Global optimization of mixed-integer nonlinear programs: A theoretical and computational study
This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototypical branch-and-bound algorithm. In the theoretical/algorithmic part of the paper, we begin by developing novel strategies for constructing linear relaxations of mixed-integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties. We then use Lagrangian/linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming. This theory leads to new range reduction schemes, including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch-and-bound tree. Finally, we incorporate these relaxation and reduction strategies in a branch-and-bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems. In the computational part of the paper, we describe our implementation discussing, wherever appropriate, the use of suitable data structures and associated algorithms. We present computational experience with benchmark separable concave quadratic programs, fractional 0–1 programs, and mixed-integer nonlinear programs from applications in synthesis of chemical processes, engineering design, just-in-time manufacturing, and molecular design.
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- 1.Ahmed, S., Tawarmalani, M., Sahinidis, N.V.: A finite branch and bound algorithm for two-stage stochastic integer programs. Mathematical Programming. Submitted, 2000Google Scholar
- 4.Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming, Theory and Algorithms. Wiley Interscience, Series in Discrete Math. Optim. 2nd edition, 1993Google Scholar
- 12.Floudas, C.A.: Deterministic Global Optimization: Theory, Algorithms and Applications. Kluwer Academic Publishers, Dordrecht, 1999Google Scholar
- 14.Gruber, P.M.: Aspects of approximation of convex bodies. In: Gruber, P. M. Gruber, Wills, J. M., (eds.), Handbook of Convex Geometry. North-Holland, 1993Google Scholar
- 15.Gruber, P.M., Kenderov, P.: Approximation of convex bodies by polytopes. Rendiconti Circ. Mat. Palermo, Serie II. 31, 195–225 (1982)Google Scholar
- 19.Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer Verlag, Berlin, Third edition, 1996Google Scholar
- 26.McCormick, G.P.: Nonlinear Programming: Theory, Algorithms and Applications. John Wiley & Sons, 1983Google Scholar
- 27.Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley Interscience, Series in Discrete Math. Opt., 1988Google Scholar
- 30.Ryoo, H.S., Sahinidis, N.V.: Global optimization of nonconvex NLPs and MINLPs with applications in process design. Computers & Chemical Engineering 19, 551–566 (1995)Google Scholar
- 37.Smith, E.M.B., Pantelides, C.C.: Global optimisation of general process models. In: Grossmann, I.E., (ed.), Global Optimization in Engineering Design. Kluwer Academic Publishers, Boston, MA, 1996, pp. 355–386Google Scholar
- 38.Tawarmalani, M.: Mixed Integer Nonlinear Programs: Theory, Algorithms and Applications. PhD thesis, Department of Mechanical & Industrial Engineering. University of Illinois, Urbana-Champaign, IL, August 2001Google Scholar
- 42.Visweswaran, V., Floudas, C.A.: Computational results for an efficient implementation of the GOP algorithm and its variants. In: Grossmann, I.E., (ed.), Global Optimization in Engineering Design. Kluwer Academic Publishers, Boston, MA, 1996, pp. 111–153Google Scholar