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A reformulation framework for global optimization. (English) Zbl 1277.90102
Summary: We present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions, two versions of the so-called \(\alpha\)BB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem forms a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.

90C26 Nonconvex programming, global optimization
90C11 Mixed integer programming
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[1] Adjiman, C.S.; Androulakis, I.P.; Floudas, C.A., Global optimization of mixed-integer nonlinear problems, AIChE J., 46, 1769-1797, (2000)
[2] Adjiman, C.S.; Dallwig, S.; Floudas, C.A.; Neumaier, A., A global optimization method, \(α\)BB, for general twice-differentiable constrained NLPs—I. theoretical advances, Comput. Chem. Eng., 22, 1137-1158, (1998)
[3] Adjiman, C.S.; Dallwig, S.; Floudas, C.A.; Neumaier, A., A global optimization method, \(α\)BB, for general twice differentiable NLPs—II. implementation and computional results, Comput. Chem. Eng., 22, 1159-1179, (1998)
[4] Adjiman, C.S.; Androulakis, I.P.; Floudas, C.A., Global optimization of MINLP problems in process synthesis and design, Comput. Chem. Eng., 21, 445-450, (1997)
[5] Akrotirianakis, I.G.; Floudas, C.A., Computational experience with a new class of convex underestimators: box-constrained NLP problems, J. Glob. Optim., 29, 249-264, (2004) · Zbl 1133.90420
[6] Akrotirianakis, I.G.; Floudas, C.A., A new class of improved convex underestimators for twice continuously differentiable constrained nlps, J. Glob. Optim., 30, 367-390, (2004) · Zbl 1082.90090
[7] Androulakis, I.P.; Maranas, C.D.; Floudas, C.A., \(α\)BB: a global optimization method for general constrained nonconvex problems, J. Glob. Optim., 7, 337-363, (1995) · Zbl 0846.90087
[8] Björk, K.-M.: A global optimization method with some heat exchanger network applications. Ph.D. thesis, Åbo Akademi University (2002) · Zbl 0369.90112
[9] Brönnimann, H.; Melquiond, G.; Pion, S., The design of the boost interval arithmetic library, Theor. Comput. Sci., 351, 111-118, (2006) · Zbl 1086.65046
[10] Dembo, R.S., Current state of the art of algorithms and computer software for geometric programming, J. Optim. Theory Appl., 26, 149-183, (1978) · Zbl 0369.90121
[11] Floudas, C.A.: Deterministic global optimization. Theory, methods and applications. Number 37 in Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, Dordrecht (2000) · Zbl 1173.90483
[12] Floudas, C.A.; Kreinovich, V., On the functional form of convex underestimators for twice continuously differentiable functions, Optim. Lett., 1, 187-192, (2007) · Zbl 1133.49030
[13] Floudas, C.A.; Kreinovich, V.; Törn, A. (ed.); Zilinskas, J. (ed.), Towards optimal techniques for solving global optimization problems: symmetry-based approach, 21-42, (2007), Boston, MA · Zbl 1267.90105
[14] Horst, R., Pardalos, P.M., Romeijn, H.E.: Handbook of global optimization. Number 2 in Nonconvex Optimization and Its Applications. Kluwer Academic Publishers, Dordrecht (2002)
[15] Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Dordrecht (2000) · Zbl 0966.90073
[16] Jeroslow, R.G., Lowe, J.K.: Modelling with integer variables. In: Mathematical Programming at Oberwolfach II, vol. 22 of Mathematical Programming Studies, pp. 167-184. Springer, Berlin (1984) · Zbl 0554.90081
[17] Li, H.-L.; Tsai, J.F.; Floudas, C.A., Convex underestimation for posynomial functions of positive variables, Optim. Lett., 2, 333-340, (2008) · Zbl 1152.90610
[18] Liberti, L.; Cafieri, S.; Tarissan, F.; Abraham, A. (ed.); Hassanien, A.-E. (ed.); Siarry, P. (ed.); Engelbrecht, A. (ed.), Reformulations in mathematical programming: a computational approach, 153-234, (2009), Berlin
[19] Lin, M.-H.; Tsai, J.-F., Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems, Eur. J. Oper. Res., 216, 17-25, (2012) · Zbl 1242.90177
[20] Liu, W.B.; Floudas, C.A., A remark on the GOP algorithm for global optimization, J. Glob. Optim., 3, 519-521, (1993) · Zbl 0785.90089
[21] Lundell, A.: Transformation techniques for signomial functions in global optimization. Ph.D. thesis, Åbo Akademi University (2009) · Zbl 1242.90137
[22] Lundell, A.; Westerlund, J.; Westerlund, T., Some transformation techniques with applications in global optimization, J. Glob. Optim., 43, 391-405, (2009) · Zbl 1169.90453
[23] Lundell, A.; Westerlund, T., Optimization of power transformations in global optimization, Chem. Eng. Trans., 11, 95-100, (2007)
[24] Lundell, A., Westerlund, T.: Exponential and power transformations for convexifying signomial terms in MINLP problems. In: Bruzzone, L. (ed.) Proceedings of the 27th IASTED International Conference on Modelling, Identification and Control, pp. 154-159. ACTA Press, Anaheim · Zbl 1169.90453
[25] Lundell, A.; Westerlund, T., Convex underestimation strategies for signomial functions, Optim. Methods Softw., 24, 505-522, (2009) · Zbl 1178.90278
[26] Lundell, A., Westerlund, T.: Implementation of a convexification technique for signomial functions. In: Jezowski, J., Thullie, J. (eds.) Proceedings of the 19th European Symposium on Computer Aided Process Engineering, vol. 26 of Computer Aided Chemical Engineering, pp. 579-583. Elsevier, Amsterdam (2009) · Zbl 1243.90181
[27] Lundell, A.; Westerlund, T., On the relationship between power and exponential transformations for positive signomial functions, Chem. Eng. Trans., 17, 1287-1292, (2009)
[28] Lundell, A., Westerlund, T.: Optimization of transformations for convex relaxations of MINLP problems containing signomial functions. In: de Brito Alves, R.M., do Nascimento, C.A.O., Biscaia, E.C. (eds.) Proceedings of the 10th International Symposium on Process Systems Engineering: Part A, vol. 27 of Computer Aided Chemical Engineering, pp. 231-236. Elsevier, Amsterdam (2009)
[29] Lundell, A.; Westerlund, T.; Lee, J. (ed.); Leyffer, S. (ed.), Global optimization of mixed-integer signomial programming problems, 349-369, (2012), New York, NY · Zbl 1242.90137
[30] Maranas, C.D.; Floudas, C.A., Finding all solutions of nonlinearly constrained systems of equations, J. Glob. Optim., 7, 143-182, (1995) · Zbl 0841.90115
[31] Maranas, C.D.; Floudas, C.A., Global optimization in generalized geometric programming, Comput. Chem. Eng., 21, 351-369, (1997)
[32] Meyer, C.A.; Floudas, C.A., Convex underestimation of twice continuously differentiable functions by piecewise quadratic perturbation: spline \(α\)BB underestimators, J. Glob. Optim., 32, 221-258, (2005) · Zbl 1080.90059
[33] Pardalos, P.M.; Romeijn, H.E.; Tuy, H., Recent developments and trends in global optimization, J. Comput. Appl. Math., 124, 209-228, (2000) · Zbl 0969.90067
[34] Pardalos, P.M., Rosen, J.B.: Constrained Global Optimization: Algorithms and Applications, vol. 268 of Lecture notes in Computer Science. Springer, Berlin (1987) · Zbl 0638.90064
[35] Peterson, E.L., The origins of geometric programming, Ann. Oper. Res., 105, 15-19, (2001) · Zbl 1024.90002
[36] Pörn, R.; Björk, K.-M.; Westerlund, T., Global solution of optimization problems with signomial parts, Discret. Optim., 5, 108-120, (2008) · Zbl 1134.90041
[37] Pörn, R.; Harjunkoski, I.; Westerlund, T., Convexification of different classes of non-convex MINLP problems, Comput. Chem. Eng., 23, 439-448, (1999)
[38] Rijckaert, M.J.; Martens, X.M., Comparison of generalized geometric programming algorithms, J. Optim. Theory Appl., 26, 205-242, (1978) · Zbl 0369.90112
[39] Rosenthal R.E.: GAMS—A User’s Guide. GAMS Development Corporation, Washington, DC (2008)
[40] Skjäl, A.; Lundell, A.; Westerlund, T., Global optimization with \(C\)\^{}{2} constraints by convex reformulations, Chem. Eng. Trans., 24, 373-378, (2011)
[41] Tsai, J.F.; Lin, M.-H., Global optimization of signomial mixed-integer nonlinear programming problems with free variables, J. Glob. Optim., 42, 39-49, (2008) · Zbl 1173.90483
[42] Tsai, J.F.; Lin, M.-H., An efficient global approach for posynomial geometric programming problems, INFORMS J. Comput., 23, 483-492, (2011) · Zbl 1243.90181
[43] Westerlund, T.; Liberti, L. (ed.); Maculan, N. (ed.), Some transformation techniques in global optimization, 47-74, (2005), Berlin
[44] Westerlund, T.; Lundell, A.; Westerlund, J., On convex relaxations in nonconvex optimization, Chem. Eng. Trans., 24, 331-336, (2011)
[45] Westerlund, T.; Westerlund, J., GGPECP—an algorithm for solving non-convex MINLP problems by cutting plane and transformation techniques, Chem. Eng. Trans., 3, 1045-1050, (2003)
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