Efficient multilevel mixed integer nonlinear programming strategies for solving large combinatorial problems in engineering

This paper reports on the experience gained in solving large combinatorial problems by using the Outer-Approximation/Equality-Relaxation (OA/ER) algorithm in two multilevel MINLP strategies. The first one is a Linked Multilevel Hierarchical Strategy (LMHS) and the second one is a Reduced Integer Space (RIS) strategy. Both strategies are used to decompose the original MINLP problem in a hierarchical manner into several MINLP levels that are much easier to solve than the original one. While the first LMHS strategy can be applied to problems that contain only simple mixed-integer constraints, e.g. standard dimensions, the RIS strategy can be used to solve problems with more complex mixed-integer constraints, e.g. different design equations for alternative units. The LMHS strategy is rigorous and can solve complex problems to global optimal solutions. On the other hand, when the RIS strategy is applied for the solution of large combinatorial problems, the global optimality cannot be guaranteed, but very good solutions can be obtained. The synthesis problem of a roller steel gate for a hydroelectric power station with 19623 binary variables is presented to illustrate the LMHS strategy, whilst the synthesis problem of a heat exchanger network comprising different types of exchangers with 1782 binary variables is presented to present the RIS strategy.