RIME Optimization Algorithm (RIME)

rime ice

RIME Optimization Algorithm

The RIME (Rime-Ice) optimization algorithm is a cutting-edge approach inspired by the natural formation of rime-ice, a unique frost that forms when supercooled water vapor in the atmosphere deposits and freezes on surfaces such as tree branches. This phenomenon results in two distinct forms of rime: soft-rime and hard-rime .

Soft-rime forms under conditions of light wind and variable environmental factors, resulting in a frost with a more random and diffuse structure. In contrast, hard-rime develops in high-wind environments with a consistent directional flow, leading to more structured and regular formations .

The RIME algorithm leverages these natural processes to develop a novel metaheuristic optimization technique. It mimics the soft-rime's random and comprehensive search ability to explore the solution space effectively and the hard-rime's directional growth to refine and exploit promising solutions. This innovative approach aims to address complex optimization problems by combining exploration with exploitation in a manner akin to the physical processes of rime-ice formation.

By drawing parallels between natural rime-ice growth and algorithmic optimization, the RIME algorithm offers a fresh perspective on solving optimization problems, making it a valuable tool in fields such as engineering design, machine learning, and logistics.

Mathematical Model

The growth process of rime ice is influenced by factors such as temperature, wind speed, and humidity. The RIME algorithm simulates the formation and growth of rime-ice by considering these factors in its mathematical model.

Rime Cluster Initialization

Each rime agent is treated as a search agent, with the rime population representing the algorithm's population. The rime population $R$ consists of $n$ rime agents $S_{i}$, each comprising $d$ rime particles $x_{ij}$. The rime population $R$ can be represented by:

$R={\begin{bmatrix}S_{1}\\S_{2}\\\vdots \\S_{i}\end{bmatrix}}\quad {\text{where}}\quad S_{i}=[x_{i1},x_{i2},\ldots ,x_{ij}]$

In this representation, $F(S_{i})$ denotes the fitness value of the rime-agent $S_{i}$ in the meta-heuristic algorithm.

Soft-Rime Search Strategy

In a breezy environment, soft-rime growth is highly random. This behavior is simulated by the following equation for the position update of each rime particle:

$R_{ij}^{new}=R_{best,j}+r_{1}\cdot \cos(\theta )\cdot \beta \cdot (h\cdot (Ub_{ij}-Lb_{ij})+Lb_{ij})$

where $R_{ij}^{new}$ is the new position of the updated particle, and $i$ and $j$ denote the $j$-th particle of the $i$-th rime-agent. $R_{best,j}$ is the $j$-th particle of the best rime-agent in the rime population $R$. The parameter $r_{1}$ is a random number in the range $(-1,1)$ and controls the direction of particle movement, while $\cos(\theta )$ adjusts the direction based on the number of iterations. $\beta$ and $h$ are environmental factors that vary with the number of iterations and control the distance between particles.

The angle $\theta$ is defined as:

$\theta ={\frac {\pi \cdot t}{10\cdot T}}$

where $t$ is the current number of iterations and $T$ is the maximum number of iterations. The environmental factor $\beta$ is given by:

$\beta =1-{\frac {(w\cdot t)}{T\cdot w}}$

where $w$ is used to control the step function of $\beta$, with a default value of 5. The condensation coefficient $E$ is defined as:

$E={\sqrt {\frac {t}{T}}}$

where $r_{2}$ is a random number in the range $[0,1]$ that, along with $E$, controls the probability of particle condensation.

Hard-Rime Puncture Mechanism

In strong gale conditions, hard-rime growth is regular and follows the wind direction. The position update for each particle is given by:

$R_{ij}^{new}=R_{best,j},\quad {\text{if }}r_{3}<F_{norm}(S_{i})$

where $R_{ij}^{new}$ is the new position of the updated particle, $R_{best,j}$ is the $j$-th particle of the best rime-agent in $R$, and $F_{norm}(S_{i})$ denotes the normalized fitness value of the current agent $S_{i}$. $r_{3}$ is a random number in the range $[0,1]$.

Positive Greedy Selection Mechanism

The positive greedy selection mechanism improves global exploration efficiency by comparing the updated fitness value of an agent with its previous value. If the updated value is better, both the fitness value and the agent are replaced. The pseudo-code for the positive greedy selection mechanism is:

Algorithm 3: Positive Greedy Selection Mechanism
Initialize the rime population R
Get the current optimal agent and optimal fitness
While t ≤ T
    For i = 1 to n
        If F(R_i^{new}) < F(R_i)
            F(R_i) = F(R_i^{new})
            R_i = R_i^{new}
            If F(R_i^{new}) < F(R_{best})
                F(R_{best}) = F(R_i^{new})
                R_{best} = R_i^{new}
            End If
        End If
    End For
    t = t + 1
End While

RIME Algorithm

The RIME algorithm combines the soft-rime search strategy, hard-rime puncture mechanism, and positive greedy selection mechanism to optimize search performance. The pseudo-code of the RIME algorithm is:

Algorithm 4: RIME Algorithm
Initialize the rime population R
Get the current optimal agent and optimal fitness
While t ≤ T
    Coefficient of adherence E = \sqrt{\frac{t}{T}}
    If r_2 < E
        Update rime agent location by the soft-rime search strategy
    End If
    If r_3 < Normalize fitness of S_i
        Cross updating between agents by the hard-rime puncture mechanism
    End If
    If F(R_i^{new}) < F(R_i)
        Select the optimal solution and replace the suboptimal solution using the positive greedy selection mechanism
    End If
    t = t + 1
End While