Global Optimal Trajectory in Chaos and NP-Hardness

Abstract

This paper presents a new canonical duality methodology for solving general nonlinear dynamical systems. Instead of the conventional iterative methods, the discretized nonlinear system is first formulated as a global optimization problem via the least squares method. The 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 Runge-Kutta type of linear iterations 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.Comment: 10 pages, 22 figure

On topology optimization and canonical duality method

“…. , n. Thus, the density distribution ρ β can be analytically obtained by substituting ( 49) and ( 50) into (45). By the fact that lim β→∞ Π d β (ς) = Π d u (ς), there must exists a β c > 0 such that…”

“…Therefore, the canonical dual problem (P d kp ) has a unique solution in S + a . Correspondingly, the primal problem (P kp ) has a unique solution defined by either (41) or (45).…”

On modeling and complete solutions to general fixpoint problems in multi-scale systems with applications

“…Otherwise, the system could have chaotic solutions. The relation between chaos in nonlinear dynamical systems and NP-hardness in computer science is discovered recently [33].…”

“…Canonical duality-triality is a methodological theory which can be used not only for modeling complex systems within a unified framework, but also for solving real-world problems with a unified methodology. This theory was developed originally from Gao and Strang's work in nonconvex mechanics [25] and has been applied successfully for solving a large class of challenging problems in both nonconvex analysis/mechancis and global optimization, such as chaotic dynamics [33,36], phase transitions in solids [27], post-buckling of large deformed beam [1], nonconvex and discrete optimization [4,12,14,20,23,42]. A comprehensive review on this theory and breakthrough from recent challenges are given in [19].…”

On unified modeling, theory, and method for solving multi-scale global optimization problems

“…and a canonical primal-dual algorithm has been developed with successful applications for solving sensor network optimization problems [19] and chaotic dynamics [15].…”