\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in $\mathbb{R}^n $

Abstract Related Papers Cited by
  • The main purpose of this research note is to show that the triality theory can always be used to identify both global minimizer and the biggest local maximizer in global optimization. An open problem left on the double-min duality is solved for a nonconvex optimization problem with double-well potential in $\mathbb{R}^n $, which leads to a complete set of analytical solutions. Also a convergency theorem is proved for linear perturbation canonical dual method, which can be used for solving global optimization problems with multiple solutions. The methods and results presented in this note pave the way towards the proof of the triality theory in general cases.
    Mathematics Subject Classification: Primary: 90C26, 90C30, 90C46.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    V. I. Arnold, "Mathematical Methods of Classical Mechanics," 2nd edition, Springer-Verlag, Berlin, Heidelberg, 1989.doi: 10.1007/978-1-4757-2063-1.

    [2]

    C. A. Desoer and B. H. Whalen, A note on pseudoinverses, Journal of the Society for Industrial and Applied Mathematics, 11 (1963), 442-447.doi: 10.1137/0111031.

    [3]

    D. Y. Gao, "Duality Principles in Nonconvex Systems. Theory Methods and Applications," Kluwer Academic Publishers, Dordrecht/Boston/London, 2000.doi: 10.1007/978-1-4757-3176-7.

    [4]

    D. Y. Gao, Perfect duality theory and complete solutions to a class of global optimization problems, Optim., 52 (2003), 467-493.doi: 10.1080/02331930310001611501.

    [5]

    D. Y. Gao, Nonconvex semi-linear problems and canonical duality solutions, Advances in Mechanics and Mathematics (eds. D. Y. Gao and R. W. Ogden), Kluwer, (2003), 261-311.

    [6]

    D. Y. Gao, Solutions and optimality to box constrained nonconvex minimization problems, J. Ind. Manag. Optim., 3 (2007), 293-304.doi: 10.3934/jimo.2007.3.293.

    [7]

    D. Y. Gao, Canonical duality theory: theory, method, and applications in global optimization, Comput. Chem., 33 (2009), 1964-1972.doi: 10.1016/j.compchemeng.2009.06.009.

    [8]

    D. Y. Gao and R. W. Ogden, Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation, Quart. J. Mech. Appl. Math., 61 (2008), 497-522.doi: 10.1093/qjmam/hbn014.

    [9]

    D. Y. Gao and N. Ruan, Solutions to quadratic minimization problems with box and integer constraints, J. Glob. Optim., 47 (2010), 463-484.doi: 10.1007/s10898-009-9469-0.

    [10]

    D. Y. Gao and G. Strang, Geometric nonlinearity: Potential energy, complementary energy, and the gap function, Quart. Appl. Math., 47 (1989), 487-504.

    [11]

    D. Y. Gao and C. Z. Wu, On the triality theory in global optimization, J. Industrial and Manegement Optimization, 8 (2012), 229-242. Also published in arXiv:1104.2970v1 at http://arxiv.org/abs/1104.2970.

    [12]

    D. Y. Gao and C. Z. WuTriality theory for general unconstrained global optimization problems, To appear in J. Global Optimization.

    [13]
    [14]

    D. M. Morales-Silva and D. Y. GaoCanonical duality theory and triality for solving general nonconstrained global optimization problems, To be submitted.

    [15]

    G. Peters and J. H. Wilkinson, The least squares problem and pseudo-inverses, The Computer Journal, 13 (1970), 309-316.doi: 10.1093/comjnl/13.3.309.

    [16]

    N. Ruan, D. Y. Gao and Y. Jiao, Canonical dual least square method for solving general nonlinear systems of quadratic equations, Comput Optim Appl, 47 (2010), 335-347.doi: 10.1007/s10589-008-9222-5.

    [17]

    M. J. Sewell, "Maximum and Minimum Principles," Cambridge University Press, Cambridge, New York, Port Chester, Melbourne, Sydney, 1987.

    [18]

    Z. B. Wang, S. C. Fang, D. Y. Gao and W. X. Xing, Canonical dual approach to solving the maximum cut problem, J. Global Optimization, 54 (2012), 341-352.doi: 10.1007/s10898-012-9881-8.

    [19]

    C. Wu, C. J. Li and D. Y. GaoCanonical primal-dual method for solving non-convex minimization problems, arXiv:1212.6492, http://arxiv.org/pdf/1212.6492v1.pdf.

    [20]

    R. K. P. Zia, E. F. Redish and S. R. McKay, Making sense of the Legendre transform, American Journal of Physics, 77 (2009), 614-622.doi: 10.1119/1.3119512.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(82) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return