×

zbMATH — the first resource for mathematics

Some transformation techniques with applications in global optimization. (English) Zbl 1169.90453
Summary: In this paper some transformation techniques, based on power transformations, are discussed. The techniques can be applied to solve optimization problems including signomial functions to global optimality. Signomial terms can always be convexified and underestimated using power transformations on the individual variables in the terms. However, often not all variables need to be transformed. A method for minimizing the number of original variables involved in the transformations is, therefore, presented. In order to illustrate how the given method can be integrated into the transformation framework, some mixed integer optimization problems including signomial functions are finally solved to global optimality using the given techniques.

MSC:
90C30 Nonlinear programming
90C11 Mixed integer programming
Software:
GGPECP; AlphaECP
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Beale E.M.L. and Forrest J.J.H. (1976). Global optimization using special ordered sets. Math. Program. 10: 52–69 · Zbl 0331.90056
[2] Björk, K.-M.: A global optimization method with some heat exchanger network applications, Ph.D. thesis, Åbo Akademi University (2002)
[3] Björk K.-M., Lindberg P.O. and Westerlund T. (2003). Some convexifications in global optimization of problems containing signomial terms. Comp. Chem. Eng. 27: 669–679
[4] Lundell, A.: Optimization techniques in global optimization. Master’s thesis, Åbo Akademi University (2007)
[5] Maranas C.D. and Floudas C.A. (1995). Finding all solutions of nonlinearly constrained systems of equations. J. Global Optim. 7: 143–182 · Zbl 0841.90115
[6] Maranas C.D. and Floudas C.A. (1997). Global optimization in generalized geometric programming. Comp. Chem. Eng. 21: 351–370
[7] Pörn, R.: Mixed integer non-linear programming: convexification techniques and algorithm development, Ph.D. thesis, Åbo Akademi University (2000)
[8] Rijckaert M.J. and Martens X.M. (1978). Comparison of generalized geometric programming algorithms. J. Optim.Theory Appl. 26(2): 205–242 · Zbl 0369.90112
[9] Westerlund, T.: Some transformation techniques in global optimization. Global Optimization: From Theory to Implementation. In: Liberti, L., Maculan, N. (eds.), pp. 47–74 Springer (2005) · Zbl 1100.90037
[10] Westerlund T. and Pörn R. (2002). Solving pseudo-convex mixed-integer problems by cutting plane techniques. Optim. Eng. 3: 253–280 · Zbl 1035.90051
[11] Westerlund T. and Westerlund J. (2003). GGPECP–an algorithm for solving non-convex MINLP problems by cutting plane and transformation techniques. Chem. Eng. Trans. 3: 1045–1050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.