On Unified Modeling, Canonical Duality-Triality Theory, Challenges and Breakthrough in Optimization

Abstract

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.