Triality Theory for General Unconstrained Global Optimization Problems

  • David Yang Gao
  • Changzhi Wu
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 37)


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.



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 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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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculty of Science and TechnologyFederation University AustraliaBallaratAustralia

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