost all modern mixed integer programming solvers.

The book presents recent developments in theories and algorithms for designing Lagrangian global optimizers, and suggests further improvements. Some notable contributions of the book include several estimates on the duality gap; a method for generating polyhedral inner and outer approximations of general MINLPs; a new decomposition-based method for solving the dual of general mixed integer all-quadratic programs; a new lower bounding method for multivariate polynomials; decomposition-based lower bounds and box-reduction techniques for MINLPs; a new adopting method for simultaneously generating discretizations and computing the relaxation of infinite dimensional MINLPs; and a Lagrangian heuristic for MINLPs.

The first two chapters introduce structured MINLPs, and discuss ways of reformulating a MINLP. Theory and computational methods for generating Lagrangian and convex relaxations are covered in chapters 3 through 7. The author presents global optimality cuts and a new method for refining discretizations of infinite dimensional MINLPs in the next two chapters. An overview of existing global optimization methods is presented in chapter 10. Deformation, rounding and partitioning, and Lagrangian heuristics are described in the next two chapters. Branch-cut-and-price algorithms for general MINLPs are described in chapter 13. The last chapter describes the MINLP solver, along with a Lagrangian global optimizer. The book contains two appendices, discussing future perspectives on MINLPs (Appendix A) and presenting a numerical experiment (Appendix B).