Global optimum search for nonconvex NLP and MINLP problems

Abstract

Nonconvex nonlinear programming problems (NLPs) and mixed-integer nonlinear programming problems (MINLPs) abound in the synthesis, design and control of chemical processes. The presently available mathematical programming techniques very often do not lead to the global solution of these classes of problems. A new approach for global optimum search is presented in this paper which involves a decomposition of the variable set into two sets—complicating and noncomplicating variables. This results in a decomposition of the constraint set leading to two subproblems. The decomposition of the original problem induces special structure in the resulting subproblems and a series of these subproblems are then solved to determine the optimal solution. Appendix A presents a systematic approach, based on graph theory, for determining the various possibilities of decomposing the variable set. The key idea is to combine a judicious selection of the complicating variables with suitable transformations leading to subproblems which can attain their respective global solutions at each iteration. Mathematical properties of the proposed approach are presented and its effectiveness is illustrated through a number of nonconvex NLP and MINLP example problems.