State transition algorithm

Part of a series on the
Evolutionary algorithm
Artificial development Cellular evolutionary algorithm Cultural algorithm Differential evolution Effective fitness Evolution strategy Gaussian adaptation Evolutionary multimodal optimization Grammatical evolution Particle swarm optimization Memetic algorithm Natural evolution strategy Promoter based genetic algorithm Spiral optimization algorithm
Genetic algorithm
Chromosome Clonal selection algorithm Crossover Mutation Genetic memory Genetic fuzzy systems Selection Fly algorithm
Genetic programming
Cartesian genetic programming Linear genetic programming Multi expression programming Schema Eurisko Parity benchmark

In global optimization, a state transition algorithm (STA) is an iterative method that generates a sequence of continually improved approximations to provide a solution to an optimization problem. It was first proposed by Zhou, et al.

State transitions

STA is a stochastic global optimization method that aims to find a solution in a reasonable amount of time. In the context of the STA, a solution to an optimization problem is regarded as a state, and updating a solution can be regarded as a state transition.

Using the state–space representation, STA uniformly describes solution updates, the execution operators to update solutions being expressed as state transition matrices, which make STA easy to understand and flexible to implement:

$\mathbf {x} _{k+1}=A_{k}\mathbf {x} _{k}+B_{k}\mathbf {u} _{k}$

$\mathbf {y} _{k+1}=f(\mathbf {x} _{k+1})$

where:

$\mathbf {x} _{k}$ stands for a current state, corresponding to a solution to an optimization problem;

$\mathbf {u} _{k}$ is a function of $\mathbf {x} _{k}$ and historical states;

$\mathbf {y} _{k}$ is the fitness value at $\mathbf {x} _{k}$;

$\mathbf {A} _{k},\mathbf {B} _{k}$ are state transformation matrices, which can be considered as execution operators;

$f(\cdot )$ is the objective function or evaluation function.

As a stochastic global optimization method, STA has the following properties:

• globality – the ability to search the whole space;
• optimality – can guarantee an optimal solution;
• convergence – the sequence generated converges;
• rapidity – reduces computational complexity;
• controllability – the search space can be flexibly controlled.

Continuous state transition algorithm (CSTA)

In continuous STA, $\mathbf {x} _{k}\in \mathbb {R} ^{n}$ is a continuous variable that four special state transformation operators act on to generate candidate solutions.

State transformation operators

(1) Rotation transformation operator (RT) is defined as

$\mathbf {x} _{k+1}=\mathbf {x} _{k}+\alpha {\frac {1}{n\|\mathbf {x} _{k}\|_{2}}}R_{r}\mathbf {x} _{k}$

where $\alpha$ is a positive constant, called the rotation factor; $R_{r}\in \mathbb {R} ^{n\times n}$ is a random matrix containing uniformly distributed random variables defined on the interval [-1,1]; and $\|\cdot \|$ is the 2-norm of a vector.

This operator can search a hypersphere with a maximum radius of $\alpha$ ($\|\mathbf {x} _{k+1}-\mathbf {x} _{k}\|_{2}\leq \alpha$).

(2) Translation transformation operator (TT) is defined as

$\mathbf {x} _{k+1}=\mathbf {x} _{k}+\beta R_{t}{\frac {\mathbf {x} _{k}-\mathbf {x} _{k-1}}{\|\mathbf {x} _{k}-\mathbf {x} _{k-1}\|_{2}}}$

where $\beta$ is a positive constant, called the translation factor, and $R_{t}\in \mathbb {R}$ is a uniformly distributed random variable defined on the interval [0,1].

This operator can search along a line from $\mathbf {x} _{k-1}$ to $\mathbf {x} _{k}$ from the starting point $\mathbf {x} _{k}$ with a maximum length of $\beta$.

(3) Expansion transformation operator (ET) is defined as

$\mathbf {x} _{k+1}=\mathbf {x} _{k}+\gamma R_{e}\mathbf {x} _{k}$

where $\gamma$ is a positive constant, called the expansion factor, and $R_{e}\in \mathbb {R} ^{n\times n}$ is a random diagonal matrix whose entries obey the Gaussian distribution.

This operator can expand the entries in $\mathbf {x} _{k}$ to $[-\infty ,+\infty ]$, searching the entire space.

(4) Axesion transformation operator (AT) is defined as

$\mathbf {x} _{k+1}=\mathbf {x} _{k}+\delta R_{a}\mathbf {x} _{k}$

where $\delta$ is a positive constant, called the axesion factor, and $R_{a}\in \mathbb {R} ^{n\times n}$ is a random diagonal matrix whose entries obey the Gaussian distribution and with only one random position having a nonzero value.

This operator searches along the axes.

Regular neighbourhood and sampling

For a given variable $\mathbf {x} _{k}$, a candidate solution $\mathbf {x} _{k+1}$ is generated using one of the aforementioned state transformation operators. Since the state transition matrix for each state transformation is random, the generated candidate solution is not unique. It is not difficult to imagine that, for a particular candidate solution, a "regular neighbourhood" will be automatically formed when using certain state transformation operators.

Since, for any solution, the entries in a state transition matrix obey certain stochastic distributions, 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 state transformations together (called the degree of search enforcement, $SE$ for short), based on the given variable, are performed by a certain state transformation operator, yielding $SE$ 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 an evaluation function denoted as $newBest$.

Then, an update strategy using the "greedy criterion" is used to update the incumbent best solution:

${\text{Best}}={\text{newBest}}$, if $f({\text{newBest}})<f({\text{Best}})$

${\text{Best}}={\text{Best}}$, otherwise

Algorithm procedure of the basic continuous STA

Using the state transformation operators, sampling technique, and update strategy, the basic continuous STA can be described as follows:

Step 1: Assume a random incumbent $Best$ solution; and set $\alpha =\alpha _{\max }=1,\alpha _{\min }=10^{-4},$ $\beta =1,\gamma =1,\delta =1,fc=2,k=0$.

Step 2: Generate $SE$ samples based on incumbent $Best$, using expansion transformation (ET), and then update the incumbent $Best$ using the greedy criterion incorporating $SE$ samples and incumbent $Best$. Let us denote $newBest$ the best solution in the $SE$ samples. If $f(newBest)<f(Best)$, then perform translation transformation (TT) to update the incumbent $Best$.

Step 3: Generate $SE$ samples based on incumbent $Best$ using rotation transformation (RT), and then update the incumbent $Best$ the using greedy criterion incorporating $SE$ samples and incumbent $Best$. If $f(newBest)<f(Best)$, then perform translation transformation (TT) to update the incumbent $Best$.

Step 4: Generate $SE$ samples based on incumbent $Best$ using axesion transformation (AT), and then update the incumbent $Best$ using the greedy criterion incorporating $SE$ samples and incumbent $Best$. If $f(newBest)<f(Best)$, then perform translation transformation (TT) to update the incumbent $Best$.

Step 5: Set $k=k+1$. If $\alpha <\alpha _{\min }$, set $\alpha =\alpha _{\max }$, else set $\alpha =\alpha /fc$, and return to Step 2 until the maximum number of iterations is done.

Philosophy behind the continuous STA

• Expansion transformation contributes to globality by searching the entire space;
• Rotation transformation benefits optimality, since when $\alpha$ is sufficiently small, the incumbent best solution becomes a local optimal solution;
• The update strategy, based on the greedy criterion, contributes to convergence: the sequence $\{f({\text{Best}}_{k})_{k=1}^{\infty }\}$ is convergent due to $f({\text{Best}}_{k+1})\leq f({\text{Best}}_{k})$ and the monotone convergence theorem;
• The sampling technique, which obviates the need for complete enumeration, and the alternate use of state transformation operators help to reduce computational complexity;
• Parameters such as $\alpha ,\beta ,\gamma ,\delta$ make for easy control of the search space.

Applications of STA

STA has found a variety of applications, such as image segmentation, task assignment, energy consumption in the alumina evaporation process, resolution of overlapping linear sweeps of voltametric peaks, PID controller design, volterra series identification, and system modeling; and it is shown that STA is comparable with most existing global optimization methods.

External links

• A Matlab Toolbox for Continuous STA
• A Continuous STA for Nonlinear System Identification
• A Discrete STA for Traveling Salesman Problem

This article "State transition algorithm" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:State transition algorithm. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

📰 Article(s) of the same category(ies)

• Meiburg's paradox
• Garment sewing data
• Electimize
• INFORMS Transactions on Education
• Danish Operations Research Association
• PDFO
• Similarity-based-TOPSIS

---