Decentralized Differential Evolutionary Algorithm for Large-Scale Networked Systems

Abstract

The optimization of a complex system with multiple subsystems is a tough problem. In this paper, a Decentralized differential evolutionary algorithm (DDEA) is proposed. The simulations for both DDEA and centralized DE on three benchmark functions are carried out. The numerical results show that DDEA is efficient to solve decentralized optimization problems. On these problems, the proposed DDEA outperforms centralized DE in convergence.

Appendix

In this appendix, the three benchmark problems are listed, followed by the corresponding decentralized problems on subsystems.

g01

$$ \mathop {\text{minimize}}\limits_{{{\mathbf{x}} \in {\mathbb{R}}^{10} }} f\left( {\mathbf{x}} \right) = 10d + \mathop \sum \limits_{i = 1}^{d} x_{i}^{2} - 10{ \cos }\left( {2\pi x_{i} } \right) $$

(13)

where $x_{i} \in \left[ { - 5.12,5.12} \right]$. ${\mathbf{x}}^{*} = 0$ and $f\left( {{\mathbf{x}}^{*} } \right) = 0$.

The problem of $i{\text{th}}$ subsystem is

$$ \mathop {\text{minimize}}\limits_{{x_{i} \in {\mathbb{R}}}} f(x_{i} |x_{i + 1} ) = 10 + x_{i}^{2} - 10{ \cos }\left( {2\pi x_{i + 1} } \right) $$

(14)

where $i = 1,2, \ldots ,10$, and $x_{11} = x_{1}$.

g02

The problem of $i{\text{th}}$ subsystem is

$$ \mathop {\text{minimize}}\limits_{{{\mathbf{x}} \in {\mathbb{R}}^{20} }} f\left( {\mathbf{x}} \right) = \mathop \sum \limits_{i = 1}^{d - 1} 100 (x_{i + 1} - x_{i}^{2} )^{2} + (x_{i} - 1)^{2} $$

(15)

where $x_{i} \in \left[ { - 2.048,2.048} \right]$, $x_{i}^{ *} = 1,\forall i$ and $f\left( {{\mathbf{x}}^{ *} } \right) = 0$.

The problem of $i{\text{th}}$ subsystem is

$$ \begin{array}{*{20}r} \hfill {\mathop {\text{minimize}}\limits_{{x_{i} \in {\mathbb{R}}}} f(x_{i} |x_{i + 1} ) = 100(x_{i + 1} - x_{i}^{2} )^{2} + (x_{i} - 1)^{2} } \\ \end{array} $$

(16)

where i = 1,2, …, 19.

g03

$$ \begin{array}{*{20}c} {\mathop {\text{minimize}}\limits_{{{\mathbf{x}} \in {\mathbb{R}}^{13} }} \quad f\left( {\mathbf{x}} \right) = 5\mathop \sum \limits_{i = 1}^{4} x_{i} - \mathop \sum \limits_{i = 1}^{4} x_{i}^{2} - \mathop \sum \limits_{i = 5}^{13} x_{i} } \\ {subject\,to\quad 2x_{1} + 2x_{2} + x_{10} + x_{11} - 10 \le 0} \\ {\quad \quad \quad \quad \quad 2x_{1} + 2x_{3} + x_{10} + x_{12} - 10 \le 0} \\ {\quad \quad \quad \quad \quad 2x_{2} + 2x_{3} + x_{11} + x_{12} - 10 \le 0} \\ {\quad \quad \quad \quad \quad - 8x_{1} + x_{10} \le 0} \\ {\quad \quad \quad \quad \quad - 8x_{2} + x_{11} \le 0} \\ {\quad \quad \quad \quad \quad - 8x_{3} + x_{12} \le 0} \\ {\quad \quad \quad \quad - 2x_{4} - x_{5} + x_{10} \le 0} \\ {\quad \quad \quad \quad - 2x_{6} - x_{7} + x_{11} \le 0} \\ {\quad \quad \quad \quad - 2x_{8} - x_{9} + x_{12} \le 0} \\ \end{array} $$

(17)

where $0 \le x_{i} \le 1\left( {i = 1, \ldots ,9} \right)$, $0 \le x_{i} \le 100\left( {i = 10,11,12} \right)$, and $0 \le x_{13} \le 1$. ${\mathbf{x}}^{ *} = \left( {1,1,1,1,1,1,1,1,1,3,3,3,1} \right)$ and $f\left( {{\mathbf{x}}^{ *} } \right) = - 15$.

For this problem, 13 subsystems have different forms of problems. For example, the problem of the first subsystem is

$$ \begin{array}{*{20}c} {\mathop {\text{minimize}}\limits_{{x_{1} \in {\mathbb{R}}}} } & {f\left( {x_{1} } \right) = 5x_{1} - x_{1}^{2} } \\ {subject\,to} & {g_{1} (x_{1} |x_{2} ,x_{10} ,x_{11} ) = 2x_{1} + 2x_{2} + x_{10} + x_{11} - 10 \le 0} \\ {} & {g_{2} (x_{1} |x_{3} ,x_{10} ,x_{12} ) = 2x_{1} + 2x_{3} + x_{10} + x_{12} - 10 \le 0} \\ {} & {g_{3} (x_{1} |x_{10} ) = - 8x_{1} + x_{10} \le 0} \\ \end{array} $$

(18)