Abstract
Triality theory is proved for a general unconstrained global optimization problem. The method adopted is simple but mathematically rigorous. Results show that if the primal problem and its canonical dual have the same dimension, the triality theory holds strongly in the tri-duality form as it was originally proposed. Otherwise, both the canonical min-max duality and the double-max duality still hold strongly, but the double-min duality holds weakly in a super-symmetrical form as it was expected. Additionally, a complementary weak saddle min-max duality theorem is discovered. Therefore, an open problem on this statement left in 2003 is solved completely. This theory can be used to identify not only the global minimum, but also the largest local minimum, maximum, and saddle points. Application is illustrated. Some fundamental concepts in optimization and remaining challenging problems in canonical duality theory are discussed.
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Notes
- 1.
In continuum physics, complementary variational principle means perfect duality since any duality gap will violate certain physical laws. The existence of a complementary variational principle was a well-known debate existing for several decades in large deformation theory (see [31]). This problem was partially solved by Gao and Strang’s work, and solved completely in 1999 [9].
- 2.
The Lagrangian form was first introduced by W. Hamilton in classical mechanics and denoted by \(L = T - U\), which is the standard notation extensively used from dynamical systems to quantum field theory (see [30]).
- 3.
The equilibrium equation \( D^* \varvec{y}^* = \varvec{f}\) in Newtonian systems is an invariant under the Galilean transformation, which is the combination of Newton’s three laws, see Chap. 2, [10]); while for Einsteins special relativity theory, this abstract equation is an invariant under the Lorentz transformation.
- 4.
In continuum physics, complementary variational principle means perfect duality since any duality gap will violate certain physical laws. The existence of a complementary variational principle was a well-known debate existing for several decades in large deformation theory (see [31]). This problem was partially solved by Gao and Strang’s work, and solved completely in 1999 [9].
- 5.
In this paper \(G^{-1}\) should be understood as a generalized inverse if \(\det G = 0\) [11].
- 6.
- 7.
It should be emphasized here that to find the largest local maximum of \(f(\varvec{x})\) is not simply equivalent to solve the problem \(\min \{ - f(\varvec{x}) | \;\varvec{x}\in \mathscr {X}\}\).
- 8.
See the web page at http://en.wikipedia.org/wiki/Mathematical_optimization.
- 9.
The skew symmetric matrix \(A_s =\frac{1}{2}(A -A^T) \) does not store energy since \(\varvec{x}^T A_s \varvec{x}\equiv 0\).
- 10.
The Hellinger–Reissner energy was first proposed by Hellinger in 1914. After the external energy \({\bar{F}}(u)\) and the boundary conditions in the statically admissible space \({\mathscr {U}}_k = \{ u\in {\mathscr {U}}_a | e = \bar{\varLambda }(u) \in {\mathscr {E}}_a \}\) were fixed by Reissner in 1953, the associated variational statement has been known as the Hellinger–Reissner principle. However, the extremality condition of this principle was an open problem, and also the existence of pure complementary variational principles has been a well-known debate existing for over several decades in large deformation mechanics (see [31]). This open problem was partially solved by Gao and Strang’s work and completely solve by the triality theory. While the pure complementary energy principle was formulated by Gao in 1999 [9].
References
Arnold, V.I.: On teaching mathematics. Uspekhi Mat. Nauk. 53(1), 229–234 (1998); English translation. Russ. Math. Surv. 53(1), 229–236 (1998)
Ekeland, I., Temam, R.: Convex Analysis and Variational Problems. North-Holland, Amsterdam (1976)
Fang, S.C., Gao, D.Y., Sheu, R.L., Wu, S.Y.: Canonical dual approach for solving 0–1 quadratic programming problems. J. Ind. Manag. Optim. 4, 125–142 (2008)
Filly, J.A., Donniell, E.F.: The Moore-Penrose generalized inverse for sums of matrices. SIAM J. Matrix. Anal. Appl. 21(2), 629–635 (1999)
Floudas, C.A.: Deterministic Global Optimization: Theory. Methods and Applications. Kluwer Academic, Dordrecht (2000)
Floudas, C.A.: Systems approaches in bioinformatics and computational genomics. In: Challenges for the Chemical Sciences in the 21th Century, Information and Communications Workshop, National Research Council of the National Academies, National Academies Press, pp. 116–125 (2003)
Gao, D.Y.: Post-buckling analysis and anomalous dual variational problems in nonlinear beam theory. In: Godoy, L.A., Suarez, L.E. (eds.) Applied Mechanics in Americans, Proceedings of the Fifth Pan American Congress of Applied Mechanics, vol. 4. The University of Iowa, Iowa city (1996)
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(11), S64–S71 (1997)
Gao, D.Y.: General analytic solutions and complementary variational principles for large deformation nonsmooth mechanics. Meccanica 34, 169–198 (1999)
Gao, D.Y.: Duality Principles in Nonconvex Systems: Theory. Methods and Applications. Kluwer Academic, Dordrecht (2000)
Gao, D.Y.: Canonical dual transformation method and generalized triality theory in nonsmooth global optimization. J. Glob. Optim. 17(1/4), 127–160 (2000)
Gao, D.Y.: Perfect duality theory and complete solutions to a class of global optimization problems. Optimization 52(4–5), 467–493 (2003)
Gao, D.Y.: Nonconvex semi-linear problems and canonical dual solutions. In: Gao, D.Y., Ogden, R.W. (eds.) Advances in Mechanics and Mathematics, vol. II, p. 261312. Kluwer Academic, Dordrecht (2003)
Gao, D.Y.: Solutions and optimality to box constrained nonconvex minimization problems. J. Indust. Manag. Optim. 3(2), 293–304 (2007)
Gao, D.Y.: Canonical duality theory: theory, method, and applications in global optimization. Comput. Chem. 33, 1964–1972 (2009)
Gao, D.Y., Cheung, Y.K.: On the extremum complementary energy principles for nonlinear elastic shells. Int. J. Solids Struct. 26, 683–693 (1989)
Gao, D.Y., Ogden, R.W.: Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation. Q. J. Mech. Appl. Math 61(4), 497–522 (2008)
Gao, Y., Onat, E.T.: Rate variational extremum principles for finite elasto- plasticity. Appl. Math. Mech. 11(7), 659–667 (1990)
Gao, D.Y., Ruan, N.: Solutions to quadratic minimization problems with box and integer constraints. J. Glob. Optim. 47(3), 463–484 (2010)
Gao, D.Y., Ruan, N., Pardalos, P.M.: Canonical dual solutions to sum of fourth-order polynomials minimization problems with applications to sensor network localization. In: Pardalos, P.M., Ye, Y.Y., Boginski, V., Commander, C. (eds.) Sensors: Theory, Algorithms and Applications, vol. 61, pp. 37–54. Springer, Berlin (2012)
Gao, D.Y. Sherali, H.D.: Canonical duality: connection between nonconvex mechanics and global optimization. In: Advances in Applied Mathematics and Global Optimization, pp. 249–316. Springer, Berlin (2009)
Gao, D.Y., Strang, G.: Geometric nonlinearity: potential energy, complementary energy, and the gap function. Quart. Appl. Math. 47(3), 487–504 (1989)
Gao, D.Y., Watson, L.T., Easterling, D.R., Thacker, W.I., Billups, S.C.: Solving the canonical dual of box- and integer-constrained nonconvex quadratic programs via a deterministic direct search algorithm. Optim. Methods Softw. 28(2), 313–326 (2013). doi:10.1080/10556788.2011.641125
Gao, D.Y. Wu, C.Z.: On the triality theory in global optimization (2010). arXiv:1104.2970v1
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)
Gao, D.Y., Yang, W.-H.: Multi-duality in minimal surface type problems. Stud. Appl. Math. 95, 127–146 (1995)
Gao, D.Y., Yu, H.F.: Multi-scale modelling and canonical dual finite element method in phase transitions of solids. Int. J. Solids Struct. 45, 3660–3673 (2008)
Holzapfel, G.A.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley, New York (2000). ISBN 978-0471823193
Holzapfel, G.A., Ogden, R.W. (eds.): Mechanics of Biological Tissue, vol. 12. Springer, Heidelberg (2006)
Landau, L.D., Lifshitz, E.M.: Mechanics, vol. 1, 3rd edn. Butterworth-Heinemann, London (1976)
Li, S.F., Gupta, A.: On dual configuration forces. J. Elast. 84, 13–31 (2006)
Moreau, J.J.: La notion de sur-potentiel et les liaisons unilatérales en élastostatique. C. R. Acad. Sc. Paris 267A, 954–957 (1968)
Ogden, R.W.: Non-Linear Elastic Deformations, 1st edn, p. 544. Dover Publications, New York (1997)
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)
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). doi:10.1016/j.ijnonlinmec.2011.05.012
Sewell, M.J.: Maximum and Minimum Principles: A Unified Approach, with Applications. Cambridge University Press, Cambridge (1987)
Silva, D.M.M., Gao, D.Y.: Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in \({\mathbb{R}^n}\) (2011). arXiv:1110.0285v1
Strang, G.: Introduction to Applied Mathematics. Wellesley-Cambridge Press, Wellesley (1986)
Strugariu, R., Voisei, M.D., Zalinescu, C.: Counter-examples in bi-duality, triality and tri-duality. Discrete and Continuous Dynamical Systems - Series A, vol. 31, No. 4, December 2011
Tonti, E.: On the mathematical structure of a large class of physical theories, pp. 49–56. Accad. Naz. dei Lincei, Serie VIII, LII (1972)
Voisei, M.D., Zalinescu, C.: Some remarks concerning Gao-Strang’s complementary gap function. Appl. Anal. (2010). doi:10.1080/00036811.2010.483427
Voisei, M.D., Zalinescu, C.: Counterexamples to some triality and tri-duality results. J. Global Optim. 49, 173–183 (2011)
Wang, Z.B., Fang, S.C., Gao, D.Y., and Xing, W.X.: Canonical dual approach to solving the maximum cut problem, to appear in, J. Glob. Optim
Yau, S.-T., Gao, D.Y.: Obstacle problem for von Karman equations. Adv. Appl. Math. 13, 123–141 (1992)
Zhang, J., Gao, D.Y., Yearwood, J.: A novel canonical dual computational approach for prion AGAAAAGA amyloid fibril molecular modeling. J. Theor. Biol. 284, 149–157 (2011). doi:10.1016/j.jtbi.2011.06.024
Acknowledgements
The main results of this paper were announced at the 2nd World Congress of Global Optimization, July 3–7, 2011, Chania, Greece. The paper was posted online on April 15, 2011 at https://arXiv.org/abs/1104.2970. The authors are gratefully indebted with Professor Hanif Sherali at Virginia Tech for his detailed remarks and important suggestions. This paper has benefited from three anonymous referees’ constructive comments. David Gao’s research is supported by US Air Force Office of Scientific Research under the grants FA2386-16-1-4082 and FA9550-17-1-0151.
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Gao, D.Y., Wu, C. (2017). Triality Theory for General Unconstrained 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_6
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