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Mathematics 2019, 7(4), 330; https://doi.org/10.3390/math7040330

Topology Structure Implied in β-Hilbert Space, Heisenberg Uncertainty Quantum Characteristics and Numerical Simulation of the DE Algorithm

1,2,† and 2,*,†
1
School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China
2
Ningxia Province Key Laboratory of Intelligent Information and Data Processing, Yinchuan 750021, China
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Received: 9 March 2019 / Revised: 30 March 2019 / Accepted: 1 April 2019 / Published: 4 April 2019
(This article belongs to the Special Issue Evolutionary Computation)
PDF [318 KB, uploaded 4 April 2019]

Abstract

The differential evolutionary ( D E ) algorithm is a global optimization algorithm. To explore the convergence implied in the H i l b e r t space with the parameter β of the D E algorithm and the quantum properties of the optimal point in the space, we establish a control convergent iterative form of a higher-order differential equation under the conditions of P - ε and analyze the control convergent properties of its iterative sequence; analyze the three topological structures implied in H i l b e r t space of the single-point topological structure, branch topological structure, and discrete topological structure; and establish and analyze the association between the H e i s e n b e r g uncertainty quantum characteristics depending on quantum physics and its topological structure implied in the β -Hilbert space of the D E algorithm as follows: The speed resolution Δ v 2 of the iterative sequence convergent speed and the position resolution Δ x β ε of the global optimal point with the swinging range are a pair of conjugate variables of the quantum states in β -Hilbert space about eigenvalues λ i R , corresponding to the uncertainty characteristics on quantum states, and they cannot simultaneously achieve bidirectional efficiency between convergent speed and the best point precision with any procedural improvements. Where λ i R is a constant in the β -Hilbert space. Finally, the conclusion is verified by the quantum numerical simulation of high-dimensional data. We get the following important quantitative conclusions by numerical simulation: except for several dead points and invalid points, under the condition of spatial dimension, the number of the population, mutated operator, crossover operator, and selected operator are generally decreasing or increasing with a variance deviation rate + 0 . 50 and the error of less than ± 0 . 5 ; correspondingly, speed changing rate of the individual iterative points and position changing rate of global optimal point β exhibit a inverse correlation in β -Hilbert space in the statistical perspectives, which illustrates the association between the H e i s e n b e r g uncertainty quantum characteristics and its topological structure implied in the β -Hilbert space of the D E algorithm.
Keywords: DE algorithm; β-Hilbert space; topology structure; quantum uncertainty property; numerical simulation DE algorithm; β-Hilbert space; topology structure; quantum uncertainty property; numerical simulation
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Wang, K.; Gao, Y. Topology Structure Implied in β-Hilbert Space, Heisenberg Uncertainty Quantum Characteristics and Numerical Simulation of the DE Algorithm. Mathematics 2019, 7, 330.

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