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].
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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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