Abstract
Many problems in systems and control engineering can be formulated as constrained optimization problems with multivariate polynomial objective functions. We propose algorithms based on polynomial B-spline form for constrained global optimization of multivariate polynomial functions. The proposed algorithms are based on a branch-and-bound framework. We tested the proposed basic constrained global optimization algorithms by considering three test problems from systems and control. The obtained results agree with those reported in literature.
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References
Henrion, D., Lasserre, J.-B.: Solving Nonconvex Optimization Problems. IEEE Control Systems 24, 72–83 (2004)
Lin, Q., Rokne, J.G.: Methods for Bounding the Range of a Polynomial. J. Computational and Applied Mathematics 58, 193–199 (1995)
Lin, Q., Rokne, J.G.: Interval Approximation of Higher Order to the Ranges of Functions. Computers & Mathematics with Applications 31, 101–109 (1996)
Michel, D., Mraoui, H., Sbibih, D., Zidna, A.: Computing the Range of Values of Real Functions using B-spline Form. Applied Mathematics and Computation 233, 85–102 (2014)
http://www.sc.iitb.ac.in/~ddgawali/Proof_of_Lemma.pdf (2014)
http://www.sc.iitb.ac.in/~ddgawali/Algorithm_pdf_file.pdf (2014)
Floudas, C.A.: Handbook of test problems in local and global optimization (1999)
Chesi, G., Garulli, A., Tesi, A.: Solving Quadratic Distance Problems: An LMI Based Approach. IEEE Trans. Automatic Control 48, 200–212 (2003)
Chesi, G., Tesi, A., Vicinio, A., Genesio, R.: An LMI Approach to Constrained Optimization with Homogenous Forms. Systems and Control Letters 42, 11–19 (2001)
Chesi, G.: Estimating the Domain of Attraction via Union of Continuous Families of Lyapunov Estimates. Systems and Control Letters 56, 326–333 (2007)
Sastry, S.: Nonlinear Systems Analysis, Stability and Control. Springer, Berlin (1999)
Parrilo, P.A.: Structured Semidefinite Programs and Semialgebric Geometric Methods in Robustness and Optimization. PhD thesis, California Insitiute of Technology, Pasadena, California (2000)
Horst, R., Pardalos, P.M.: Handbook of Global Optimization (1995)
Hansen, E., Walster, G.W.: Global Optimization using Interval Analysis: Revised and Expanded, vol. 264. CRC Press (2003)
Jaulin, L.: Applied Interval Analysis: with Examples in Parameter and State Estimation, Robust Control and Robotics, vol. 1. Springer (2001)
Kearfott, R.B.: Rigorous Global Search Continuous Problems. Springer (1996)
Vaidyanathan, R., El-Halwagi, M.: Global Optimization of Nonconvex Nonlinear Programs via Interval Analysis. Computers & Chemical Engineering 18, 889–897 (1994)
Lyche, T., Morken, K.: Spline Methods Draft. Department of Informatics, Centre of Mathematics for Applications, University of Oslo (2008)
Park, S.: Approximate Branch-and-Bound Global Optimization using B-spline Hypervolumes. Advances in Engineering Software 45, 11–20 (2012)
Ratschek, H., Rokne, J.: Computer Methods for the Range of Functions. Ellis Horwood Limited Publishers, Chichester (1984)
Patil, B.V., Nataraj, P.S.V., Bhartiya, S.: Global Optimization of Mixed-Integer Nonlinear (Polynomial) Programming Problems: the Bernstein Polynomial Approach. Computing 94, 325–343 (2012)
Ratschek, H., Rokne, J.: New Computer Methods for Global Optimization. Ellis Horwood series in Mathematics and its Applications. Horwood (1988)
Rogers, D.F.: An Introduction to NURBS: With Historical Perspective. Morgan Kaufmann (2001)
Mathworks Inc. MATLAB version 8.0.0.783 (R 2012 b) (2012)
DeVore, R.A., Lorentz, G.G.: Constructive Approximation, vol. 303. Springer Science & Business Media (1993)
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Gawali, D., Zidna, A., Nataraj, P.S.V. (2015). Solving Nonconvex Optimization Problems in Systems and Control: A Polynomial B-spline Approach. In: Le Thi, H., Pham Dinh, T., Nguyen, N. (eds) Modelling, Computation and Optimization in Information Systems and Management Sciences. Advances in Intelligent Systems and Computing, vol 359. Springer, Cham. https://doi.org/10.1007/978-3-319-18161-5_40
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DOI: https://doi.org/10.1007/978-3-319-18161-5_40
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18160-8
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