app.py

import numpy as np
import math
import matplotlib.pyplot as plt 
from matplotlib.colors import LogNorm
from io import BytesIO
from PIL import Image
import gradio as gr
from mpl_toolkits.mplot3d import Axes3D

class Objective:
    def Evaluate(self, p):
        return -5.0*np.exp(-0.5*((p[0]+2.2)**2/0.4+(p[1]-4.3)**2/0.4)) + -2.0*np.exp(-0.5*((p[0]-2.2)**2/0.4+(p[1]+4.3)**2/0.4))

# Create an instance of the Objective class
obj = Objective()

# Evaluate the fitness of a position
position = np.array([-2.2, 4.3])
fitness = obj.Evaluate(position)

print(f"The fitness of the position {position} is {fitness}")

class Bounds:
    def __init__(self, lower, upper, enforce="clip"):
        self.lower = np.array(lower)
        self.upper = np.array(upper)
        self.enforce = enforce.lower()
    
    def Upper(self): 
        return self.upper
    
    def Lower(self): 
        return self.lower
    
    def Limits(self, pos): 
        npart, ndim = pos.shape 
        for i in range(npart):
            for j in range(ndim):
                if pos[i, j] < self.lower[j]:
                    if self.enforce == "clip":
                        pos[i, j] = self.lower[j]
                    elif self.enforce == "resample":
                        pos[i, j] = self.lower[j] + np.random.random() * (self.upper[j] - self.lower[j])
                elif pos[i, j] > self.upper[j]:
                    if self.enforce == "clip":
                        pos[i, j] = self.upper[j]
                    elif self.enforce == "resample":
                        pos[i, j] = self.lower[j] + np.random.random() * (self.upper[j] - self.lower[j])
            pos[i] = self.Validate(pos[i]) 
        return pos
    
    def Validate(self, pos):
        return pos

# Define the bounds
lower_bounds = [-6, -6]
upper_bounds = [6, 6]

# Create an instance of the Bounds class
bounds = Bounds(lower_bounds, upper_bounds, enforce="clip")

# Define a set of positions
positions = np.array([[15, 15], [-15, -15], [5, 15], [15, 5]])

# Enforce the bounds on the positions
valid_positions = bounds.Limits(positions)

print(f"Valid positions: {valid_positions}")

# Define the bounds
lower_bounds = [-6, -6]
upper_bounds = [6, 6]

# Create an instance of the Bounds class
bounds = Bounds(lower_bounds, upper_bounds, enforce="resample")

# Define a set of positions
positions = np.array([[15, 15], [-15, -15], [5, 15], [15, 5]])

# Enforce the bounds on the positions
valid_positions = bounds.Limits(positions)

print(f"Valid positions: {valid_positions}")

class QuasiRandomInitializer:
    def __init__(self, npart=10, ndim=2, bounds=None, k=1, jitter=0.0):
        self.npart = npart
        self.ndim = ndim
        self.bounds = bounds
        self.k = k
        self.jitter = jitter
        self.primes = [
            2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
            101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197,
            199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313,
            317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439,
            443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571,
            577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659
        ]

    def Halton(self, i, b):
        f = 1.0
        r = 0
        while i > 0:
            f = f / b
            r = r + f * (i % b)
            i = math.floor(i / b)
        return r

    def InitializeSwarm(self):
        self.swarm = np.zeros((self.npart, self.ndim))
        lo = np.zeros(self.ndim)
        hi = np.ones(self.ndim)
        if self.bounds is not None:
            lo = self.bounds.Lower()
            hi = self.bounds.Upper()

        for i in range(self.npart):
            for j in range(self.ndim):
                h = self.Halton(i + self.k, self.primes[j % len(self.primes)])
                q = self.jitter * (np.random.random() - 0.5)
                self.swarm[i, j] = lo[j] + (hi[j] - lo[j]) * h + q

        if self.bounds is not None:
            self.swarm = self.bounds.Limits(self.swarm)

        return self.swarm

# Define the bounds
lower_bounds = [-6, -6]
upper_bounds = [6, 6]
bounds = Bounds(lower_bounds, upper_bounds, enforce="clip")

# Create an instance of the QuasiRandomInitializer class
init = QuasiRandomInitializer(npart=50, ndim=2, bounds=bounds)

# Initialize the swarm
swarm_positions = init.InitializeSwarm()

print(f"Initial swarm positions: {swarm_positions}")

# Define the bounds
lower_bounds = [-6, -6]
upper_bounds = [6, 6]
bounds = Bounds(lower_bounds, upper_bounds, enforce="resample")

# Create an instance of the QuasiRandomInitializer class
init = QuasiRandomInitializer(npart=50, ndim=2, bounds=bounds)

# Initialize the swarm
swarm_positions = init.InitializeSwarm()

print(f"Initial swarm positions: {swarm_positions}")

class GWO:
    def __init__(self, obj, eta=2.0, npart=10, ndim=2, max_iter=200,tol=None,init=None,done=None,bounds=None):   
            self.obj = obj
            self.npart = npart
            self.ndim = ndim
            self.max_iter = max_iter
            self.init = init
            self.done = done
            self.bounds = bounds
            self.tol = tol
            self.eta = eta
            self.initialized = False
        
    def Initialize(self):
        """Set up the swarm"""

        self.initialized = True
        self.iterations = 0
       
        self.pos = self.init.InitializeSwarm()  # initial swarm positions
        self.vpos= np.zeros(self.npart)
        for i in range(self.npart):
            self.vpos[i] = self.obj.Evaluate(self.pos[i])
            
        # Initialize the list to store positions at each iteration
        self.all_positions = []
        self.all_positions.append(self.pos.copy())  # Store the initial positions

        #  Swarm bests
        self.gidx = []
        self.gbest = []
        self.gpos = []
        self.giter = []
        idx = np.argmin(self.vpos)
        self.gidx.append(idx)
        self.gbest.append(self.vpos[idx])
        self.gpos.append(self.pos[idx].copy())
        self.giter.append(0)

        #  1st, 2nd, and 3rd best positions
        idx = np.argsort(self.vpos)
        self.alpha = self.pos[idx[0]].copy()
        self.valpha= self.vpos[idx[0]]
        self.beta  = self.pos[idx[1]].copy()
        self.vbeta = self.vpos[idx[1]]
        self.delta = self.pos[idx[2]].copy()
        self.vdelta= self.vpos[idx[2]]

    # *** Gradio app method optimize created [leveraged vis-a-vis optimize function on the outside of the underlying anatomy of GWO class] ***
    def optimize(self):
        """
        Run a full optimization and return the best positions and fitness values.
        This method is designed to be used with Gradio.
        """
        # Initialize the swarm
        self.Initialize()

        # Lists to store the best positions and fitness values at each step
        best_positions = []
        best_fitness = []

        # Main loop
        while not self.Done():
            self.Step()  # Perform an optimization step
            # Update best_positions and best_fitness with the current best values
            best_positions.append(self.gbest[-1])
            best_fitness.append(self.gpos[-1])

        # Convert the list of best positions to a NumPy array
        best_positions_array = np.array(best_positions)
        np.save('best_positions_array', best_positions_array)

       

        # Print the best positions and fitness found
        print("Best Positions:", best_positions)
        print("Best Fitness:", best_fitness)

        # Return the best positions and fitness after the optimization
        return best_positions, best_fitness

    def Step(self):
        """Do one swarm step"""

        #  a from eta ... zero (default eta is 2)
        a = self.eta - self.eta*(self.iterations/self.max_iter)

        #  Update everyone
        for i in range(self.npart):
            A = 2*a*np.random.random(self.ndim) - a
            C = 2*np.random.random(self.ndim)
            Dalpha = np.abs(C*self.alpha - self.pos[i]) 
            X1 = self.alpha - A*Dalpha

            A = 2*a*np.random.random(self.ndim) - a
            C = 2*np.random.random(self.ndim)
            Dbeta = np.abs(C*self.beta - self.pos[i]) 
            X2 = self.beta - A*Dbeta

            A = 2*a*np.random.random(self.ndim) - a
            C = 2*np.random.random(self.ndim)
            Ddelta = np.abs(C*self.delta - self.pos[i]) 
            X3 = self.delta - A*Ddelta 
            
            self.pos[i,:] = (X1+X2+X3) / 3.0

        #  Keep in bounds
        if (self.bounds != None):
            self.pos = self.bounds.Limits(self.pos)

        #  Get objective function values and check for new leaders
        for i in range(self.npart):
            self.vpos[i] = self.obj.Evaluate(self.pos[i])

            #  new alpha?
            if (self.vpos[i] < self.valpha):
                self.vdelta = self.vbeta
                self.delta = self.beta.copy()
                self.vbeta = self.valpha
                self.beta = self.alpha.copy()
                self.valpha = self.vpos[i]
                self.alpha = self.pos[i].copy()

            #  new beta?
            if (self.vpos[i] > self.valpha) and (self.vpos[i] < self.vbeta):
                self.vdelta = self.vbeta
                self.delta = self.beta.copy()
                self.vbeta = self.vpos[i]
                self.beta = self.pos[i].copy()
            
            #  new delta?
            if (self.vpos[i] > self.valpha) and (self.vpos[i] < self.vbeta) and (self.vpos[i] < self.vdelta):
                self.vdelta = self.vpos[i]
                self.delta = self.pos[i].copy()

            #  is alpha new swarm best?
            if (self.valpha < self.gbest[-1]):
                self.gidx.append(i)
                self.gbest.append(self.valpha)
                np.save('best_fitness.npy', np.array(self.gbest))
                self.gpos.append(self.alpha.copy())
                np.save('best_positions.npy', np.array(self.gpos))
                # Save the positions at the current iteration
                self.all_positions.append(self.pos.copy())
                np.save('all_positions.npy', np.array(self.all_positions), allow_pickle=True)
                self.giter.append(self.iterations)

        self.iterations += 1
    
    def Done(self):
        """Check if we are done"""

        if (self.done == None):
            if (self.tol == None):
                return (self.iterations == self.max_iter)
            else:
                return (self.gbest[-1] < self.tol) or (self.iterations == self.max_iter)
        else:
            return self.done.Done(self.gbest,
                        gpos=self.gpos,
                        pos=self.pos,
                        max_iter=self.max_iter,
                        iteration=self.iterations)
    
    def Evaluate(self, pos):
        p = np.zeros(self.npart) 
        for i in range(self.npart):
            p[i] = self.obj.Evaluate(pos[i]) 
        return p
    
    def Results(self):
        if (not self.initialized):
            return None 
        return {
            "npart": self.npart,
            "ndim": self.ndim,
            "max_iter": self.max_iter,
            "iterations": self.iterations,
            "tol": self.tol,
            "eta": self.eta,
            "gbest": self.gbest,
            "giter": self.giter,
            "gpos": self.gpos,
            "gidx": self.gidx,
            "pos": self.pos,
            "vpos": self.vpos
            }
    def plot_contour_and_wolves(self, wolf_positions):
         # Ensure wolf_positions is a 2D array
        best_positions_1D = np.load('best_positions.npy')

        wolf_positions = best_positions_1D
            
        # Define the objective function
        def objective_function(x, y):
            return -5.0*np.exp(-0.5*((x+2.2)**2/0.4+(y-4.3)**2/0.4)) + -2.0*np.exp(-0.5*((x-2.2)**2/0.4+(y+4.3)**2/0.4))

        # Determine the search space boundaries based on the wolf positions
        x_min, x_max = wolf_positions[:, 0].min() - 1, wolf_positions[:, 0].max() + 1
        y_min, y_max = wolf_positions[:, 1].min() - 1, wolf_positions[:, 1].max() + 1

        # Generate a grid of points within the determined search space
        x = np.linspace(x_min, x_max, 100)
        y = np.linspace(y_min, y_max, 100)
        X, Y = np.meshgrid(x, y)

        # Evaluate the objective function on the grid
        Z = objective_function(X, Y)

        # Plot the contour map
        plt.figure(figsize=(10, 8))
        contour = plt.contour(X, Y, Z, levels=20, cmap='magma')
        plt.colorbar(contour)

        # Plot the wolf positions
        plt.plot(wolf_positions[:, 0], wolf_positions[:, 1], 'ro', markersize=5, label='Wolves')

        # Set plot title and labels
        plt.title('Contour Map of Wolves Oscillating Over Search Space')
        plt.xlabel('x')
        plt.ylabel('y')
        plt.legend()

        # Set the x and y limits of the plot based on the determined search space
        plt.xlim(x_min, x_max)
        plt.ylim(y_min, y_max)

        # Convert the plot to a PIL Image and return it
        buf = BytesIO()
        plt.savefig(buf, format='png')
        buf.seek(0)
        plt.close()  # Close the figure to free up memory
        return Image.open(buf)
    
    #def Dispersion(self):
        #"""Calculate the dispersion of the swarm"""
        #x, y = self.gpos[:, 0], self.gpos[:, 1]
        #dx = x.max() - x.min()
        #dy = y.max() - y.min()
        #return (dx + dy) / 2.0

    #def Dispersion(self):
        # """Calculate the dispersion of the swarm"""
        #dispersion = np.std(self.gpos[:, 0]) + np.std(self.gpos[:, 1])
        #return dispersion

    def Dispersion(self):
        """Calculate the dispersion of the swarm"""
        # Ensure self.gpos is a NumPy array
        if not isinstance(self.gpos, np.ndarray):
            self.gpos = np.array(self.gpos)
        
        # Now self.gpos should be a NumPy array, so we can calculate the dispersion
        x, y = self.gpos[:, 0], self.gpos[:, 1]
        dx = x.max() - x.min()
        dy = y.max() - y.min()
        return (dx + dy) / 2.0

    def plot_dispersion_heatmap(self, x_range, y_range, resolution=100):
        # Create a grid of points within the specified range
        x = np.linspace(*x_range, resolution)
        y = np.linspace(*y_range, resolution)
        X, Y = np.meshgrid(x, y)
        positions = np.vstack([X.ravel(), Y.ravel()]).T

        # Calculate the dispersion for each position in the grid
        dispersion_values = np.array([self.Dispersion() for _ in positions])
        Z = dispersion_values.reshape(X.shape)

        # Plot the dispersion heatmap
        plt.figure(figsize=(10, 8))
        plt.pcolormesh(X, Y, Z, cmap='viridis')
        plt.colorbar(label='Dispersion')

        # Set plot title and labels
        plt.title('Dispersion Heatmap')
        plt.xlabel('x')
        plt.ylabel('y')

        # Convert the plot to a PIL Image and return it
        buf = BytesIO()
        plt.savefig(buf, format='png')
        buf.seek(0)
        plt.close()  # Close the figure to free up memory
        return Image.open(buf)

    def plot_dispersion(self):
        """Plot the dispersion over time"""
        # Assuming self.giter stores the iteration number at which each best position was found
        # and self.gbest stores the corresponding best fitness values
        dispersion_values = [self.Dispersion() for _ in range(self.max_iter)]

        plt.figure(figsize=(10, 6))
        plt.plot(range(self.max_iter), dispersion_values, label='Dispersion')
        plt.xlabel('Iteration')
        plt.ylabel('Dispersion')
        plt.title('Evolution of Dispersion')
        plt.legend()
        plt.show()

        # Convert the plot to a PIL Image and return it
        buf = BytesIO()
        plt.savefig(buf, format='png')
        buf.seek(0)
        plt.close()  # Close the figure to free up memory
        return Image.open(buf)

    #def plot_dispersion_heatmap(self, x_range, y_range, grid_size=100):
        #"""Plot a heatmap of dispersion over a grid of positions"""
        #x_values = np.linspace(x_range[0], x_range[1], grid_size)
        #y_values = np.linspace(y_range[0], y_range[1], grid_size)
        #X, Y = np.meshgrid(x_values, y_values)
        #positions = np.vstack([X.ravel(), Y.ravel()]).T

        # Calculate dispersion for each position in the grid
        #dispersion_values = np.array([self.Dispersion(pos) for pos in positions])
        #Z = dispersion_values.reshape(X.shape)

        #plt.figure(figsize=(10, 8))
        #plt.imshow(Z, extent=(x_range[0], x_range[1], y_range[0], y_range[1]), origin='lower', cmap='viridis')
        #plt.colorbar(label='Dispersion')
        #plt.title('Heatmap of Dispersion')
        #plt.xlabel('X')
        #plt.ylabel('Y')
        #plt.show()

        # Convert the plot to a PIL Image and return it
        #buf = BytesIO()
        #plt.savefig(buf, format='png')
        #buf.seek(0)
        #plt.close()  # Close the figure to free up memory
        #return Image.open(buf)

    def plot_positions(self, iterations=[0, 2, 8, 10]):
        """Plot the positions of the particles over the specified iterations"""
        # Load the all_positions data from the .npy file
        all_positions = np.load('all_positions.npy', allow_pickle=True)
        
        # Create a figure with the correct number of subplots
        num_iterations = len(iterations)
        fig, axs = plt.subplots(num_iterations, figsize=(6, 4 * num_iterations))
        
        # If there is only one subplot, make it an array to simplify the loop
        if num_iterations == 1:
            axs = [axs]
        
        # Iterate over the subplots and the specified iterations to plot
        for i, ax in enumerate(axs):
            # Plot the particles' positions at the specified iteration
            iteration = iterations[i]
            if iteration < len(all_positions):
                positions = all_positions[iteration]
                ax.scatter(positions[:, 0], positions[:, 1], c='r', alpha=0.5)

                # Set plot title and labels
                ax.set_title(f'Iteration {iteration}')
                ax.set_xlabel('X')
                ax.set_ylabel('Y')
            else:
                ax.axis('off')  # Hide the subplot if the iteration is out of range

        plt.tight_layout()

        # Save the plot to a BytesIO object
        buf = BytesIO()
        plt.savefig(buf, format='png')
        buf.seek(0)
        image = Image.open(buf)
        plt.close()

        return image

    def plot_3d_meshgrid_contour(self, obj):
        """Plot the 3D meshgrid with a 2D contour on the base"""
        # Define the range for the x and y axis
        x_range = np.linspace(self.bounds.Lower()[0], self.bounds.Upper()[0], 100)
        y_range = np.linspace(self.bounds.Lower()[1], self.bounds.Upper()[1], 100)

        # Create a grid of points
     
        X, Y = np.meshgrid(x_range, y_range)
        Z = np.zeros_like(X)

        # Evaluate the objective function on the grid
        for i in range(X.shape[0]):
            for j in range(X.shape[1]):
                Z[i, j] = obj.Evaluate(np.array([X[i, j], Y[i, j]]))

        # Create a 3D plot with subplots for each rotation
        fig, axs = plt.subplots(2, 2, figsize=(14, 10), subplot_kw={'projection': '3d'})
        plt.subplots_adjust(wspace=0.1, hspace=0.1)

        # Define the rotations for each subplot
        rotations = [(30, 45), (30, 135), (30, -45), (30, -135)]

        for i, ax in enumerate(axs.flatten()):
            # Plot the meshgrid
            ax.plot_surface(X, Y, Z, cmap='viridis', alpha=0.5)

            # Plot the contour
            ax.contour(X, Y, Z, levels=20, colors='k', alpha=0.5)

            # Plot the best positions found by the algorithm
            best_positions = np.array(self.gpos)
            ax.plot(best_positions[:, 0], best_positions[:, 1], self.gbest, 'g*', markersize=4)

            # Set labels and title
            ax.set_xlabel('X Position')
            ax.set_ylabel('Y Position')
            ax.set_zlabel('Z Position')
            ax.set_title(f'GWO Optimization Process in 3D - Rotation {i+1}')

            # Set the limits of the plot to match the bounds
            ax.set_xlim(self.bounds.Lower()[0], self.bounds.Upper()[0])
            ax.set_ylim(self.bounds.Lower()[1], self.bounds.Upper()[1])
            ax.set_zlim(np.min(Z), np.max(Z))

            # Change the background color to light blue
            ax.set_facecolor('lightblue')

            # Apply the rotation
            ax.view_init(*rotations[i])

        # Show the plot
        plt.show()

        # Save the plot to a BytesIO object
        buf = BytesIO()
        plt.savefig(buf, format='png')
        buf.seek(0)
        image = Image.open(buf)
        plt.close()

        return image



def optimize(npart, ndim, max_iter):
    # Initialize the GWO algorithm with the provided parameters
    gwo = GWO(obj=obj, npart=npart, ndim=ndim, max_iter=max_iter, init=init, bounds=bounds)
    
    best_positions, best_fitness= gwo.optimize()
    
    # Convert best_fitness and best_positions to NumPy arrays if necessary
    best_fitness_npy = np.array(best_fitness)
    best_positions_npy = np.array(best_positions)

    # Calculate dispersion
    dispersion = gwo.Dispersion()
    dispersion_text = f"Dispersion: {dispersion}"

    # Load best_positions_loaded_array
    best_positions_array =  np.load('best_positions_array.npy') 
    
    # Plot the contour_and_wolves 
    plot_contour_and_wolves = gwo.plot_contour_and_wolves(best_positions_array)

    # Plot the dispersion over time
    dispersion_plot = gwo.plot_dispersion()

    # Plot the dispersion heatmap
    dispersion_heatmap_plot = gwo.plot_dispersion_heatmap(x_range=(-6,6), y_range=(-6,6))

    # Plot the plot_positions 
    plot_positions = gwo.plot_positions(iterations=[0, 2, 8, 10])

    # Plot the plot_3d_meshgrid_contour
    plot_3d_meshgrid_contour = gwo.plot_3d_meshgrid_contour(obj=obj)

    # Format the output strings
    best_fitness_text = f"Best Fitness: {best_fitness_npy}"
    best_positions_text = f"Best Positions: {best_positions_npy}"

    # Return the images and text
    return plot_contour_and_wolves, dispersion_plot, dispersion_heatmap_plot, plot_positions, plot_3d_meshgrid_contour, best_fitness_text, best_positions_text, best_fitness_npy, best_positions_npy, dispersion_text




# Define the Gradio interface
iface = gr.Interface(
    fn=optimize,  # Pass the optimize function object
    inputs=[
        gr.components.Slider(10, 50, 50, step=1, label="Number of Wolves [Default = 50 Wolves]"),
        gr.components.Slider(2, 2, 2, step=1, label="Two-Dimensional Search Space"),
        gr.components.Slider(100, 200, 200, step=1, label="Maximum Iterations [Default = 200 Epochs]"),
    ],
    outputs=[
        gr.components.Image(type="pil", label="Contour Map of Wolves Oscillating Over Search Space"),
        gr.components.Image(type="pil", label="Dispersion Plot"),
        gr.components.Image(type="pil", label="Dispersion Heatmap Plot"),
        gr.components.Image(type="pil", label="Best Positions Plot"),
        gr.components.Image(type="pil", label="Plot 3D Meshgrid Contour"),
        gr.components.Textbox(label="Dispersion Values"),
        gr.components.Textbox(label="Best Positions"),
        gr.components.Textbox(label="Best Fitness")
        
    ],
    title="Grey Wolf Optimizer",
    description=r"""
    ## Grey Wolf Optimizer
    
    The Grey Wolf Optimizer (GWO) is a population-based metaheuristic optimization algorithm inspired by the social behavior of grey wolves in nature.
    
    The objective function to be optimized is given by the following formula:
    
    ```math
    f(p) = -5.0 \cdot \exp \left( -0.5 \cdot \left( \frac{(x+2.2)^2}{0.4} + \frac{(y-4.3)^2}{0.4} \right) \right) + -2.0 \cdot \exp \left( -0.5 \cdot \left( \frac{(x-2.2)^2}{0.4} + \frac{(y+4.3)^2}{0.4} \right) \right)
    ```
    Or in a more readable form:
    
    $$
    f(p) = -5.0 \cdot \exp \left( -0.5 \cdot \left( \frac{(x+2.2)^2}{0.4} + \frac{(y-4.3)^2}{0.4} \right) \right) + -2.0 \cdot \exp \left( -0.5 \cdot \left( \frac{(x-2.2)^2}{0.4} + \frac{(y+4.3)^2}{0.4} \right) \right)
    $$
    """,
    article="## Grey Wolf Optimizer\nThe Grey Wolf Optimizer (GWO) is a population-based metaheuristic optimization algorithm inspired by the social behavior of grey wolves in nature."
)

# Launch the Gradio interface
iface.launch()