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Solving systems of nonlinear equations with continuous GRASP. (English) Zbl 1163.90750
Summary: A method for finding all roots of a system of nonlinear equations is described. Our method makes use of C-GRASP, a recently proposed continuous global optimization heuristic. Given a nonlinear system, we solve a corresponding adaptively modified global optimization problem multiple times, each time using C-GRASP, with areas of repulsion around roots that have already been found. The heuristic makes no use of derivative information. We illustrate the approach using systems found in the literature.

MSC:
90C30 Nonlinear programming
90C15 Stochastic programming
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[1] ()
[2] Bazaraa, M.S.; Jarvis, J.J.; Sherali, H.D., Linear programming and network flows, (1990), John Wiley and Sons · Zbl 0722.90042
[3] Blackman, S.; Banh, N., Track association using correction bias and missing data, (), 529-539
[4] Cohen, H., A course in computational algebraic number theory, (1993), Springer-Verlag Berlin · Zbl 0786.11071
[5] Cox, D.A.; Little, J.B.; O’Shea, D., Ideals, varieties, and algorithms, (1997), Springer-Verlag New York
[6] Cox, D.A.; Little, J.B.; O’Shea, D., Using algebraic geometry, (2005), Springer-Verlag New York
[7] Feo, T.A.; Resende, M.G.C., Greedy randomized adaptive search procedures, Journal of global optimization, 6, 109-133, (1995) · Zbl 0822.90110
[8] Festa, P.; Resende, M.G.C., (), 325-367
[9] Floudas, C.A., Recent advances in global optimization for process synthesis, design, and control: enclosure of all solutions, Computers and chemical engineering, (1999), S: 963-973
[10] Floudas, C.A.; Pardalos, P.M.; Adjiman, C.; Esposito, W.; Gumus, Z.; Harding, S.; Klepeis, J.; Meyer, C.; Schweiger, C., Handbook of test problems in local and global optimization, (1999), Kluwer Academic Publishers Dordrecht · Zbl 0943.90001
[11] Golub, G.H.; Van Loan, C.F., Matrix computations, (1996), Johns Hopkins University Press · Zbl 0865.65009
[12] Hirsch, M.J.; Meneses, C.N.; Pardalos, P.M.; Resende, M.G.C., Global optimization by continuous GRASP, Optimization letters, 1, 2, 201-212, (2007) · Zbl 1149.90119
[13] M.J. Hirsch, P.M. Pardalos, M.G.C. Resende, Continuous GRASP: Enhancements and sequential stopping rules (2006) (submitted for publication)
[14] M.J. Hirsch, P.M. Pardalos, M.G.C. Resende, Sensor registration in a sensor network by Continuous GRASP, in: IEEE Proceedings of the Military Communications Conference, Wash. D.C., October 2006
[15] Hoffman, K.; Kunze, R., Linear algebra, (1971), Prentice Hall · Zbl 0212.36601
[16] Kubicek, M.; Hofmann, H.; Hlavacek, V.; Sinkule, J., Multiplicity and stability in a sequence of two nonadiabatic nonisothermal CSTR, Chemical engineering sceinces, 35, 987-996, (1980)
[17] Levedahl, M., An explicit pattern matching assignment algorithm, (), 461-469
[18] Martinez, J.M., Algorithms for solving nonlinear systems of equations, (), 81-108 · Zbl 0828.90125
[19] Matsumoto, M.; Nishimura, T., Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator, ACM transactions on modeling and computer simulation, 8, 1, 3-30, (1998) · Zbl 0917.65005
[20] J.P. Merlet, The COPRIN examples page. http://www-sop.inria.fr/coprin/logiciels/ALIAS/Benches/benches.html, 2006
[21] Moré, J.J., A collection of nonlinear model problems, (), 723-762
[22] Nielson, J.; Roth, B., On the kinematic analysis of robotic mechanisms, The international journal of robotics research, 18, 12, 1147-1160, (1999)
[23] Pinter, J.D., Computational global optimization in nonlinear systems: an interactive tutorial, (2001), Lionhart Publishing · Zbl 1079.90003
[24] Pramanik, S., Kinematic synthesis of a six-member mechanism for automotive steering, ASME journal of mechanical design, 124, 642-645, (2002)
[25] Royden, H.L., Real analysis, (1988), Macmillan Publishing · Zbl 0704.26006
[26] Wikipedia, Ackerman Steering Geometry. http://en.wikipedia.org/wiki/Ackermann_steering_geometry, 2006
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