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F_coulomb.py

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+import numpy as np
+
+def normaliazed_direction(ori, des):
+    dir_ = des-ori
+    nor_dir_ = dir_/np.sqrt(np.sum(dir_**2))
+    nor_dir_=np.nan_to_num(nor_dir_)
+    return nor_dir_
+
+def distance(ori, des):
+    dis_ = des-ori
+    return dis_
+
+def acc(F, m):
+    Npar=F.shape[0]
+    Nvar=F.shape[1]
+    # Acceleration Matrix
+    Ac = np.full((Npar, Nvar), 0.0)
+    for ind_r, row in enumerate(Ac):
+        for ind_c, col in enumerate(row):
+            Ac[ind_r][ind_c]=F[ind_r][ind_c]/m[ind_r]
+    return Ac
+
+def F_coulomb(x, v, q, P, minimize=True, k=0.0001, weight_method='Linear', t=1.0, max_q=100.0): 
+    """ The procedure that generates the new positions of the particles is based on how particles with positive and negative charges move
+        in electric fields. Thus, the procedure will be as follows:
+        
+        1. n particles are created at random positions within the search space.
+        2. Each particle is assigned a positive charge type.
+        3. The magnitude of each particle's charge is directly related to the objective function.
+        3.1 Assignment Methods -> Linear, Exponential, Quadratic (better assignment methods can be explored)
+        3.2 Simply put, a solution range is established and adjusted to a charge range between 0 to max_q according to some assignment method.
+        4. Each particle holds a velocity. These start at rest but will change over the iterations of the algorithm.
+        5. The P particles with the best values obtained from the iteration will remain at rest and emit an electric field of opposite sign (negative) attracting the rest.
+        5.1 Let E = k*Q/r^2 be the magnetic field at a point located at a distance r from the source.
+        5.2 Let Fe be the electric force between the magnetic field and the particle -> Fe = E*q0 where q0 is the particle's charge.
+        5.3 Let Fij be the force produced between particle i and j -> Fij = k*qi*qj/(r^2) where r is the distance between the particles, qi and qj are the charges of i and j.
+        5.4 Thus, the resultant forces of particle i are FRi = sum_{p=1}^P k*Qp*Qi/(rip^2) + sum_{j=P+1}^n k*Qi*Qj/(rij^2) 
+        5.5 Moreover, by Newton's Law, F=m*a -> Fe=m*a. Thus, the velocities of each particle are calculated vf=vi+a*t -> vf = vi + (FRi/m)*t
+        5.6 The positions are updated as follows x = vi*t + (a*t^2)/2 -> x = vi*t + (FRi/m)*(t^2)/2
+        
+        Finally, if the particles are too close, it could cause a problem. If rij^2 << 0, then the electric force would be too strong, leading
+        the particles to abruptly separate from each other. To avoid this behavior, a small distance (0.00000001) is established. If the distance
+        between two particles is smaller than that distance, it will be considered that both particles have collided. Then, if more than half of 
+        the particles have collided, we will apply a random function to generate new solutions over a reduced search space.
+    
+    Args:
+        x (array(size=(Npar, Nvar)): current particle positions
+        v (array(size=(Npar, Nvar)): current particle velocity
+        q (array(size=(Npar, 1))): current electric charge of each particle
+        P (int): quantity of electric fields (the best P particles become electric fields)
+        minimize (bool, optional): solver objective. Defaults to True.
+        k (float, optional): electric constant. Defaults to 0.0001.
+        weight_method (str, optional): method to reassign electric charges. Defaults to 'Linear'. Options=['Linear', 'Quadratic', 'Exponential'].
+        t (float, optional): time. Defaults to 1.0.
+        max_q (float, optional): upper bound of electric charge to assing. Defaults to 100.0.
+
+    Returns:
+        F_total, Ac, vf, new_pos: returns the force obtained on each particle, their acceleration, their velocities
+                                  and their new positions
+    """
+
+    Npar=x.shape[0] 
+    Nvar=x.shape[1]
+
+    # Distance Matrix
+    dis = []
+    for ind_r, row in enumerate(x):
+        for ind_c, col in enumerate(x):
+            dis.append(distance(x[ind_r], x[ind_c]))
+    dis=np.array(dis).reshape((Npar,Npar,Nvar))
+
+    # Direction Matrix
+    d = []
+    for ind_r, row in enumerate(x):
+        for ind_c, col in enumerate(x):
+            d.append(normaliazed_direction(x[ind_r], x[ind_c]))
+    d=np.array(d).reshape((Npar,Npar,Nvar))
+    
+    colisioned_part = []
+    r_2 = np.zeros((Npar, Npar))
+    for ind_r, row in enumerate(dis):
+        for ind_c, col in enumerate(dis):
+            value = dis[ind_r][ind_c]
+            value_2 = value**2
+            value_sum = np.sum(value_2)
+            if value_sum < 0.00000001: # Particles have practically collided. Notice later that fe=0.0 ahead.
+                r_2[ind_r][ind_c] = 0.0
+                if ind_r != ind_c:
+                    colisioned_part.append(ind_c)
+            else:
+                r_2[ind_r][ind_c] = value_sum
+    colisioned_part_ = np.unique(np.array(colisioned_part))
+
+
+    # Each particle is assigned an electric charge magnitude based on the solution provided by the particle.
+    q=np.array(q)
+    min_value = q[0]
+    max_value = q[-1]
+    if minimize:
+        q = -1.0*(q-max_value-1.0)
+        max_value = q[0]
+
+    if weight_method=="Linear":
+        # We adjust the charges according to the following range [0, max_q].
+        q = (q/max_value)*max_q
+        reverse_ind = [len(q)-i-1 for i in range(len(q))] # It is inverted to give more charge to the particles that are further away.
+        q = q[reverse_ind]
+        
+    elif weight_method=="Quadratic":
+        # We adjust the charges according to the following range [0, max_q].
+        q_2=q**2
+        q=(q_2/np.max(q_2))*max_q
+        reverse_ind = [len(q)-i-1 for i in range(len(q))] # It is inverted to give more charge to the particles that are further away.
+        q = q[reverse_ind]
+        
+    elif weight_method=="Exponential":
+        # We adjust the charges according to the following range [0, max_q].
+        q_exp=np.array([np.exp(i) for i in q])
+        q=(q_exp/np.max(q_exp))*max_q
+        reverse_ind = [len(q)-i-1 for i in range(len(q))] # It is inverted to give more charge to the particles that are further away.
+        q = q[reverse_ind]
+        
+    Npar=d.shape[0]
+    Nvar=d.shape[2]
+    F=np.full((Npar, Npar, Nvar), 0.0)
+    for ind_r, row in enumerate(dis):
+        for ind_c, col in enumerate(row):
+            # The magnitude of the electric force Fe is calculated.            
+            d_2=r_2[ind_r][ind_c]
+            if d_2==0:
+                Fe=0.0
+            else:
+                q1=q[ind_r]
+                q2=q[ind_c]
+                Fe=float(k*q1*q2/d_2)
+                if ind_r >= P and ind_c >= P: # Repulsive forces are generated between particles of the same sign.
+                    F[ind_r][ind_c]=-1.0*Fe*d[ind_r][ind_c]
+                else: # There is attraction between particles and electric fields.
+                    F[ind_r][ind_c]=Fe*d[ind_r][ind_c]
+    F[:P, :P] = 0.0
+    F_total = np.sum(F, axis=1)
+    F_total[:P]=0.0
+    # Remember that F=ma -> m1*a1 = m2*a2. So, if m1>m2 -> a2>a1 -> Particles with greater mass have less acceleration.
+    # For this method, the weight of the particle is not important, so we will set them all to be equal to 1.0.
+    m=np.ones(Npar)
+    # vf = acc + vi*t
+    Ac=acc(F_total, m)
+    vf = Ac + t*v
+    Ac[:P]=0.0
+    vf[:P]=0.0 # The velocity of the magnetic fields is set to 0.0.
+    # Finally, Xf = xi + vi*t + 0.5*acc*(t^2) (Final Position)
+    x__=v*t + 0.5*Ac*(t**2)
+    new_pos = x + x__
+    
+    # If more than half of the particles have collided, those that have collided will move randomly within the reduced search space.
+    max_crash_part=Npar/2
+    if len(colisioned_part_)>max_crash_part:
+        lo = np.min(new_pos, axis=0)
+        up = np.max(new_pos, axis=0)
+        for i in colisioned_part_:
+            new_pos[i] = np.random.uniform(low=lo, high=up, size=(1, Nvar))
+
+    return F_total, Ac, vf, new_pos
+

F_newton.py

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+import numpy as np
+
+def normaliazed_direction(ori, des):
+    dir_ = des-ori
+    nor_dir_ = dir_/np.sqrt(np.sum(dir_**2))
+    nor_dir_=np.nan_to_num(nor_dir_)
+    return nor_dir_
+
+def distance(ori, des):
+    dis_ = des-ori
+    return dis_
+
+def acc(F, m):
+    Npar=F.shape[0]
+    Nvar=F.shape[1]
+    # Matriz de aceleracion
+    Ac = np.full((Npar, Nvar), 0.0)
+    for ind_r, row in enumerate(Ac):
+        for ind_c, col in enumerate(row):
+            Ac[ind_r][ind_c]=F[ind_r][ind_c]/m[ind_r]
+    return Ac
+
+def F_newton(x_, m, minimize=True, G=0.0001, keep_best=False, weight_method='Linear', t=1.0, max_mass=100.0): 
+    """ The procedure that generates the new positions of the particles is based on how bodies are attracted to each other
+        through an attractive force (universal law of gravitation). Thus, the procedure will be as follows:
+        
+        1. n particles are created at random positions within the search space.
+        2. Each particle is assigned a mass based on the solution provided by the particle.
+            2.1 The better the solution provided by the particle, the greater the assigned mass.
+            2.2 Assignment methods -> Linear, Exponential, Quadratic (further discussion on better assignment methods can be explored).
+            2.2.1 In simple terms, a range of solutions is established and adjusted to a charge range between 0 to max_mass according to some assignment method.
+        3. Let Fij be the force produced between particle i and j -> Fij = G*mi*mj/(r^2) where r is the distance between the particles, mi and mj are the masses of i and j.
+            3.1 Thus, the resulting force on particle i is FRi = Σ{j=1} G*mi*mj/(rij^2).
+            3.2 Additionally, by Newton's Law, F = m*a -> a = F/m. Thus, the velocities of each particle are calculated as vf = vi + a*t -> vf = vi + (F/m)*t.
+            3.3 Finally, the positions are updated as follows: x' = x + vi*t + (a*t^2)/2 -> x' = x + vi*t + (F/m)*(t^2)/2.
+            3.4 Applying the assumption vi=0 -> x' = x + (F/m)*(t^2)/2.
+
+    Args:
+        x_ (array(size=(Npar, Nvar)): current particle positions
+        m (array(size=(Npar, 1))): current mass of each particle
+        minimize (bool, optional): solver objective. Defaults to True.
+        G (float, optional): gravitational constant. Defaults to 0.0001.
+        keep_best (bool, optional): It will save the best value obtained in each iteration. Defaults to False.
+        weight_method (str, optional): method to reassign mass. Defaults to 'Linear'. Options=['Linear', 'Quadratic', 'Exponential'].
+        t (float, optional): time. Defaults to 1.0.
+        max_mass (float, optional): upper bound of mass to assing. Defaults to 100.0.
+
+    Returns:
+        F_total, Ac, new_pos: returns the force obtained on each particle, their acceleration, 
+                              and their new positions
+    """
+    Npar=x_.shape[0]
+    Nvar=x_.shape[1]
+
+    # Distance Matrix
+    dis = []
+    for ind_r, row in enumerate(x_):
+        for ind_c, col in enumerate(x_):
+            dis.append(distance(x_[ind_r], x_[ind_c]))
+    dis=np.array(dis).reshape((Npar,Npar,Nvar))
+
+    # Direction Matrix
+    d = []
+    for ind_r, row in enumerate(x_):
+        for ind_c, col in enumerate(x_):
+            d.append(normaliazed_direction(x_[ind_r], x_[ind_c]))
+    d=np.array(d).reshape((Npar,Npar,Nvar))
+
+    colisioned_part = []
+    r_2 = np.zeros((Npar, Npar))
+    for ind_r, row in enumerate(dis):
+        for ind_c, col in enumerate(dis):
+            value = dis[ind_r][ind_c]
+            value_2 = value**2
+            value_sum = np.sum(value_2)
+            if value_sum < 0.00000001: # Particles have practically collided. Notice later that F_=0.0 ahead.
+                r_2[ind_r][ind_c] = 0.0
+                if ind_r != ind_c:
+                    colisioned_part.append(ind_c)
+            else:
+                r_2[ind_r][ind_c] = value_sum
+    colisioned_part_ = np.unique(np.array(colisioned_part))
+
+
+    # Each particle is assigned a mass magnitude based on the solution provided by the particle.
+    m=np.array(m)
+    min_value = m[0]
+    max_value = m[-1]
+
+    if minimize:
+        m = -1.0*(m-max_value-1.0)
+        max_value = m[0]
+
+    if weight_method=="Linear":
+        # We adjust the mass according to the following range [0, max_mass].
+        m = (m/max_value)*max_mass
+        reverse_ind = [len(m)-i-1 for i in range(len(m))] 
+        m = m[reverse_ind]
+        
+    elif weight_method=="Quadratic": 
+        # We adjust the mass according to the following range [0, max_mass].
+        m_2=m**2
+        m=(m_2/np.max(m_2))*max_mass
+        reverse_ind = [len(m)-i-1 for i in range(len(m))] 
+        m = m[reverse_ind]
+        
+    elif weight_method=="Exponential": 
+        # We adjust the mass according to the following range [0, max_mass].
+        m_exp=np.array([np.exp(i) for i in m])
+        m=(m_exp/np.max(m_exp))*max_mass
+        reverse_ind = [len(m)-i-1 for i in range(len(m))] 
+        m = m[reverse_ind]
+        
+    m = np.nan_to_num(m, nan=0.0001)
+    Npar=d.shape[0]
+    Nvar=d.shape[2]
+    F=np.full((Npar, Npar, Nvar), 0.0)
+    for ind_r, row in enumerate(dis):
+        for ind_c, col in enumerate(row):
+            # The magnitude of the attraction force F_ is calculated.            
+            d_2=r_2[ind_r][ind_c]
+            if d_2==0:
+                F_=0.0
+            else:
+                m1=m[ind_r]
+                m2=m[ind_c]
+                F_=float(G*m1*m2/d_2)
+            F[ind_r][ind_c]=F_*d[ind_r][ind_c]
+    F_total = np.sum(F, axis=1)
+    # Note: Between two particles, the attractive forces will be the same, but not their accelerations.
+    # Remember that F = M * A -> m1 * a1 = m2 * a2, so if m1 > m2 -> a2 > a1 -> Particles with greater mass have lower acceleration.Ac=acc(F_total, m)
+    Ac=acc(F_total, m)
+    Ac = np.nan_to_num(Ac, nan=0.0)
+    
+    # Finally, Xf = xi + vi * t + 0.5 * acc * (t^2) (Final Position), but if the particles always start at rest (vi=0) -> xf = 0.5 * acc * (t^2).
+    x__ = 0.5*Ac*(t**2)
+    new_pos=x_ + x__
+    
+    if keep_best==True:
+        best_ind = np.argmin(m)
+        Ac[best_ind]=0.0
+        new_pos=x_ + 0.5*Ac*(t**2)
+    
+    # If more than half of the particles have collided, those that have collided will move randomly within the reduced search space.
+    max_crash_part=Npar/2
+    if len(colisioned_part_)>max_crash_part:
+        lo = np.min(new_pos, axis=0)
+        up = np.max(new_pos, axis=0)
+        for i in colisioned_part_:
+            new_pos[i] = np.random.uniform(low=lo, high=up, size=(1, Nvar))
+
+    return F_total, Ac, new_pos

custom_pso.py

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+'''
+    --------------------------------------------------------------
+    Custom PSO algorithm for continuous domains
+    --------------------------------------------------------------
+    Autor: Rodrigo Araya 
+    Email: [email protected]
+'''
+
+import math
+import numpy as np
+import pandas as pd
+from F_newton import F_newton
+from F_coulomb import F_coulomb
+import plotly.graph_objects as go
+import plotly.express as px
+
+
+def Random_(problem, x_, auto_reduce_search_space=False):
+    """ This solver generates a random solution for each particle. Additionally, it is also possible to 
+        apply an exploitation method where the search space is reduced on each iteration.
+
+    Args:
+        problem (dic): Dictionary that include: objective function, lower bound and upper bound of each variable and optimal solution
+        x_ (array(size=())): Current position of each particle
+        auto_reduce_search_space (bool, optional): If True, it update the lower and upper bounds in a way to reduce the search space. Defaults to False.
+
+    Returns:
+        Stemp, S_f: New positions for each particle and their performance after being evaluated with the objective function.
+    """
+    
+    
+    nPart = x_.shape[0]
+    nVar = x_.shape[1]
+    
+    if auto_reduce_search_space:
+        Lo = np.min(x_, axis=0)
+        Up = np.max(x_, axis=0)
+    else:
+        Lo = np.zeros(nVar)
+        Up = np.ones(nVar)
+
+    new_x = np.random.uniform(low=Lo, high=Up, size=(nPart, nVar))
+    
+    Stemp = np.zeros((nPart,nVar))
+    for k in range(nPart):
+        for i in range(nVar):
+            Stemp[k][i] = new_x[k][i]
+            if Stemp[k][i] > Up[i]:
+                Stemp[k][i] = Up[i]
+            elif Stemp[k][i] < Lo[i]:
+                Stemp[k][i] = Lo[i]
+    f,S_r,maximize = mp_evaluator(problem, Stemp)
+
+    S_f = np.zeros((nPart,1))
+    
+    for i in range(len(S_r)):
+        S_f[i] = f[i]
+    return Stemp, S_f
+    
+def F_newton_(problem, x_, m, Lo, Up, G=0.00001, keep_best=True, weight_method='Linear', t=1.0, max_mass=100.0):
+    _, _, new_x = F_newton(x_, m, minimize=True, G=G, keep_best=keep_best, weight_method=weight_method, t=t, max_mass=max_mass)
+    nPart=x_.shape[0]
+    nVar=x_.shape[1]
+    Stemp = np.zeros((nPart,nVar))
+    for k in range(nPart):
+        for i in range(nVar):
+            Stemp[k][i] = new_x[k][i]
+            if Stemp[k][i] > Up[i]:
+                Stemp[k][i] = Up[i]
+            elif Stemp[k][i] < Lo[i]:
+                Stemp[k][i] = Lo[i]
+    f,S_r,maximize = mp_evaluator(problem, Stemp)
+
+    S_f = np.zeros((nPart,1))
+    
+    for i in range(len(S_r)):
+        S_f[i] = f[i]
+    return Stemp, S_f
+    
+def F_coulomb_(problem, x_, v, q, P, Lo, Up, k=0.00001, weight_method='Linear', t=1.0, max_q=100.0):
+    _, _, v, new_x = F_coulomb(x_, v, q, P=P, minimize=True, k=k, weight_method=weight_method, t=t, max_q=max_q)
+    nPart=x_.shape[0]
+    nVar=x_.shape[1]
+    Stemp = np.zeros((nPart,nVar))
+    for k in range(nPart):
+        for i in range(nVar):
+            Stemp[k][i] = new_x[k][i]
+            if Stemp[k][i] > Up[i]:
+                Stemp[k][i] = Up[i]
+            elif Stemp[k][i] < Lo[i]:
+                Stemp[k][i] = Lo[i]
+    f,S_r,maximize = mp_evaluator(problem, Stemp)
+
+    S_f = np.zeros((nPart,1))
+    
+    for i in range(len(S_r)):
+        S_f[i] = f[i]
+    return Stemp, v, S_f
+
+def evaluator(problem, x):
+    # Dado que se trabaja bajo el dominio [0, 1] para cada variable, se debe crear una funcion que regrese a los valores a los dominios originales
+    # Ejemplo (ndim=2): si se tiene el dominio original [-5, 5] y [-2, 2] para la variables x1, x2 y se cambio a [0, 1] y [0, 1] se vuelve al original tras:
+    # y1=ax+b -> -5 = a*0+b -> b=-5
+    # y1=ax+b -> 5 = a*1+-5 -> a=10
+    # y1=10*x+-5
+    # y2=ax+b -> -2 = a*0+b -> b=-2
+    # y2=ax+b -> 2 = a*1+-2 -> a=4
+    # y2=4*x+-2
+    # Luego se aplica y1(x1) e y2(x2) respectivamente. Extendible a n dimensiones.
+    # Dado que [0, 1] no cambia, se generaliza la formula b=lb y a=ub-lb -> y_{i} = (ub-lb)_{i}*x + lb_{i}
+    lb = [var[0] for var in problem['bounds']]
+    ub = [var[1] for var in problem['bounds']]
+    x = [(ub[ind]-lb[ind])*i+lb[ind] for ind, i in enumerate(x)]
+    # calculate fitness
+    f = problem['f'](x)
+    fitness = dict(Obj=f)
+    return fitness
+
+def mp_evaluator(problem, x):
+    results = [evaluator(problem, c) for c in x]
+    f = [r['Obj'] for r in results]
+    # maximization or minimization problem
+    maximize = False
+    return (f, [r for r in results],maximize)
+
+def correct_x(problem, x_):
+    lb = [var[0] for var in problem['bounds']]
+    ub = [var[1] for var in problem['bounds']]
+    x = np.empty(x_.shape)
+    for ind, row in enumerate(x_):
+        corr_row = np.array([(ub[ind]-lb[ind])*i+lb[ind] for ind, i in enumerate(row)])
+        x[ind]=corr_row
+    return x
+
+
+
+def custom_pso(problem, nPart, nVar, maxiter, G=0.00001, k=0.0001, keep_best=True, weight_method='Linear', t=1.0, seed=0, max_mass=100.0, solver='Newton', P=3, max_q=100.0, auto_reduce_search_space=False, dinamic=False):
+    """The procedure of this algorithm is as follows:
+    1. Generate a file of size nSize=nPart.
+    2. Generate nPart initial solutions. Each solution contains nVar variables.
+    3. Evaluate the nPart solutions and add them to the file ordered from best to worst solution.
+    4. While the termination condition is not met (iterations > maxiter):
+        4.1 Generate new solutions using some solver. Options = ['Random', 'Newton', 'Coulomb']
+        4.2 Add new solutions to the file.
+        4.3 Remove duplicate solutions.
+        4.4 Evaluate solutions and sort from best to worst solution.
+        4.5 Keep in the file the nPar best solutions.
+        4.6 Save iteration results in a history (dataframe).
+        4.7 Evaluate termination condition. If negative, return to step 4.1.
+    
+    Args:
+        problem (dic): Dictionary that include: objective function, lower bound and upper bound of each variable and optimal solution
+        nPart (_type_): Quantity of particles
+        nVar (_type_): Quantity of variables
+        maxiter (_type_): Maximum number of iterations.
+        seed (int, optional): set the generation of random numbers. Defaults to 0.
+        solver (str, optional): solver to apply. Defaults to 'Newton'. Options=['Random', 'Newton', 'Coulomb'].
+        
+        Random Solver:
+            auto_reduce_search_space (bool, optional): If True, it update the lower and upper bounds in a way to reduce the search space. Defaults to False.
+
+        Newton Solver:
+            G (float, optional): Gravitational constant. Defaults to 0.00001.
+            keep_best (bool, optional): It will keep the best value obtained in each iteration. Defaults to True.
+            weight_method (str, optional): method to reassign particle mass. Defaults to 'Linear'. Options=['Linear', 'Quadratic', 'Exponential'].
+            t (float, optional): time. Defaults to 1.0.
+            max_mass (float, optional): upper bound of mass to assing. Defaults to 100.0.
+            dinamic (bool, optional): It will change the max_mass value depending on the current iteration and difference between best and worst value obteined. Defaults to False.
+            
+        Coulomb Solver:
+            P (int): quantity of electric fields (the best P particles become electric fields)
+            max_q (float, optional): upper bound of electric charge to assing. Defaults to 100.0.
+            weight_method (str, optional): method to reassign electric charges. Defaults to 'Linear'. Options=['Linear', 'Quadratic', 'Exponential'].
+            k (float, optional): electric constant. Defaults to 0.0001.
+            t (float, optional): time. Defaults to 1.0.
+            dinamic (bool, optional): It will change the max_q value depending on the current iteration and difference between best and worst value obteined. Defaults to False.
+            
+        
+    Returns:
+        df, best_var, best_sol: dataframe contening data from all iterations, the best variables and the best value obtained
+    """
+    
+    # number of variables
+    parameters_v = [f'x{i+1}' for i in range(nVar)]
+    
+    # number of variables
+    nVar = len(parameters_v)
+
+    # size of solution archive
+    nSize = nPart
+
+    # number of Particules
+    nPart = nPart
+
+    # maximum iterations
+    maxiter = maxiter
+
+    # bounds of variables
+    Up = [1]*nVar
+    Lo = [0]*nVar
+    
+    # dinamic q
+    din_max_q=max_q
+    
+    # dinamic mass
+    din_max_mass = max_mass
+
+    # initilize matrices
+    S = np.zeros((nSize,nVar))
+    S_f = np.zeros((nSize,1))
+
+    # initial velocity
+    v = np.zeros((nSize,nVar))
+
+    # history
+    columns_ = ['iter']
+    for i in parameters_v: columns_.append(i)
+    columns_.append('f')
+    df = pd.DataFrame(columns=columns_)
+    
+    # generate first random solution
+    #np.random.seed(seed)
+    Srand = np.random.uniform(low=0,high=1,size=(nPart, nVar))
+    f,S_r,maximize = mp_evaluator(problem, Srand)
+
+
+    for i in range(len(S_r)):
+        S_f[i] = f[i]
+        
+    # add responses and "fitness" column to solution
+    S = np.hstack((Srand, v, S_f))
+    # sort according to fitness (last column)
+    S = sorted(S, key=lambda row: row[-1],reverse = maximize)
+    S = np.array(S)
+
+    # save each iteration
+    iter_n = np.full((nPart, 1), 0.0)
+    x_ = S[:, 0:nVar]
+    x = correct_x(problem, x_)
+    res = np.reshape(S[:, -1], (nPart, 1))
+    rows=np.hstack((iter_n, x, res))
+    df_new = pd.DataFrame(rows, columns=columns_)
+    df = pd.concat([df, df_new])
+
+    iterations=1
+    
+    # iterations
+    while True:
+        # get a new solution
+        if solver == "Random":
+            Stemp, S_f = Random_(problem, S[:, :nVar], auto_reduce_search_space=auto_reduce_search_space)
+        
+        elif solver=="Newton":
+            din_mass=0
+            if dinamic: 
+                din_mass = ((maxiter-iterations)/(maxiter))*(np.max(S_f)-np.min(S_f))
+            else:
+                din_mass=max_mass
+            
+            m = np.reshape(S[:, -1], (nPart, 1))
+            Stemp, S_f = F_newton_(problem, S[:, :nVar], m, Lo, Up, G=G, keep_best=keep_best, weight_method=weight_method, t=t, max_mass=din_mass)
+
+        elif solver == "Coulomb":
+            din_q=0
+            if dinamic:
+                din_q = ((maxiter-iterations)/(maxiter))*(np.max(S_f)-np.min(S_f))
+            else:
+                din_q=max_q
+            q = np.reshape(S[:, -1], (nPart, 1))
+            Stemp, v, S_f = F_coulomb_(problem, S[:, :nVar], v, q, P, Lo, Up, k=k, weight_method=weight_method, t=t, max_q=din_q)
+        
+        # add responses and "fitness" column to solution
+        Ssample = np.hstack((Stemp, v, S_f))
+
+        # add new solutions in the solutions table
+        Solution_temp = np.vstack((S,Ssample))
+
+        # delate duplicated rows
+        Solution_temp = np.unique(Solution_temp, axis=0)
+
+        # sort according to "fitness"
+        Solution_temp = sorted(Solution_temp, key=lambda row: row[-1],reverse = maximize)
+        Solution_temp = np.array(Solution_temp)
+
+        # keep best solutions
+        S = Solution_temp[:nSize][:]
+
+        # save each iteration
+        iter_n = np.full((nPart, 1), iterations)
+        x_ = S[:, 0:nVar]
+        x = correct_x(problem, x_)
+        res = np.reshape(S[:, -1], (nPart, 1))
+        rows=np.hstack((iter_n, x, res))
+        df_new = pd.DataFrame(rows, columns=columns_)
+        df = pd.concat([df, df_new])
+
+        iterations += 1
+        if iterations > maxiter:
+            break
+
+    best_sol = np.min(df['f'])
+    ind_best_sol = np.argmin(df['f'])
+    best_var = df.iloc[ind_best_sol, 1:len(parameters_v)+1]
+    return df, best_var, best_sol
+
+# Test Functions.
+# Adapted from "https://www.sfu.ca/~ssurjano/optimization.html"
+
+def Ackley(x, a=20.0, b=0.2, c=2.0*np.pi):
+    d = len(x)
+    sum1 = np.sum(np.square(x))
+    sum2 = np.sum(np.array([np.cos(c*i) for i in x]))
+    term1 = -a * np.exp(-b * np.sqrt(sum1 / d))
+    term2 = -np.exp(sum2 / d)
+    return term1 + term2 + a + np.exp(1)
+
+def sixth_Bukin(x):
+    return 100 * np.sqrt(np.abs(x[1] - 0.01*x[0]**2)) + 0.01*np.abs(x[0] + 10)
+
+def Cross_in_Tray(x):
+    return -0.0001 * math.pow(np.abs(np.sin(x[0]) * np.sin(x[1]) * np.exp(np.abs(100.0 - np.sqrt(x[0]**2 + x[1]**2)/np.pi)))+1.0, 0.1)
+
+def Drop_Wave(x):
+    return -1.0*(1.0 + np.cos(12.0 * np.sqrt(x[0]**2 + x[1]**2))) / (0.5 * (x[0]**2 + x[1]**2) + 2.0)
+
+def Eggholder(x):
+    return -(x[1] + 47.0) * np.sin(np.sqrt(np.abs(x[1] + x[0]/2 + 47.0))) - x[0] * np.sin(np.sqrt(np.abs(x[0] - (x[1] + 47.0))))
+
+def Griewank(x):
+    sum_part=0.0
+    prod_part=1.0
+    for i, xi in enumerate(x):
+        sum_part += xi/4000.0
+        prod_part *= np.cos(xi/np.sqrt(i+1))
+    return sum_part - prod_part + 1.0
+
+def Holder_Table(x):
+    return -np.abs(np.sin(x[0])*np.cos(x[1])*np.exp(np.abs(1.0-(np.sqrt(x[0]**2 + x[1]**2)/np.pi))))
+
+def Levy(x):
+    # d dimensions
+    w1 = 1.0 + (x[0]-1.0)/4.0
+    wd = 1.0 + (x[-1]-1.0)/4.0
+    sum_part = 0.0
+    for i, xi in enumerate(x):
+        wi = 1.0 + (xi-1.0)/4.0
+        sum_part += ((wi-1.0)**2)*(1.0 + 10.0*math.pow(np.sin(np.pi*wi+1.0), 2))
+    return math.pow(np.sin(np.pi*w1), 2) + sum_part + ((wd-1.0)**2)*(1.0 + math.pow(np.sin(2*np.pi*wd), 2))
+
+def Rastrigin(x):
+    # d dimensions
+    d = len(x)
+    sum_part = 0.0
+    for i, xi in enumerate(x):
+        sum_part += (xi**2) - 10.0*np.cos(2*np.pi*xi)
+    return 10*d + sum_part
+
+def second_Schaffe(x):
+    return 0.5 + (math.pow(np.sin(x[0]**2 + x[1]**2), 2)-0.5)/(1.0 + 0.001*(x[0]**2 + x[1]**2))**2
+
+def fourth_Schaffer(x):
+    return 0.5 + (math.pow(np.cos(np.abs(x[0]**2 - x[1]**2)), 2)-0.5)/(1.0 + 0.001*(x[0]**2 + x[1]**2))**2
+
+def Schwefel(x):
+    # d dimensions
+    d = len(x)
+    sum_part = 0.0
+    for xi in x:
+        sum_part += xi*np.sin(np.sqrt(np.abs(xi)))
+    return 418.9829*d - sum_part
+
+def Shubert(x):
+    sum_1 = 0.0
+    sum_2 = 0.0
+    for i in np.arange(5):
+        i = float(i + 1)
+        sum_1 += i*np.cos((i+1)*x[0] + i)
+        sum_2 += i*np.cos((i+1)*x[1] + i)
+    return sum_1*sum_2
+
+def Styblinski_Tang(x):
+    f = (sum([math.pow(i,4)-16*math.pow(i,2)+5*i for i in x])/2)
+    return f
+
+def Easom(x):
+    f = np.cos(x[0])*np.cos(x[1])*np.exp(-(math.pow(x[0]-np.pi, 2) + math.pow(x[1]-np.pi, 2)))
+    return f
+
+def Bohachevsky(x):
+    f = x[0]**2 + 2*x[1]**2 -0.3*np.cos(3*np.pi*x[0]) - 0.4*np.cos(4*np.pi*x[1]) + 0.7
+    return f
+
+def Perm_d_beta(x, d=2, beta=4):
+    f = 0.0
+    for i in range(d):
+        for j in range(d):
+            f += ((j+1) + beta)*(x[j]**(i+1) - (1/(j+1)**(i+1)))**2
+    return f
+    
+def Rotated_Hyper_Ellipsoid(x, d=2):
+    f = 0.0
+    for i in range(d):
+        for j in range(i+1):
+            f += x[j]**2
+    return f
+            
+def Sum_of_Different_Powers(x, d=2):
+    f = 0.0
+    for i in range(d):
+        f += np.abs(x[i])**i+1+1
+    return f
+        
+def SUM_SQUARES(x, d=2):    
+    f = 0.0
+    for i in range(d):
+        f += (i+1)*x[i]**2
+    return f
+
+def TRID(x, d=2):
+    sum_1 = 0.0
+    sum_2 = 0.0
+    for i in range(d):
+        sum_1 += (x[i] - 1)**2
+    for i in range(d-1):
+        sum_2 += x[i+1]*x[i]   
+    f = sum_1 + sum_2
+    return f 
+
+def BOOTH(x):
+    f = (x[0] + 2*x[1] -7)**2 + (2*x[0] + x[1] - 5)**2
+    return f
+
+def Matyas(x):
+    f = 0.26*(x[0]**2 + x[1]**2) - 0.48*x[0]*x[1]
+    return f
+
+def MCCORMICK(x):
+    f = np.sin(x[0] + x[1]) + (x[0] - x[1])**2 - 1.5*x[0] + 2.5*x[1] + 1
+    return f
+
+def Power_Sum(x, d=2, b=[8, 18, 44, 114]):
+    f = 0.0
+    for i in range(d):
+        sum_1 = 0.0
+        for j in range(d):
+            sum_1 += x[j]**(i+1)
+        f += (sum_1 - b[i])**2
+    return f
+            
+def Zakharov(x, d=2):
+    f = 0.0
+    sum_1 = 0.0
+    sum_2 = 0.0
+    sum_3 = 0.0
+    for i in range(d):
+        sum_1 += x[i]**2
+        sum_2 += 0.5*(i+1)*x[i]
+        sum_3 += 0.5*(i+1)*x[i]
+    f = sum_1 + sum_2**2 + sum_3**4
+    return f
+
+def THREE_HUMP_CAMEL(x):
+    f = 2*x[0]**2 - 1.05*x[0]**4 + (x[0]**6/6) + x[0]*x[1] + x[1]**2
+    return f
+
+def SIX_HUMP_CAMEL(x):
+    f = (4 - 2.1*x[0] + (x[0]**4)/3)*x[0]**2 + x[0]*x[1] + (-4 + 4*x[1]**2)*x[1]**2
+    return f
+
+def DIXON_PRICE(x, d=2):
+    sum_1 = 0.0
+    for i in range(d-1):
+        i = i + 1
+        sum_1 += (i+1)*(2*x[i]**2 - x[i-1])**2
+    f = (x[0] - 1)**2 + sum_1
+    return f 
+    
+def ROSENBROCK(x, d=2):
+    f = 0.0 
+    for i in range(d-1):
+        f += 100*(x[i+1] - x[i]**2)**2 + (x[i] - 1)**2
+    return f    
+
+def DE_JONG(x):
+    a = [[-32, -16, 0, 16, 32, -32, -16, 0, 16, 32, -32, -16, 0, 16, 32, -32, -16, 0, 16, 32, -32, -16, 0, 16, 32], 
+         [-32, -32, -32, -32, -32, -16, -16, -16, -16, -16, 0, 0, 0, 0, 0, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32]]
+    sum_1 = 0.0
+    for i in range(25):
+        sum_1 += 1/((i+1) + (x[0] - a[0][i])**6 + (x[1] - a[1][i])**6)
+    f = (0.002 + sum_1)**(-1)
+    return f
+
+def MICHALEWICZ(x, d=2, m=10):
+    f = 0.0
+    for i in range(d):
+        f += np.sin(x[i])*np.sin(((i+1)*x[i]**2)/np.pi)**(2*m) 
+    f = -f
+    return f
+  
+def BEALE(x):
+    f = (1.5 - x[0] + x[0]*x[1])**2 + (2.25 - x[0] + x[0]*x[1]**2)**2 + (2.625 - x[0] + x[0]*x[1]**3)**2
+    return f
+    
+def BRANIN(x):
+    a=1
+    b=5.1/(4*(np.pi)**2)
+    c = 5/np.pi
+    r = 6
+    s = 10
+    t = 1/(8*np.pi)
+    f = a*(x[1] - b*x[0]**2 + c*x[0] - r)**2 + s*(1 - t)*np.cos(x[0]) + s
+    return f
+    
+def GOLDSTEIN_PRICE(x):
+    f = (1 + ((x[0] + x[1] + 1)**2)*(19 - 14*x[0] + 3*x[0]**2 - 14*x[1] + 6*x[0]*x[1] + 3*x[1]**2))*(30 + ((2*x[0] - 3*x[1])**2)*(18 - 32*x[0] + 12*x[0]**2 + 48*x[1] - 36*x[0]*x[1] + 27*x[1]**2))    
+    return f
+
+def PERM_D_BETA(x, d=2, beta=1):
+    f = 0.0
+    for i in range(d):
+        sum_1 = 0
+        for j in range(d):
+            sum_1 += ((j+1)**(i+1) + beta)*((x[j]/(j+1))**(i+1) - 1)
+        f += sum_1
+    return f
+
+class test_functions():
+    def __init__(self) -> None:
+        self.Ackley_ = {'name':'Ackley', 'f':Ackley, 'bounds':[[-32.768, 32.768], [-32.768, 32.768]], 'opt':[[0.0, 0.0], 0.0]}
+        self.sixth_Bukin_ = {'name':'sixth_Bukin', 'f':sixth_Bukin, 'bounds':[[-15.0, -5.0], [-3.0, 3.0]], 'opt':[[-10.0, 1.0], 0.0]}
+        self.Cross_in_Tray_ = {'name':'Cross_in_Tray', 'f':Cross_in_Tray, 'bounds':[[-10.0, 10.0], [-10.0, 10.0]], 'opt':[[[1.3491, -1.3491], [1.3491, 1.3491], [-1.3491, 1.3491], [-1.3491, -1.3491]], -2.06261]}
+        self.Drop_Wave_ = {'name':'Drop_Wave', 'f':Drop_Wave, 'bounds':[[-5.12, 5.12], [-5.12, 5.12]], 'opt':[[0, 0], -1.0]}
+        self.Eggholder_ = {'name':'Eggholder', 'f':Eggholder, 'bounds':[[-512.0, 512.0], [-512.0, 512.0]], 'opt':[[512.404, 512.404], -959.6407]}
+        self.Griewank_ = {'name':'Griewank', 'f':Griewank, 'bounds':[[-600.0, 600.0], [-600.0, 600.0]], 'opt':[[0.0, 0.0], 0.0]}
+        self.Holder_Table_ = {'name':'Holder_Table', 'f':Holder_Table, 'bounds':[[-10.0, 10.0], [-10.0, 10.0]], 'opt':[[[8.05502, 9.66459], [8.05502, -9.66459], [-8.05502, 9.66459], [-8.05502, -9.66459]], -19.2085]}
+        self.Levy_ = {'name':'Levy', 'f':Levy, 'bounds':[[-10.0, 10.0], [-10.0, 10.0]], 'opt':[[1.0, 1.0], 0.0]}
+        self.Rastrigin_ = {'name':'Rastrigin', 'f':Rastrigin, 'bounds':[[-5.12, 5.12], [-5.12, 5.12]], 'opt':[[0.0, 0.0], 0.0]}
+        self.second_Schaffe_ = {'name':'second_Schaffe', 'f':second_Schaffe, 'bounds':[[-100.0, 100.0], [-100.0, 100.0]], 'opt':[[0.0, 0.0], 0.0]}
+        self.fourth_Schaffer_ = {'name':'fourth_Schaffer', 'f':fourth_Schaffer, 'bounds':[[-100.0, 100.0], [-100.0, 100.0]], 'opt':[[0.0, 0.0], 0.0]}
+        self.Schwefel_ = {'name':'Schwefel', 'f':Schwefel, 'bounds':[[-500.0, 500.0], [-500.0, 500.0]], 'opt':[[420.9687, 420.9687], 0.0]}
+        self.Shubert_ = {'name':'Shubert', 'f':Shubert, 'bounds':[[-10.0, 10.0], [-10.0, 10.0]], 'opt':[[0.0, 0.0], -186.7309]}
+        self.Styblinski_Tang_ = {'name':'Styblinski_Tang', 'f':Styblinski_Tang, 'bounds':[[-5, 5], [-5, 5]]}
+        self.Easom_ = {'name':'Easom', 'f':Easom, 'bounds':[[-3, 3], [-3, 3]]}
+        self.Bohachevsky_ = {'name':'Bohachevsky', 'f':Bohachevsky, 'bounds':[[-100.0, 100.0], [-100.0, 100.0]], 'opt':[[0, 0], 0]}
+        self.Perm_d_beta_ = {'name':'Perm_d_beta', 'f':Perm_d_beta, 'bounds':[[-2.0, 2.0], [-2.0, 2.0]], 'opt':[[1, 0.5], 0]}
+        self.Rotated_Hyper_Ellipsoid_ = {'name':'Rotated_Hyper_Ellipsoid', 'f':Rotated_Hyper_Ellipsoid, 'bounds':[[-65.536, 65.536], [-65.536, 65.536]], 'opt':[[0.0, 0.0], 0]}
+        self.Sum_of_Different_Powers_ = {'name':'Sum_of_Different_Powers', 'f':Sum_of_Different_Powers, 'bounds':[[-1, 1], [-1, 1]], 'opt':[[0.0, 0.0], 0]}
+        self.SUM_SQUARES_ = {'name':'SUM_SQUARES', 'f':SUM_SQUARES, 'bounds':[[-10, 10], [-10, 10]], 'opt':[[0.0, 0.0], 0]}
+        self.TRID_ = {'name':'TRID', 'f':TRID, 'bounds':[[-4, 4], [-4, 4]], 'opt':[[2, 2], -2]}
+        self.BOOTH_ = {'name':'BOOTH', 'f':BOOTH, 'bounds':[[-10, 10], [-10, 10]], 'opt':[[1, 3], 0]}
+        self.Matyas_ = {'name':'Matyas', 'f':Matyas, 'bounds':[[-10, 10], [-10, 10]], 'opt':[[0, 0], 0]}
+        self.MCCORMICK_ = {'name':'MCCORMICK', 'f':MCCORMICK, 'bounds':[[-1.5, 4], [-3, 4]], 'opt':[[-0.54719, -1.54719], -1.9133]}
+        self.Power_Sum_ = {'name':'Power_Sum', 'f':Power_Sum, 'bounds':[[0, 2], [0, 2]]}
+        self.Zakharov_ = {'name':'Zakharov', 'f':Zakharov, 'bounds':[[-5, 10], [-5, 10]], 'opt':[[0.0, 0.0], 0.0]}
+        self.THREE_HUMP_CAMEL_ = {'name':'THREE_HUMP_CAMEL', 'f':THREE_HUMP_CAMEL, 'bounds':[[-5, 5], [-5, 5]], 'opt':[[0.0, 0.0], 0.0]}
+        self.SIX_HUMP_CAMEL_  = {'name':'SIX_HUMP_CAMEL', 'f':SIX_HUMP_CAMEL, 'bounds':[[-3, 3], [-2, 2]], 'opt':[[0.0898, -0.7126], -1.0316]}
+        self.DIXON_PRICE_ = {'name':'DIXON_PRICE', 'f':DIXON_PRICE, 'bounds':[[-10, 10], [-10, 10]], 'opt':[[1, 1/np.sqrt(2)], 0]}
+        self.ROSENBROCK_ = {'name':'ROSENBROCK', 'f':ROSENBROCK, 'bounds':[[-5, 10], [-5, 10]], 'opt':[[1, 1], 0]}
+        self.DE_JONG_ = {'name':'DE_JONG', 'f':DE_JONG, 'bounds':[[-65.536, 65.536], [-65.536, 65.536]]}
+        self.MICHALEWICZ_ = {'name':'MICHALEWICZ', 'f':MICHALEWICZ, 'bounds':[[0, np.pi], [0, np.pi]], 'opt':[[2.2, 1.57], -1.8013]}
+        self.BEALE_ = {'name':'BEALE', 'f':BEALE, 'bounds':[[-4.5, 4.5], [-4.5, 4.5]], 'opt':[[3, 0.5], 0]}
+        self.BRANIN_ = {'name':'BRANIN', 'f':BRANIN, 'bounds':[[-5, 10], [0, 15]], 'opt':[[-np.pi, 12.275], 0.397887]}
+        self.GOLDSTEIN_PRICE_ = {'name':'GOLDSTEIN_PRICE', 'f':GOLDSTEIN_PRICE, 'bounds':[[-2, 2], [-2, 2]], 'opt':[[0, -1], 3]}
+        self.PERM_D_BETA_ = {'name':'PERM_D_BETA', 'f':PERM_D_BETA, 'bounds':[[-2, 2], [-2, 2]], 'opt':[[1, 2], 0]}
+        self.dictionary = {'Ackley': self.Ackley_, 
+                           'sixth_Bukin':self.sixth_Bukin_, 
+                           'Cross_in_Tray':self.Cross_in_Tray_, 
+                           'Drop_Wave':self.Drop_Wave_, 
+                           'Eggholder':self.Eggholder_, 
+                            'Griewank':self.Griewank_, 
+                            'Holder_Table':self.Holder_Table_, 
+                            'Levy':self.Levy_, 
+                            'Rastrigin':self.Rastrigin_, 
+                            'second_Schaffe':self.second_Schaffe_, 
+                            'fourth_Schaffer':self.fourth_Schaffer_, 
+                            'Schwefel':self.Schwefel_, 
+                            'Shubert':self.Shubert_, 
+                            'Styblinski_Tang':self.Styblinski_Tang_, 
+                            'Easom':self.Easom_, 
+                            'Bohachevsky':self.Bohachevsky_, 
+                            'Perm_d_beta':self.Perm_d_beta_, 
+                            'Rotated_Hyper_Ellipsoid':self.Rotated_Hyper_Ellipsoid_, 
+                            'Sum_of_Different_Powers': self.Sum_of_Different_Powers_, 
+                            'SUM_SQUARES':self.SUM_SQUARES_, 
+                            'TRID':self.TRID_, 
+                            'BOOTH': self.BOOTH_, 
+                            'Matyas':self.Matyas_, 
+                            'MCCORMICK': self.MCCORMICK_, 
+                            'Power_Sum':self.Power_Sum_, 
+                            'Zakharov':self.Zakharov_, 
+                            'THREE_HUMP_CAMEL' :self.THREE_HUMP_CAMEL_, 
+                            'SIX_HUMP_CAMEL': self.SIX_HUMP_CAMEL_, 
+                            'DIXON_PRICE': self.DIXON_PRICE_, 
+                            'ROSENBROCK_': self.ROSENBROCK_, 
+                            'DE_JONG':self.DE_JONG_, 
+                            'MICHALEWICZ': self.MICHALEWICZ_, 
+                            'BEALE':self.BEALE_, 
+                            'BRANIN': self.BRANIN_, 
+                            'GOLDSTEIN_PRICE':self.GOLDSTEIN_PRICE_, 
+                            'PERM_D_BETA':self.PERM_D_BETA_} 
+        self.whole_list = list(self.dictionary.keys())
+
+
+def plotly_graph(problem, df=None):
+    function=problem['f']
+    x_lb=problem['bounds'][0][0]
+    x_ub=problem['bounds'][0][1]
+    y_lb=problem['bounds'][1][0]
+    y_ub=problem['bounds'][1][1]
+    
+    x = np.linspace(x_lb, x_ub, 100)
+    y = np.linspace(y_lb, y_ub, 100)
+    z = np.empty((100, 100))
+    for ind_y, j in enumerate(y):
+        for ind_x, i in enumerate(x):
+            z[ind_y][ind_x] = function(np.array([i, j]))
+    
+    steps_ = int(np.max(df['iter']))
+
+    fig1 = go.Figure(data=[go.Surface(x=x, y=y, z=z)])
+    for step in range(steps_):
+        points = df[df['iter']==step]
+        points_x = list(points['x1'])
+        points_y = list(points['x2'])
+        points_z = list(points['f'])
+        fig1.add_scatter3d(x=np.array(points_x), y=np.array(points_y), z=np.array(points_z), mode='markers', visible=False, marker=dict(size=5, color="white", line=dict(width=1, color="black")))
+        fig1.update_layout(title=f"f = {step}")
+    
+    # Create figure
+    fig = go.Figure(data=[go.Scatter3d(x=[], y=[], z=[],
+                                mode="markers",
+                                marker=dict(size=5, color="white", line=dict(width=1, color="black"))
+                                ), fig1.data[0]]
+                    )
+        
+    # Frames
+    frames = [go.Frame(data=[go.Scatter3d(x=k['x'],
+                                        y=k['y'],
+                                        z=k['z']
+                                        ), fig1.data[0]
+                            ],
+                    traces= [0],
+                    name=f'frame{ind}'      
+                    ) for ind, k  in  enumerate(fig1.data[1:])
+            ]
+
+    fig.update(frames=frames)
+
+    def frame_args(duration):
+        return {
+                "frame": {"duration": duration},
+                "mode": "immediate",
+                "fromcurrent": True,
+                "transition": {"duration": duration, "easing": "linear"},
+                }
+
+
+    sliders = [
+        {"pad": {"b": 10, "t": 60},
+        "len": 0.9,
+        "x": 0.1,
+        "y": 0,
+        
+        "steps": [
+                    {"args": [[f.name], frame_args(0)],
+                    "label": str(k),
+                    "method": "animate",
+                    } for k, f in enumerate(fig.frames)
+                ]
+        }
+            ]
+
+    fig.update_layout(
+
+        updatemenus = [{"buttons":[
+                        {
+                            "args": [None, frame_args(150)],
+                            "label": "Play", 
+                            "method": "animate",
+                        },
+                        {
+                            "args": [[None], frame_args(150)],
+                            "label": "Pause", 
+                            "method": "animate",
+                    }],
+                        
+                    "direction": "left",
+                    "pad": {"r": 10, "t": 70},
+                    "type": "buttons",
+                    "x": 0.1,
+                    "y": 0,
+                }
+            ],
+            sliders=sliders
+        )
+
+    fig.update_layout(sliders=sliders)
+    fig.write_html('animation.html')
+    return fig
+
+        
+problem=test_functions().PERM_D_BETA_  
+df, best_var, best_sol = custom_pso(problem, 20, 2, maxiter=100, solver="Newton", k=0.000000001, G=0.00000001, t=1.0, max_q=0.01, auto_reduce_search_space=True, dinamic=True)
+#plotly_graph(problem, df)
+
+print(best_sol)