Continuous reformulation of MINLP problems

Abstract:
The solution of mixed-integer nonlinear programming (MINLP) problems often suffers from a lack of robustness, reliability, and efficiency due to the combined computational challenges of the discrete nature of the decision variables and the nonlinearity or even nonconvexity of the equations. By means of a continuous reformulation, the discrete decision variables can be replaced by continuous decision variables and the MINLP can then be solved by reliable NLP solvers. In this work, we reformulate 98 representative test problems of the MINLP library MINLPLib with the help of Fischer-Burmeister (FB) NCP-functions and solve the reformulated problems in a series of NLP steps while a relaxation parameter is reduced. The solution properties are compared to the MINLP solution with branch & bound and outer approximation solvers. Since a large portion of the reformulated problems yield local optima of poor quality or cannot even be solved to a discrete solution, we propose a reinitialization and a post-processing procedure. Extended with these procedures, the reformulation achieved a comparable performance to the MINLP solvers SBB and DICOPT for the 98 test problems. Finally, we present a large-scale example from synthesis of distillation systems which we were able to solve more efficiently by continuous reformulation compared to MINLP solvers.