title: applying "tighter" bounds in scipy.optimize.curve_fit
body: <p>I have a dataset that I am trying to fit with parameters (a,b,c,d) that are within +/- 5% of the true fitting parameters. However, when I do this with scipy.optimize.curve_fit I <code>get the error "ValueError: 'x0' is infeasible."</code> inside the least squares package.</p>
  
  <p>If I relax my bounds, then optimize.curve_fit seems to work as desired. I have also noticed that my parameters that are larger seem to be more flexible in applying bounds (i.e. I can get these to work with tighter constraints, but never below 20%).  The following code is an MWE and has two sets of bounds (variable <code>B</code>), one that works and one that returns an error.</p>
  
  <pre><code># %% import modules
  import IPython as IP
  IP.get_ipython().magic('reset -sf')
  import matplotlib.pyplot as plt
  import os as os
  import numpy as np
  import scipy as sp
  import scipy.io as sio
  
  plt.close('all')
  
  
  #%% Load the data
  capacity = np.array([1.0,9.896265560165975472e-01,9.854771784232364551e-01,9.823651452282157193e-01,9.797717842323651061e-01,9.776970954356846155e-01,9.751037344398340023e-01,9.735477178423235234e-01,9.714730290456431439e-01,9.699170124481327759e-01,9.683609958506222970e-01,9.668049792531120401e-01,9.652489626556015612e-01,9.636929460580913043e-01,9.621369294605808253e-01,9.610995850622406911e-01,9.595435684647302121e-01,9.585062240663900779e-01,9.574688796680497216e-01,9.559128630705394647e-01,9.548755186721991084e-01,9.538381742738588631e-01,9.528008298755185068e-01,9.517634854771783726e-01,9.507261410788381273e-01,9.496887966804978820e-01,9.486514522821576367e-01,9.476141078838172804e-01,9.460580912863070235e-01,9.450207468879666672e-01,9.439834024896265330e-01,9.429460580912862877e-01,9.419087136929459314e-01,9.408713692946057972e-01,9.393153526970953182e-01,9.382780082987551840e-01,9.372406639004148277e-01,9.356846473029045708e-01,9.346473029045642145e-01,9.330912863070539576e-01,9.320539419087136013e-01,9.304979253112033444e-01,9.289419087136928654e-01,9.273858921161826085e-01,9.258298755186721296e-01,9.242738589211617617e-01,9.227178423236513938e-01,9.211618257261410259e-01,9.196058091286306579e-01,9.180497925311202900e-01,9.159751037344397995e-01,9.144190871369294316e-01,9.123443983402489410e-01,9.107883817427384621e-01,9.087136929460579715e-01,9.071576763485477146e-01,9.050829875518671130e-01,9.030082987551866225e-01,9.009336099585061319e-01,8.988589211618257524e-01,8.967842323651451508e-01,8.947095435684646603e-01,8.926348547717841697e-01,8.905601659751035681e-01,8.884854771784231886e-01,8.864107883817426980e-01,8.843360995850622075e-01,8.817427385892115943e-01,8.796680497925309927e-01,8.775933609958505022e-01,8.749999999999998890e-01,8.729253112033195094e-01,8.708506224066390189e-01,8.682572614107884057e-01,8.661825726141078041e-01,8.635892116182571909e-01,8.615145228215767004e-01,8.589211618257260872e-01,8.563278008298754740e-01,8.542531120331948724e-01,8.516597510373442592e-01,8.490663900414936460e-01,8.469917012448132665e-01,8.443983402489626533e-01,8.418049792531120401e-01,8.397302904564315496e-01,8.371369294605809364e-01,8.345435684647303232e-01,8.324688796680497216e-01,8.298755186721991084e-01,8.272821576763484952e-01,8.246887966804978820e-01,8.226141078838173915e-01,8.200207468879667783e-01,8.174273858921160540e-01,8.153526970954355635e-01,8.127593360995849503e-01,8.101659751037343371e-01,8.075726141078837239e-01,8.054979253112033444e-01,8.029045643153527312e-01,8.003112033195021180e-01,7.977178423236515048e-01,7.956431535269707922e-01,7.930497925311201790e-01,7.904564315352695658e-01,7.883817427385891863e-01,7.857883817427385731e-01,7.831950207468879599e-01,7.811203319502073583e-01,7.785269709543567451e-01,7.759336099585061319e-01,7.738589211618256414e-01,7.712655601659750282e-01,7.686721991701244150e-01,7.665975103734440355e-01,7.640041493775934223e-01,7.619294605809127097e-01,7.593360995850620965e-01,7.567427385892114833e-01,7.546680497925311037e-01,7.520746887966804906e-01,7.499999999999998890e-01,7.474066390041492758e-01,7.453319502074687852e-01,7.427385892116181720e-01,7.406639004149377925e-01,7.380705394190871793e-01,7.359958506224064667e-01,7.339211618257260872e-01,7.313278008298754740e-01,7.292531120331949834e-01,7.266597510373443702e-01,7.245850622406637687e-01,7.225103734439833891e-01,7.199170124481327759e-01,7.178423236514521744e-01,7.157676348547717948e-01,7.136929460580911933e-01,7.110995850622405801e-01,7.090248962655600895e-01,7.069502074688797100e-01,7.048755186721989974e-01,7.022821576763483842e-01,7.002074688796680046e-01,6.981327800829875141e-01,6.960580912863069125e-01,6.939834024896265330e-01,6.919087136929459314e-01,6.898340248962655519e-01,6.877593360995849503e-01])
  cycles = np.arange(0,151)
  
  
  #%% fit the capacity data
  
  # define the empicrial model to be fitted
  def He_model(k,a,b,c,d):
      return a*np.exp(b*k)+c*np.exp(d*k)
  
  # Fit the entire data set with the function
  P0 = [-40,0.005,40,-0.005]
  fit, tmp = sp.optimize.curve_fit(He_model, cycles,capacity, p0=P0,maxfev=10000000)
  capacity_fit = He_model(cycles, fit[0], fit[1],fit[2], fit[3])
  
  
  # track all four data points with a 5% bound from the best fit
  b_lim = np.zeros((4,2))
  for i in range(4):
      b_lim[i,0] = fit[i]-np.abs(0.2*fit[i]) # these should be 0.05
      b_lim[i,1] = fit[i]+np.abs(0.2*fit[i])
  
  # bounds that work    
  B  = ([b_lim[0,0],-np.inf,b_lim[2,0],-np.inf],[b_lim[0,1], np.inf, b_lim[2,1], np.inf])
  
  # bounds that do not work, but are closer to what I want.
  #B  = ([b_lim[0,0],b_lim[1,0],b_lim[2,0],b_lim[3,0]],[b_lim[0,1], b_lim[1,1], b_lim[2,1], b_lim[3,1]])
  
  
  fit_4_5per, tmp = sp.optimize.curve_fit(He_model, cycles,capacity , p0=P0,bounds=B)
  capacity_4_5per = He_model(cycles, fit_4_5per[0], fit_4_5per[1], fit_4_5per[2], fit_4_5per[3])
  
  #%% plot the results
  
  plt.figure()
  plt.plot(cycles,capacity,'o',fillstyle='none',label='data',markersize=4)
  plt.plot(cycles,capacity_fit,'--',label='best fit')
  plt.plot(cycles,capacity_4_5per,'-.',label='5 percent bounds')
  plt.ylabel('capacity')
  plt.xlabel('charge cycles')
  plt.legend()
  plt.ylim([0.70,1])
  plt.tight_layout()
  </code></pre>
  
  <p>I understand that optimize.curve_fit may need some space to explore the data set to find the optimum spot, however, I feel that 5% should be enough for it. Maybe I am wrong? Is there maybe a better way to apply bounds to a parameter?  </p>