  A unified model is addressed for general optimization problems in multi-scale
complex systems. Based on necessary conditions and basic principles in physics,
the canonical duality-triality theory is presented in a precise way to include
traditional duality theories and popular methods as special applications. Two
conjectures on NP-hardness are discussed, which should play important roles for
correctly understanding and efficiently solving challenging real-world
problems. Applications are illustrated for both nonconvex continuous
optimization and mixed integer nonlinear programming. Misunderstandings and
confusion on some basic concepts, such as objectivity, nonlinearity,
Lagrangian, and Lagrange multiplier method are discussed and classified.
Breakthrough from recent false challenges by C. Z\u{a}linescu and his
co-workers are addressed. This paper will bridge a significant gap between
optimization and multi-disciplinary fields of applied math and computational
sciences.
