Efficient state transition algorithm （ESTA）

Efficient state transition algorithm with guaranteed optimality

https://github.com/tiezhongyu2005/ESTA

State transition algorithm (STA) has been emerging as a novel metaheuristic method for global optimization in recent years. In this version, effcient state transition algorithm under an automatic termination criterion with optimality guarantees is proposed.

# Effcient state transition algorithm (ESTA)

State transition algorithm (STA) is a new metaheuristic method, which is used for global optimization with box constraints without the derivative information.

$\min f(x)$

$\mathrm{s.t.} x_{lb} \leq x \leq x_{ub}$

In ESTA, we have four state tranformation operators, namely, rotation, translation, expansion and axesion.

Rotation Tranformation(RT)

$$ s_{k+1}=s_{k}+\alpha R_{r} \frac{u_{k}}{\|u_{k}\|_{2}} $$

where, $\alpha$ is a positive constant, called rotation factor; $R_{r}$ $\in$ $\Re$, is a uniformly distributed random variable defined over the interval [-1, 1] and $u_k \in \Re^n$ is a random vector whose entries are uniformly distributed random variables within the interval [-1, 1]. The rotation transformation has the function of searching in a hypersphere.

Translation Tranformation(RT)

$$ s_{k+1}=s_{k}+ \beta R_{t} (s_k - s_{k-1})$$

where, $\beta$ is a positive constant, called translation factor; $R_{t} \in \Re$ is a uniformly distributed random variable defined over the interval [-1,1]. $s_{k-1}$ is chosen randomly from the historical best archive.

Expansion Tranformation(RT)

$$s_{k+1} = s_{k}+ \gamma R_{e}s_{k}$$

where, $\gamma$ is a positive constant, called expansion factor; $R_{e} \in \Re^{n \times n}$ is a random diagonal matrix with its elements obeying the Gaussian distribution. It is also obvious to find the expansion transformation has the function of expanding the components in $s_{k}$ to the range of $[-\infty, +\infty]$, searching in the whole space.

Axesion Tranformation(RT)

$$s_{k+1} = s_{k}+ \delta R_{a}s_{k}$$

where, $\delta$ is a positive constant, called axesion factor; $R_{a}$ $\in \Re^{n \times n}$ is a random diagonal matrix with its entries obeying the Gaussian distribution and only one random position having nonzero value. The axesion transformation aims to search along the axes and strengthens single dimensional search.

An adaptive parameter control strategy is proposed as follows.

The following paper is used:

Zhou X, Yang C, Gui W, et al. Efficient state transition algorithm with guaranteed optimality[J]. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2026. DOI: 10.1109/TSMC.2026.3688256