we found the cluster centers [- - ] [- ]and [ - ]giving the clustering depicted in figure the corresponding loss ( was found to be clustering via continuous multiextremal optimization as already mentionedthe exact minimization of the loss function ( is difficult to accomplish via standard local search methods such as gradient descentas the function is highly multimodal howevernothing is preventing us from using global optimization methods such as the ce or sco methods discussed in sections and example (clustering via cewe take the same data set as in example and cluster the points via minimization of the loss ( using the ce method the python code below is very similar to the code in example except that now we are dealing with six-dimensional optimization problem the loss function is implemented in the function sclusterwhich essentially reuses the squared distance computation of the -means code in example the ce program typically converges to loss of corresponding to the (globalminimizers [- - ] [- ]and [ - ]which slightly differs from the local minimizers for the -means algorithm clustce py import numpy as np np set_printoptions precision = xmat np genfromtxt ('clusterdata csv 'delimiter =',' nd xmat shape def scluster ( )nd xmat shape dist np zeros (( , )cc reshape ( ,kfor in range ( , )dist [ ,:np sum (xmat cc[:,it)** minvals np amin(dist , return minvals mean (numvar * mu np zeros numvar initialize centers sigma np onesnumvar )* rho nel int( *rho)eps func scluster best_trj np array numvar best_perf np inf trj np zeros shape =(nnumvar )while (np maxsigma )>eps)for in range ( numvar )