Canonical Duality Theory: Connections between Nonconvex Mechanics and Global Optimization

2009 ◽  
pp. 257-326 ◽  
Author(s):  
David Y. Gao ◽  
Hanif D. Sherali
Keyword(s):  
Global Optimization ◽  
Duality Theory ◽  
Canonical Duality Theory ◽  
Canonical Duality

scholarly journals Global Optimal Trajectory in Chaos and NP-Hardness

2016 ◽  
Vol 26 (08) ◽  
pp. 1650142 ◽  
Author(s):  
Vittorio Latorre ◽  
David Yang Gao
Keyword(s):  
Dynamical Systems ◽  
Duality Theory ◽  
Optimization Problem ◽  
Nonlinear Dynamical Systems ◽  
Global Optimization Problem ◽  
Canonical Duality Theory ◽  
Nonlinear Dynamical ◽  
Canonical Duality ◽  
Global Optimal ◽  
Np Hardness

This paper presents an unconventional theory and method for solving general nonlinear dynamical systems. Instead of the direct iterative methods, the discretized nonlinear system is first formulated as a global optimization problem via the least squares method. A newly developed canonical duality theory shows that this nonconvex minimization problem can be solved deterministically in polynomial time if a global optimality condition is satisfied. The so-called pseudo-chaos produced by linear iterative methods are mainly due to the intrinsic numerical error accumulations. Otherwise, the global optimization problem could be NP-hard and the nonlinear system can be really chaotic. A conjecture is proposed, which reveals the connection between chaos in nonlinear dynamics and NP-hardness in computer science. The methodology and the conjecture are verified by applications to the well-known logistic equation, a forced memristive circuit and the Lorenz system. Computational results show that the canonical duality theory can be used to identify chaotic systems and to obtain realistic global optimal solutions in nonlinear dynamical systems. The method and results presented in this paper should bring some new insights into nonlinear dynamical systems and NP-hardness in computational complexity theory.

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