Draft:State transition algorithm - Revision history

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2025-05-17T14:26:06Z

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2025-02-03T18:35:56Z

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← Previous revision		Revision as of 18:35, 3 February 2025
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	<ref name=Zhou2012State>		<ref name=Zhou2012State>
	{{cite journal\|last1=X.J.\|first1=Zhou\|last2=C.H.\|first2=Yang\|last3=W.H.\|first3=Gui\|title=State transition algorithm\|journal=Journal of Industrial and Management Optimization\|date=2012\|volume=8\|issue=4\|pages=1039–1056}}</ref>		{{cite journal\|last1=X.J.\|first1=Zhou\|last2=C.H.\|first2=Yang\|last3=W.H.\|first3=Gui\|title=State transition algorithm\|journal=Journal of Industrial and Management Optimization\|date=2012\|volume=8\|issue=4\|pages=1039–1056\|doi=10.3934/jimo.2012.8.1039 }}</ref>

	<ref name=Zhou2019statistical>		<ref name=Zhou2019statistical>
	{{cite journal\|last1=X.J.\|first1=Zhou\|last2=C.H.\|first2=Yang\|last3=W.H.\|first3=Gui\|title=A statistical study on parameter selection of operators in continuous state transition algorithm\|journal=IEEE Transactions on Cybernetics\|date=2019\|volume=49\|issue=10\|pages=3722–3730}}</ref>		{{cite journal\|last1=X.J.\|first1=Zhou\|last2=C.H.\|first2=Yang\|last3=W.H.\|first3=Gui\|title=A statistical study on parameter selection of operators in continuous state transition algorithm\|journal=IEEE Transactions on Cybernetics\|date=2019\|volume=49\|issue=10\|pages=3722–3730\|doi=10.1109/TCYB.2018.2850350 \|pmid=30028721 \|arxiv=1812.07812 }}</ref>

	<ref name=Zhou2014Nonlinear>		<ref name=Zhou2014Nonlinear>
	{{cite journal\|last1=X.J.\|first1=Zhou\|last2=C.H.\|first2=Yang\|last3=W.H.\|first3=Gui\|title=Nonlinear system identification and control using state transition algorithm\|journal=Applied Mathematics and Computation\|date=2014\|volume=226\|pages=169–179}}</ref>		{{cite journal\|last1=X.J.\|first1=Zhou\|last2=C.H.\|first2=Yang\|last3=W.H.\|first3=Gui\|title=Nonlinear system identification and control using state transition algorithm\|journal=Applied Mathematics and Computation\|date=2014\|volume=226\|pages=169–179\|doi=10.1016/j.amc.2013.09.055 \|arxiv=1206.0677 }}</ref>

	<ref name=Zhou2016Optimal>		<ref name=Zhou2016Optimal>
	{{cite journal\|last1=X. J.\|first1=Zhou\|last2=D.Y.\|first2=Gao\|last3=A.R.\|first3=Simpson\|title=Optimal design of water distribution networks by discrete state transition algorithm\|journal=Engineering Optimization\|date=2016\|volume=48\|issue=4\|pages=603–628}}</ref>		{{cite journal\|last1=X. J.\|first1=Zhou\|last2=D.Y.\|first2=Gao\|last3=A.R.\|first3=Simpson\|title=Optimal design of water distribution networks by discrete state transition algorithm\|journal=Engineering Optimization\|date=2016\|volume=48\|issue=4\|pages=603–628\|doi=10.1080/0305215X.2015.1025775 \|arxiv=1304.7622 \|bibcode=2016EnOp...48..603Z }}</ref>

	<ref name=Zhou2016Discrete>		<ref name=Zhou2016Discrete>
	{{cite journal\|last1=X. J.\|first1=Zhou\|last2 = D.Y.\|first2=Gao\|last3=C.H.\|first3=Yang\|last4=W.H.\|first4=Gui\|title=Discrete state transition algorithm for unconstrained integer optimization problems\|journal=Neurocomputing\|date=2016\|volume=173\|pages=864–874}}</ref>		{{cite journal\|last1=X. J.\|first1=Zhou\|last2 = D.Y.\|first2=Gao\|last3=C.H.\|first3=Yang\|last4=W.H.\|first4=Gui\|title=Discrete state transition algorithm for unconstrained integer optimization problems\|journal=Neurocomputing\|date=2016\|volume=173\|pages=864–874\|doi=10.1016/j.neucom.2015.08.041 \|hdl=1959.17/100331 }}</ref>

	<ref name=Friedland2005Control>		<ref name=Friedland2005Control>
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	<ref name=Han2017new>		<ref name=Han2017new>
	{{cite journal\|last1=J.\|first1=Han\|last2=C.H.\|first2=Yang\|last3=X.J.\|first3=Zhou\|last4=W.H.\|first4=Gui\|title=A new multi-threshold image segmentation approach using state transition algorithm\|journal=Applied Mathematical Modelling\|date=2017\|volume=44\|pages=588–601}}</ref>		{{cite journal\|last1=J.\|first1=Han\|last2=C.H.\|first2=Yang\|last3=X.J.\|first3=Zhou\|last4=W.H.\|first4=Gui\|title=A new multi-threshold image segmentation approach using state transition algorithm\|journal=Applied Mathematical Modelling\|date=2017\|volume=44\|pages=588–601\|doi=10.1016/j.apm.2017.02.015 }}</ref>

	<ref name=Wang2016new>		<ref name=Wang2016new>
	{{cite journal\|last1=C.\|first1=Wang\|last2=H.L.\|first2=Zhang\|last3=W.H.\|first3=Fan\|last4=X.C.\|first4=Fan\|title=A new wind power prediction method based on chaotic theory and Bernstein Neural Network\|journal=Energy\|date=2016\|volume=117\|pages=259–271}}</ref>		{{cite journal\|last1=C.\|first1=Wang\|last2=H.L.\|first2=Zhang\|last3=W.H.\|first3=Fan\|last4=X.C.\|first4=Fan\|title=A new wind power prediction method based on chaotic theory and Bernstein Neural Network\|journal=Energy\|date=2016\|volume=117\|pages=259–271\|doi=10.1016/j.energy.2016.10.041 \|bibcode=2016Ene...117..259W }}</ref>

	<ref name=Wang2020wind>		<ref name=Wang2020wind>
	{{cite journal\|last1=C.\|first1=Wang\|last2=H.L.\|first2=Zhang\|last3=P.\|first3=Ma\|title=Wind power forecasting based on singular spectrum analysis and a new hybrid Laguerre neural network\|journal=Applied Energy\|date=2020\|volume=259\|pages=114–139}}</ref>		{{cite journal\|last1=C.\|first1=Wang\|last2=H.L.\|first2=Zhang\|last3=P.\|first3=Ma\|title=Wind power forecasting based on singular spectrum analysis and a new hybrid Laguerre neural network\|journal=Applied Energy\|date=2020\|volume=259\|pages=114–139\|doi=10.1016/j.apenergy.2019.114139 \|bibcode=2020ApEn..25914139W }}</ref>


	<ref name=Wang2016Optimization>		<ref name=Wang2016Optimization>
	{{cite journal\|last1=Y.L.\|first1=Wang\|last2=H.M.\|first2=He\|last3=X.J.\|first3=Zhou\|last4=C.H.\|first4=Yang\|last5=Y.F.\|first5=Xie\|title=Optimization of both operating costs and energy efficiency in the alumina evaporation process by a multi-objective state transition algorithm\|journal=Canadian Journal of Chemical Engineering\|volume=94\|pages=53–65}}</ref>		{{cite journal\|last1=Y.L.\|first1=Wang\|last2=H.M.\|first2=He\|last3=X.J.\|first3=Zhou\|last4=C.H.\|first4=Yang\|last5=Y.F.\|first5=Xie\|title=Optimization of both operating costs and energy efficiency in the alumina evaporation process by a multi-objective state transition algorithm\|journal=Canadian Journal of Chemical Engineering\|date=2016 \|volume=94\|pages=53–65\|doi=10.1002/cjce.22353 }}</ref>

	<ref name = Wang2016State>		<ref name = Wang2016State>
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	<ref name = zhou2019dynamic>		<ref name = zhou2019dynamic>
	{{cite journal\|last1=X.J.\|first1=Zhou\|last2=M.\|first2=Huang\|last3=T.W.\|first3=Huang\|last4=C.H.\|first4=Yang\|last5=W.H.\|first5=Gui\|title=Dynamic optimization for copper removal process with continuous production constraints\|journal=IEEE Transactions on Industrial Informatics\|date=2020\|volume=16\|pages=104870–104883}}}</ref>		{{cite journal\|last1=X.J.\|first1=Zhou\|last2=M.\|first2=Huang\|last3=T.W.\|first3=Huang\|last4=C.H.\|first4=Yang\|last5=W.H.\|first5=Gui\|title=Dynamic optimization for copper removal process with continuous production constraints\|journal=IEEE Transactions on Industrial Informatics\|date=2020\|volume=16\|issue=12 \|pages=104870–104883\|doi=10.1109/TII.2019.2943500 }}}</ref>
	}}		}}

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2025-02-02T16:59:54Z

Altered pages. Add: bibcode, pmid, doi, date, authors 1-1. Removed parameters. Formatted dashes. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | Suggested by Eastmain | Category:AfC pending submissions by age/0 days ago | #UCB_Category 40/86

← Previous revision		Revision as of 16:59, 2 February 2025
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	<ref name=Zhou2019statistical>		<ref name=Zhou2019statistical>
	{{cite journal\|last1=X.J.\|first1=Zhou\|last2=C.H.\|first2=Yang\|last3=W.H.\|first3=Gui\|title=A statistical study on parameter selection of operators in continuous state transition algorithm\|journal=IEEE Transactions on Cybernetics\|date=2019\|volume=49\|issue=10\|pages=~~3722--3730~~}}</ref>		{{cite journal\|last1=X.J.\|first1=Zhou\|last2=C.H.\|first2=Yang\|last3=W.H.\|first3=Gui\|title=A statistical study on parameter selection of operators in continuous state transition algorithm\|journal=IEEE Transactions on Cybernetics\|date=2019\|volume=49\|issue=10\|pages=3722–3730}}</ref>

	<ref name=Zhou2014Nonlinear>		<ref name=Zhou2014Nonlinear>
	{{cite journal\|~~last~~=X.J.\|~~first~~=Zhou\|last2=C.H.\|first2=Yang\|last3=W.H.\|first3=Gui\|title=Nonlinear system identification and control using state transition algorithm\|journal=Applied Mathematics and Computation\|date=2014\|volume=226\|pages=169–179}}</ref>		{{cite journal\|last1=X.J.\|first1=Zhou\|last2=C.H.\|first2=Yang\|last3=W.H.\|first3=Gui\|title=Nonlinear system identification and control using state transition algorithm\|journal=Applied Mathematics and Computation\|date=2014\|volume=226\|pages=169–179}}</ref>

	<ref name=Zhou2016Optimal>		<ref name=Zhou2016Optimal>
Line 107:		Line 107:

	<ref name=Han2017new>		<ref name=Han2017new>
	{{cite journal\|last1=J.\|first1=Han\|last2=C.H.\|first2=Yang\|last3=X.J.\|first3=Zhou\|last4=W.H.\|first4=Gui\|title=A new multi-threshold image segmentation approach using state transition algorithm\|journal=Applied Mathematical Modelling\|date=2017\|volume=44\|pages=~~588--601~~}}</ref>		{{cite journal\|last1=J.\|first1=Han\|last2=C.H.\|first2=Yang\|last3=X.J.\|first3=Zhou\|last4=W.H.\|first4=Gui\|title=A new multi-threshold image segmentation approach using state transition algorithm\|journal=Applied Mathematical Modelling\|date=2017\|volume=44\|pages=588–601}}</ref>

	<ref name=Wang2016new>		<ref name=Wang2016new>
	{{cite journal\|last1=C.\|first1=Wang\|last2=H.L.\|first2=Zhang\|last3=W.H.\|first3=Fan\|last4=X.C.\|first4=Fan\|title=A new wind power prediction method based on chaotic theory and Bernstein Neural Network\|journal=Energy\|date=2016\|volume=117\|pages=~~259 - 271~~}}</ref>		{{cite journal\|last1=C.\|first1=Wang\|last2=H.L.\|first2=Zhang\|last3=W.H.\|first3=Fan\|last4=X.C.\|first4=Fan\|title=A new wind power prediction method based on chaotic theory and Bernstein Neural Network\|journal=Energy\|date=2016\|volume=117\|pages=259–271}}</ref>

	<ref name=Wang2020wind>		<ref name=Wang2020wind>
	{{cite journal\|last1=C.\|first1=Wang\|last2=H.L.\|first2=Zhang\|last3=P.\|first3=Ma\|title=Wind power forecasting based on singular spectrum analysis and a new hybrid Laguerre neural network\|journal=Applied Energy\|date=2020\|volume=259\|pages=~~114 - 139~~}}</ref>		{{cite journal\|last1=C.\|first1=Wang\|last2=H.L.\|first2=Zhang\|last3=P.\|first3=Ma\|title=Wind power forecasting based on singular spectrum analysis and a new hybrid Laguerre neural network\|journal=Applied Energy\|date=2020\|volume=259\|pages=114–139}}</ref>


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	<ref name = Wang2016State>		<ref name = Wang2016State>
	{{cite journal\|last1=G.W.\|first1=Wang\|last2=C.H.\|first2=Yang\|last3=H.Q.\|first3=Zhu\|last4=Y.G.\|first4=Li\|last5=X.W.\|first5=Peng\|last6=W.H.\|first6=Gui\|title=State-transition-algorithm-based resolution for overlapping linear sweep voltammetric peaks with high signal ratio\|journal=Chemometrics and Intelligent Laboratory Systems\|volume=151\|pages=61–70}}</ref>		{{cite journal\|last1=G.W.\|first1=Wang\|last2=C.H.\|first2=Yang\|last3=H.Q.\|first3=Zhu\|last4=Y.G.\|first4=Li\|last5=X.W.\|first5=Peng\|last6=W.H.\|first6=Gui\|title=State-transition-algorithm-based resolution for overlapping linear sweep voltammetric peaks with high signal ratio\|journal=Chemometrics and Intelligent Laboratory Systems\|date=2016 \|volume=151\|pages=61–70\|doi=10.1016/j.chemolab.2015.12.008 }}</ref>

	<ref name=Zhang2018fractional>		<ref name=Zhang2018fractional>
	{{cite journal\|last1=F.X.\|first1=Zhang\|last2=C.H.\|first2=Yang\|last3=X.J.\|first3=Zhou\|last4=W.H.\|first4=Gui\|title=Fractional-order PID controller tuning using continuous state transition algorithm\|journal=Neural Computing and Applications\|date=2018\|volume=29\|number=10\|pages=~~795~~ - ~~804~~}}</ref>		{{cite journal\|last1=F.X.\|first1=Zhang\|last2=C.H.\|first2=Yang\|last3=X.J.\|first3=Zhou\|last4=W.H.\|first4=Gui\|title=Fractional-order PID controller tuning using continuous state transition algorithm\|journal=Neural Computing and Applications\|date=2018\|volume=29\|number=10\|pages=795–804\|doi=10.1007/s00521-016-2605-0 }}</ref>


	<ref name=Zhang2019optimal>		<ref name=Zhang2019optimal>
	{{cite journal\|last1=F.X.\|first1=Zhang\|last2=C.H.\|first2=Yang\|last3=X.J.\|first3=Zhou\|last4=W.H.\|first4=Gui\|title=Optimal setting and control strategy for industrial process based on discrete-time fractional-order PID\|journal=IEEE Access\|date=2019\|volume=7\|pages=~~47747~~ ~~- 47761~~}}</ref>		{{cite journal\|last1=F.X.\|first1=Zhang\|last2=C.H.\|first2=Yang\|last3=X.J.\|first3=Zhou\|last4=W.H.\|first4=Gui\|title=Optimal setting and control strategy for industrial process based on discrete-time fractional-order PID\|journal=IEEE Access\|date=2019\|volume=7\|pages=47747–47761\|doi=10.1109/ACCESS.2019.2909816 }}</ref>

	<ref name=Huang2018hybrid>		<ref name=Huang2018hybrid>
	{{cite journal\|last1=Z.K.\|first1=Huang\|last2=C.H.\|first2=Yang\|last3=X.J.\|first3=Zhou\|last4=T.W.\|first4=Huang\|title=A hybrid feature selection method based on binary state transition algorithm and relieff\|journal=IEEE Journal of Biomedical and Health Informatics\|date=2018\|volume=23\|number=5\|pages=~~1888~~ - ~~1898~~}}</ref>		{{cite journal\|last1=Z.K.\|first1=Huang\|last2=C.H.\|first2=Yang\|last3=X.J.\|first3=Zhou\|last4=T.W.\|first4=Huang\|title=A hybrid feature selection method based on binary state transition algorithm and relieff\|journal=IEEE Journal of Biomedical and Health Informatics\|date=2018\|volume=23\|number=5\|pages=1888–1898\|doi=10.1109/JBHI.2018.2872811 \|pmid=30281502 }}</ref>


	<ref name = Xie2016new>		<ref name = Xie2016new>
	{{cite journal\|last1=Y.\|first1=Xie\|last2=S.\|first2=Wei\|last3=X.\|first3=Wang\|last4=S.\|first4=Xie\|last5=C.\|first5=Yang\|title=A new prediction model based on the leaching rate kinetics in the alumina digestion process\|journal=Hydrometallurgy\|date=2016\|volume=164\|pages=7–14}}</ref>		{{cite journal\|last1=Y.\|first1=Xie\|last2=S.\|first2=Wei\|last3=X.\|first3=Wang\|last4=S.\|first4=Xie\|last5=C.\|first5=Yang\|title=A new prediction model based on the leaching rate kinetics in the alumina digestion process\|journal=Hydrometallurgy\|date=2016\|volume=164\|pages=7–14\|doi=10.1016/j.hydromet.2016.05.005 \|bibcode=2016HydMe.164....7X }}</ref>

	<ref name = Liu2020sta>		<ref name = Liu2020sta>
	{{cite journal\|last1=J.P.\|first1=Liu\|last2=C.R.\|first2=Jiang\|last3=J.Z.\|first3=He\|last4=Z.H.\|first4=Tang\|last5=Y.F.\|first5=Xie\|title=STA-APSNFIS: STA-optimized adaptive pre-sparse neuro-fuzzy inference system for online soft sensor modeling\|journal=IEEE Access\|date=2020\|volume=8\|pages=~~7255 - 7263~~}}</ref>		{{cite journal\|last1=J.P.\|first1=Liu\|last2=C.R.\|first2=Jiang\|last3=J.Z.\|first3=He\|last4=Z.H.\|first4=Tang\|last5=Y.F.\|first5=Xie\|title=STA-APSNFIS: STA-optimized adaptive pre-sparse neuro-fuzzy inference system for online soft sensor modeling\|journal=IEEE Access\|date=2020\|volume=8\|pages=7255–7263}}</ref>

	<ref name = zhou2019dynamic>		<ref name = zhou2019dynamic>
	{{cite journal\|last1=X.J.\|first1=Zhou\|last2=M.\|first2=Huang\|last3=T.W.\|first3=Huang\|last4=C.H.\|first4=Yang\|last5=W.H.\|first5=Gui\|title=Dynamic optimization for copper removal process with continuous production constraints\|journal=IEEE Transactions on Industrial Informatics\|date=2020\|volume=16\|pages=~~104870 - 104883~~}}}</ref>		{{cite journal\|last1=X.J.\|first1=Zhou\|last2=M.\|first2=Huang\|last3=T.W.\|first3=Huang\|last4=C.H.\|first4=Yang\|last5=W.H.\|first5=Gui\|title=Dynamic optimization for copper removal process with continuous production constraints\|journal=IEEE Transactions on Industrial Informatics\|date=2020\|volume=16\|pages=104870–104883}}}</ref>
	}}		}}

Tiezhongyu2010: Submitting using AfC-submit-wizard

2025-02-02T01:50:20Z

Submitting using AfC-submit-wizard

← Previous revision		Revision as of 01:50, 2 February 2025
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			{{Short description\|State transition algorithm for global optimization}}
			{{Draft topics\|stem}}
			{{AfC topic\|stem}}
			{{AfC submission\|\|\|ts=20250202015020\|u=Tiezhongyu2010\|ns=118}}
	{{AfC submission\|t\|\|ts=20250129120237\|u=Tiezhongyu2010\|ns=118\|demo=}}<!-- Important, do not remove this line before article has been created. -->		{{AfC submission\|t\|\|ts=20250129120237\|u=Tiezhongyu2010\|ns=118\|demo=}}<!-- Important, do not remove this line before article has been created. -->

Tiezhongyu2010: ←Created page with '{{subst:AfC submission/draftnew}} In global optimization, '''state transition algorithm (STA)''' is an iterative method that generates a sequence of improving approximate solutions for an optimization problem. Due to its intrinsic properties, STA has the ability to find a global optimal solution in probability and can guarantee an optimal solution. State transition algorit...'

2025-01-29T12:03:11Z

←Created page with '{{subst:AfC submission/draftnew}} In global optimization, '''state transition algorithm (STA)''' is an iterative method that generates a sequence of improving approximate solutions for an optimization problem. Due to its intrinsic properties, STA has the ability to find a global optimal solution in probability and can guarantee an optimal solution. State transition algorit...'

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{{AfC submission|t||ts=20250129120237|u=Tiezhongyu2010|ns=118|demo=}}

In global optimization, '''state transition algorithm (STA)''' is an [[iterative method]] that generates a sequence of improving approximate solutions for an [[optimization problem]]. Due to its intrinsic properties, STA has the ability to find a global optimal solution in probability and can guarantee an optimal solution.

State transition algorithm <ref name=Zhou2012State/><ref name=Zhou2019statistical/><ref name=Zhou2014Nonlinear/><ref name=Zhou2016Optimal/><ref name=Zhou2016Discrete/> was firstly proposed by Zhou et al, and it is a stochastic global optimization method and aims to find a possible global or approximate optimal solution in a reasonable amount of time. In STA, a solution to an optimization problem is regarded as a state, and an update of a solution can be regarded as a state transition. Using the state-space representation,<ref name=Friedland2005Control/> in STA, it describes solutions updating in a unified framework, and the execution operators to update solutions are expressed as state transition matrices, which make it easy to understand and flexible to implement:
: <math> \mathbf{x}_{k+1} = A_k \mathbf{x}_k + B_k \mathbf{u}_k</math>
: <math> \mathbf{y}_{k+1} = f(\mathbf{x}_{k+1})</math>
where:
: <math> \mathbf{x}_k </math> stands for a current state, corresponding to a solution to an optimization problem;
: <math> \mathbf{u}_k </math> is a function of <math> \mathbf{x}_{k} </math> and historical states;
: <math> \mathbf{y}_k </math> is the fitness value at <math> \mathbf{x}_{k} </math>;
: <math> \mathbf{A}_k, \mathbf{B}_k </math> are state transformation matrices, which can be considered as execution operators;
: <math> f(\cdot) </math> is the objective function or evaluation function.
As a stochastic global optimization method, STA has the following properties:
* globality, STA has the ability to search the whole space;
* optimality, STA can guarantee to find an optimal solution;
* convergence, the sequence generated by STA is convergent;
* rapidity, inherent advantages existing in STA to reduce the computational complexity;
* controllability, STA can control the search space flexibly.

== Continuous state transition algorithm (CSTA) ==
In continuous STA, <math> \mathbf{x}_k \in \mathbb{R}^n </math> is a continuous variable, and four special state transformation operators are designed to generate new candidate solutions.

=== State transformation operators ===
(1) Rotation transformation (RT)
: <math> \mathbf{x}_{k+1} = \mathbf{x}_k + \alpha \frac{1}{n\|\mathbf{x}_k\|_2} R_r \mathbf{x}_k </math>
where <math> \alpha </math> is a positive constant, called the rotation factor, <math> R_r \in \mathbb{R}^{n \times n} </math> is a random matrix with its entries being uniformly distributed random variables defined on the interval [-1,1], and <math> \|\cdot\| </math> is the 2-norm of a vector.
The rotation transformation has the functionality to search in a hypersphere with maximal radius <math> \alpha </math> , that is to say, <math> \|\mathbf{x}_{k+1} - \mathbf{x}_{k}\|_2 \leq \alpha </math>.

(2) Translation transformation (TT)
: <math> \mathbf{x}_{k+1} = \mathbf{x}_k + \beta R_t \frac{\mathbf{x}_k - \mathbf{x}_{k-1}}{\|\mathbf{x}_k - \mathbf{x}_{k-1}\|_2} </math>
where <math> \beta </math> is a positive constant, called the translation factor, and <math> R_t \in \mathbb{R} </math>
is a uniformly distributed random variable defined on the interval [0,1]. The translation transformation has the functionality to search along a line from <math> \mathbf{x}_{k-1} </math> to <math> \mathbf{x}_k </math> at the starting point <math> \mathbf{x}_k </math> with maximal length <math> \beta </math>.

(3) Expansion transformation (ET)
: <math> \mathbf{x}_{k+1} = \mathbf{x}_k + \gamma R_e \mathbf{x}_k </math>
where <math> \gamma </math> is a positive constant, called the expansion factor, and <math> R_e \in \mathbb{R}^{n \times n} </math> is a random diagonal matrix with its entries obeying the Gaussian distribution. The expansion transformation has the functionality to expand the entries in <math> \mathbf{x}_k </math> to the range of <math> [-\infty, +\infty] </math>, searching in the whole space.

(4) Axesion transformation (AT)
: <math> \mathbf{x}_{k+1} = \mathbf{x}_k + \delta R_a \mathbf{x}_k </math>
where <math> \delta </math> is a positive constant, called the axesion factor, and <math> R_a \in \mathbb{R}^{n \times n} </math> is a random diagonal matrix with its entries obeying the Gaussian distribution and with only one random position having nonzero value. The axesion transformation aims to search along the axes.

=== Regular neighbourhood and sampling ===
For a given solution <math> \mathbf{x}_k </math>, a candidate solution <math> \mathbf{x}_{k+1} </math> is generated by using one time of the aforementioned state transformation operators. Since the state transition matrix in each state transformation is random, the generated candidate solution is not unique. Based on a given point, it is not difficult to imagine that a '''regular neighbourhood''' will be automatically formed when using certain state transformation operators.

Since the entries in state transition matrix obey certain stochastic distribution, for any given solution, the new candidate becomes a random vector and its corresponding solution (the value of a random vector) can be regarded as a '''sample'''. Considering that any two random state transition matrices in each state transformation operator are independent,
several times of state transformation (called the degree of search enforcement, <math> SE </math> for short) based on the given solution are performed for certain state transformation operator, yielding <math> SE </math> samples.

=== An update strategy ===
As mentioned above, based on the incumbent best solution, a total number of ''SE'' candidate solutions are sampled. A new best solution is selected from the candidate set by virtue of the evaluation function, denoted as <math> newBest </math>.
Then, an update strategy based on '''greedy criterion''' is used to update the incumbent best solution:

: <math> \text{Best} = \text{newBest} </math>, if <math> f(\text{newBest}) < f(\text{Best}), </math>

: <math> \text{Best} = \text{Best} </math> , otherwise

=== Algorithm procedure of the basic continuous STA ===
With the state transformation operators, sampling technique and update strategy, the basic continuous STA can be described as follows:

Step 1: Initiate a random solution <math> Best </math> and set <math> \alpha = \alpha_{\max} = 1, \alpha_{\min} = 10^{-4}, </math> <math> \beta = 1, \gamma = 1, \delta = 1, fc = 2, k = 0; </math>

Step 2: Generate <math> SE </math> samples based on incumbent <math> Best </math> using '''Expansion Transformation''', and then update the incumbent <math> Best </math> using greedy criterion incorporating <math> SE </math> samples and incumbent <math> Best </math> . Let us denote <math> newBest </math> the best solution in <math> SE </math> samples, if <math> f(newBest) < f(Best) </math>, then perform the '''Translation Transformation''' similarly to update the incumbent <math> Best </math>;

Step 3: Generate <math> SE </math> samples based on incumbent <math> Best </math> using '''Rotation Transformation''', and then update the incumbent <math> Best </math> using greedy criterion incorporating <math> SE </math> samples and incumbent <math> Best </math> . If <math> f(newBest) < f(Best) </math>, then perform the '''Translation Transformation''' similarly to update the incumbent <math> Best </math>;

Step 4: Generate <math> SE </math> samples based on incumbent <math> Best </math> using '''Axesion Transformation''', and then update the incumbent <math> Best </math> using greedy criterion incorporating <math> SE </math> samples and incumbent <math> Best </math> . If <math> f(newBest) < f(Best) </math>, then perform the '''Translation Transformation''' similarly to update the incumbent <math> Best </math>;

Step 5: set <math> k = k + 1 </math>, if <math> \alpha < \alpha_{\min} </math>, set <math> \alpha = \alpha_{\max} </math>, else set <math> \alpha = \alpha /fc </math>, and return to Step 2 until the maximum of iterations is met.

=== Philosophy behind the continuous STA ===
* The '''expansion transformation''' contributes to the globality since it has the functionality to search the whole space;
* The '''rotation transformation''' benefits the optimality since when <math> \alpha </math> is sufficiently small, the incumbent best solution becomes a local optimal solution;
* The update strategy based on '''greedy criterion''' contributes to the convergence, that is to say, the sequence <math> \{f(\text{Best}_k)_{k=1}^\infty \}</math> is convergent due to <math> f(\text{Best}_{k+1}) \leq f(\text{Best}_k) </math> and the [[monotone convergence theorem]];
* The '''sampling technique''' (it can avoid complete enumeration) and the '''alternate use''' of state transformation operators help to reduce computational complexity;
* The parameters like <math> \alpha, \beta, \gamma, \delta </math> can be adjusted to control the search space.

==Applications of STA==

STA has found a variety of applications, like image segmentation,<ref name=Han2017new/>
wind power prediction,<ref name=Wang2016new/> <ref name=Wang2020wind/> energy consumption in the alumina evaporation process,<ref name=Wang2016Optimization/> resolution of overlapping linear
sweep voltammetric peaks,<ref name = Wang2016State/> PID controller design,<ref name = Zhang2018fractional/><ref name = Zhang2019optimal/> feature selection,<ref name = Huang2018hybrid/>, system modeling,<ref name = Xie2016new/> <ref name=Liu2020sta/> and dynamic optimization <ref name = zhou2019dynamic/>and it is shown that STA is comparable to most existing global optimization methods.

==References==
{{Reflist|refs=

<ref name=Zhou2012State>
{{cite journal|last1=X.J.|first1=Zhou|last2=C.H.|first2=Yang|last3=W.H.|first3=Gui|title=State transition algorithm|journal=Journal of Industrial and Management Optimization|date=2012|volume=8|issue=4|pages=1039–1056}}</ref>

<ref name=Zhou2019statistical>
{{cite journal|last1=X.J.|first1=Zhou|last2=C.H.|first2=Yang|last3=W.H.|first3=Gui|title=A statistical study on parameter selection of operators in continuous state transition algorithm|journal=IEEE Transactions on Cybernetics|date=2019|volume=49|issue=10|pages=3722--3730}}</ref>

<ref name=Zhou2014Nonlinear>
{{cite journal|last=X.J.|first=Zhou|last2=C.H.|first2=Yang|last3=W.H.|first3=Gui|title=Nonlinear system identification and control using state transition algorithm|journal=Applied Mathematics and Computation|date=2014|volume=226|pages=169–179}}</ref>

<ref name=Zhou2016Optimal>
{{cite journal|last1=X. J.|first1=Zhou|last2=D.Y.|first2=Gao|last3=A.R.|first3=Simpson|title=Optimal design of water distribution networks by discrete state transition algorithm|journal=Engineering Optimization|date=2016|volume=48|issue=4|pages=603–628}}</ref>

<ref name=Zhou2016Discrete>
{{cite journal|last1=X. J.|first1=Zhou|last2 = D.Y.|first2=Gao|last3=C.H.|first3=Yang|last4=W.H.|first4=Gui|title=Discrete state transition algorithm for unconstrained integer optimization problems|journal=Neurocomputing|date=2016|volume=173|pages=864–874}}</ref>

<ref name=Friedland2005Control>
{{cite book|last1=Friedland |first1=Bernard |year=2005| title=Control System Design: An Introduction to State-Space Methods |publisher=Dover |isbn=0-486-44278-0}}</ref>

<ref name=Han2017new>
{{cite journal|last1=J.|first1=Han|last2=C.H.|first2=Yang|last3=X.J.|first3=Zhou|last4=W.H.|first4=Gui|title=A new multi-threshold image segmentation approach using state transition algorithm|journal=Applied Mathematical Modelling|date=2017|volume=44|pages=588--601}}</ref>

<ref name=Wang2016new>
{{cite journal|last1=C.|first1=Wang|last2=H.L.|first2=Zhang|last3=W.H.|first3=Fan|last4=X.C.|first4=Fan|title=A new wind power prediction method based on chaotic theory and Bernstein Neural Network|journal=Energy|date=2016|volume=117|pages=259 - 271}}</ref>

<ref name=Wang2020wind>
{{cite journal|last1=C.|first1=Wang|last2=H.L.|first2=Zhang|last3=P.|first3=Ma|title=Wind power forecasting based on singular spectrum analysis and a new hybrid Laguerre neural network|journal=Applied Energy|date=2020|volume=259|pages=114 - 139}}</ref>

<ref name=Wang2016Optimization>
{{cite journal|last1=Y.L.|first1=Wang|last2=H.M.|first2=He|last3=X.J.|first3=Zhou|last4=C.H.|first4=Yang|last5=Y.F.|first5=Xie|title=Optimization of both operating costs and energy efficiency in the alumina evaporation process by a multi-objective state transition algorithm|journal=Canadian Journal of Chemical Engineering|volume=94|pages=53–65}}</ref>

<ref name = Wang2016State>
{{cite journal|last1=G.W.|first1=Wang|last2=C.H.|first2=Yang|last3=H.Q.|first3=Zhu|last4=Y.G.|first4=Li|last5=X.W.|first5=Peng|last6=W.H.|first6=Gui|title=State-transition-algorithm-based resolution for overlapping linear sweep voltammetric peaks with high signal ratio|journal=Chemometrics and Intelligent Laboratory Systems|volume=151|pages=61–70}}</ref>

<ref name=Zhang2018fractional>
{{cite journal|last1=F.X.|first1=Zhang|last2=C.H.|first2=Yang|last3=X.J.|first3=Zhou|last4=W.H.|first4=Gui|title=Fractional-order PID controller tuning using continuous state transition algorithm|journal=Neural Computing and Applications|date=2018|volume=29|number=10|pages=795 - 804}}</ref>

<ref name=Zhang2019optimal>
{{cite journal|last1=F.X.|first1=Zhang|last2=C.H.|first2=Yang|last3=X.J.|first3=Zhou|last4=W.H.|first4=Gui|title=Optimal setting and control strategy for industrial process based on discrete-time fractional-order PID|journal=IEEE Access|date=2019|volume=7|pages=47747 - 47761}}</ref>

<ref name=Huang2018hybrid>
{{cite journal|last1=Z.K.|first1=Huang|last2=C.H.|first2=Yang|last3=X.J.|first3=Zhou|last4=T.W.|first4=Huang|title=A hybrid feature selection method based on binary state transition algorithm and relieff|journal=IEEE Journal of Biomedical and Health Informatics|date=2018|volume=23|number=5|pages=1888 - 1898}}</ref>

<ref name = Xie2016new>
{{cite journal|last1=Y.|first1=Xie|last2=S.|first2=Wei|last3=X.|first3=Wang|last4=S.|first4=Xie|last5=C.|first5=Yang|title=A new prediction model based on the leaching rate kinetics in the alumina digestion process|journal=Hydrometallurgy|date=2016|volume=164|pages=7–14}}</ref>

<ref name = Liu2020sta>
{{cite journal|last1=J.P.|first1=Liu|last2=C.R.|first2=Jiang|last3=J.Z.|first3=He|last4=Z.H.|first4=Tang|last5=Y.F.|first5=Xie|title=STA-APSNFIS: STA-optimized adaptive pre-sparse neuro-fuzzy inference system for online soft sensor modeling|journal=IEEE Access|date=2020|volume=8|pages=7255 - 7263}}</ref>

<ref name = zhou2019dynamic>
{{cite journal|last1=X.J.|first1=Zhou|last2=M.|first2=Huang|last3=T.W.|first3=Huang|last4=C.H.|first4=Yang|last5=W.H.|first5=Gui|title=Dynamic optimization for copper removal process with continuous production constraints|journal=IEEE Transactions on Industrial Informatics|date=2020|volume=16|pages=104870 - 104883}}}</ref>
}}

==External links==
* [http://www.mathworks.com/matlabcentral/fileexchange/52498-state-transition-algorithm A Matlab Toolbox for Continuous STA]
* [http://www.mathworks.com/matlabcentral/fileexchange/53785-state-transition-algorithm-for-nonlinear-system-identification, A Continuous STA for Nonlinear System Identification]
* [http://www.mathworks.com/matlabcentral/fileexchange/52499-discrete-state-transition-algorithm-for-traveling-salesman-problem, A Discrete STA for Traveling Salesman Problem]