Skip to main content
SpringerLink
Log in

Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems

  • Chapter
  • First Online:
Canonical Duality Theory

Part of the book series: Advances in Mechanics and Mathematics ((AMMA,volume 37))

Abstract

This paper presents a canonical duality theory for solving general nonconvex/discrete constrained minimization problems. By using the canonical dual transformation, these challenging problems can be reformulated as a unified canonical dual problem (i.e., with zero duality gap) in continuous space, which can be solved easily to obtain global optimal solution. Some basic concepts and general theory in canonical systems are reviewed. Applications to Boolean least squares problems are illustrated.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. An, L.T.H., Tao, P.D.: The DC (difference of convex functions) programming and DCA revisited with DC models with real world nonconvex optimization problems. Ann. Oper. Res. 133, 23–46 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bazaraa, M.S., Sherali, H.D.: Nonlinear Programming: Theory and Algorithms, third edn. Wiley, New York (2006)

    Google Scholar 

  3. Burke, J.V., Lewis, A.S., Overton, M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15(3), 751–779 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cai, K., Gao, D.Y., Qin, Q.H.: Post-buckling solutions of hyper-elastic beam by canonical dual finite element method. Math. Mech. Solids 19(6), 659–671 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Y., Gao, D.Y.: Global solutions to nonconvex optimization of 4th-order polynomial and log-sum-exp functions. J. Global Optim. 64, 417–431 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chiang, M.: Geometric Programming for Communication Systems. Now Publishers Inc., Hanover (2005)

    MATH  Google Scholar 

  7. Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)

    MATH  Google Scholar 

  8. Gao, D.Y.: Dual extremum principles in finite deformation theory with applications to post-buckling analysis of extended nonlinear beam theory. Appl. Mech. Rev. 50, S64–S71 (1997)

    Article  Google Scholar 

  9. Gao, D.Y.: Duality Principles in Nonconvex Systems: Theory Methods and Applications. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  10. Gao, D.Y.: Canonical dual transformation method and generalized triality theory in nonsmooth global optimization. J. Glob. Optim. 17(1/4), 127–160 (2000)

    Article  MATH  Google Scholar 

  11. Gao, D.Y.: Perfect duality theory and complete set of solutions to a class of global optimization. Optimization 52, 467–493 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gao, D.Y.: Nonconvex semi-linear problems and canonical duality solutions. In: Advances in Mechanics and Mathematics II, pp. 261–311. Kluwer Academic Publishers, Dordrecht (2003)

    Google Scholar 

  13. Gao, D.Y.: Solutions and optimality to box constrained nonconvex minimization problems. J. Ind. Manag. Optim. 3(2), 293–304 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gao, D.Y.: Canonical duality theory: unified understanding and generalized solutions for global optimization. Comput. Chem. Eng. 33, 1964–1972 (2009)

    Article  Google Scholar 

  15. Gao, D.Y.: Analytical solutions to general anti-plane shear problems in finite elasticity. Continuum Mech. Thermodyn. 28, 175–194 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gao, D.Y., Ogden, R.W.: Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem. ZAMP 59, 498–517 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gao, D.Y., Ruan, N.: Solutions to quadratic minimization problems with box and integer constraints. J. Glob. Optim. 47(3), 463–484 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gao, D.Y., Sherali, H.D.: Canonical Duality Theory: Connections between nonconvex mechanics and global optimization. In: Advance in Mechanics and Mathematics, vol. 3, pp. 257–326. Springer, (2008)

    Google Scholar 

  19. Gao, D.Y., Strang, G.: Geometric nonlinearity: Potential energy, complementary energy, and the gap function. Q. Appl. Math. 47(3), 487–504 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gao, D.Y., Wu, C.Z.: On the triality theory for a quartic polynomial optimization problem. J. Ind. Manag. Optim. 8(1), 229–242 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gao, D.Y., Ruan, N., Sherali, H.: Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality. J. Glob. Optim. 45, 473–497 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hajela, P.: Genetic search-an approach to the nonconvex optimization problem. AIAA J. 28(7), 1205–1210 (1990)

    Article  Google Scholar 

  23. Hansen, P., Jaumard, B., Ruiz, M., Xiong, J.: Global minimization of indefinite quadratic functions subjects to box constraints. Nav. Res. Logist. 40(3), 373–392 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  24. Horst, R., Pardalos, P.M., Romeijn, H.E.: Handbook of Global Optimization. Kluwer Academic Publishers, Dordrecht (2000)

    Google Scholar 

  25. Li, S.F., Gupta, A.: On dual configuration forces. J. Elast. 84, 13–31 (2006)

    Article  MATH  Google Scholar 

  26. Nesterov, Y.: Squared functional systems and optimization problems. In: Frenk, H., et al. (eds.) High Performance Optimization, pp. 405–440. Kluwer Academic Publishers, Dordrecht (2000)

    Chapter  Google Scholar 

  27. Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Kluwer Academic Publishers, Dordrecht (2000)

    MATH  Google Scholar 

  28. Ruan, N., Gao, D.Y.: Canonical duality approach for nonlinear dynamical systems. IMA J. Appl. Math. 79(2), 313–325 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ruan, N., Gao, D.Y.: Global optimal solutions to a general sensor network localization problem. Perform. Eval. 75(76), 1–16 (2014)

    Article  Google Scholar 

  30. Ruan, N., Gao, D.Y., Jiao, Y.: Canonical dual least square method for solving general nonlinear systems of quadratic equations. Comput. Optim. Appl. 47, 335–347 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Santos, H.A.F.A., Gao, D.Y.: Canonical dual finite element method for solving post-buckling problems of a large deformation elastic beam. Int. J. Nonlinear Mech. 47, 240–247 (2012)

    Article  Google Scholar 

  32. Tonti, E.: On the mathematical structure of a large class of physical theories. Accad. Naz. dei Lincei, Serie VIII, 49–56 (1972)

    Google Scholar 

  33. Strang, G.: Introducation to Applied Mathematics. Wellesley-Cambridge Press, Wellesley (1986)

    MATH  Google Scholar 

  34. Wang, Z.B., Fang, S.C., Gao, D.Y., Xing, W.X.: Canonical dual approach to solving the maximum cut problem. J. Glob. Optim. 54, 341–351 (2012)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Science and Technology, Federation University Australia, Ballarat, VIC, 3353, Australia

    Ning Ruan & David Yang Gao

Authors
  1. Ning Ruan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Yang Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ning Ruan .

Editor information

Editors and Affiliations

  1. Faculty of Science and Technology, Federation University Australia, Mt. Helen, Victoria, Australia

    David Yang Gao

  2. Department of Computer, Control, and Managment Engineering, Sapienza University of Rome, Rome, Roma, Italy

    Vittorio Latorre

  3. Faculty of Science and Technology, Federation University Australia, Mt. Helen, Victoria, Australia

    Ning Ruan

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Ruan, N., Gao, D.Y. (2017). Canonical Duality Theory for Solving Nonconvex/Discrete Constrained Global Optimization Problems. In: Gao, D., Latorre, V., Ruan, N. (eds) Canonical Duality Theory. Advances in Mechanics and Mathematics, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-319-58017-3_9

Download citation

Publish with us

Policies and ethics

Search

Navigation