The canonical duality theory (CDT) is advertised by its creator, DY Gao, as being "a breakthrough methodological theory that can be used not only for modeling complex systems within a unified framework, but also for solving a large class of challenging problems in multidisciplinary fields of engineering, mathematics, and sciences".
Recently (2017) the book "Canonical Duality Theory. Unified Methodology for Multidisciplinary Study" was published by Springer in the series "Advances in Mechanics and Mathematics".
Because together with two collaborators we published some papers containing counter-examples to "triality" results spread in several papers I was interested to read this book in order to better understand CDT. By CDT one associates the so-called "extended Lagrangian" to a class of optimization problem; this "extended Lagrangian" is quadratic in the primal variable, and applications are given to quadratic optimization problems, too.
Our first step was to study quadratic problems by the method suggested by CDT and to compare the results with those obtained by DY Gao and his collaborators (see arXiv:1809.09032). The same program was followed for constrained optimization problems (see arXiv:1810.09009) and for unconstrained ones (see arXiv:1811.04469).
So, our aim is to present the main ideas of CDT, the main results which can be obtained using this "methodological theory, and some drawbacks in getting the existing results spread in the literature.
Personal website of Constantin Zălinescu