L1 convex optimization with equality constraints in python

问题

I need to minimize L_1(x) subject to Mx = y.

x is a vector with dimension b, y is a vector with dimension a, and M is a matrix with dimensions (a,b).

After some reading I determined to use scipy.optimize.minimize:

import numpy as np
from scipy.optimize import minimize

def objective(x):     #L_1 norm objective function
    return np.linalg.norm(x,ord=1)

constraints = [] #list of all constraint functions
for i in range(a):
    def con(x,y=y,i=i):
        return np.matmul(M[i],x)-y[i]
    constraints.append(con)

#make constraints into tuple of dicts as required by scipy 
cons = tuple({'type':'eq','fun':c} for c in constraints)

#perform the minimization with sequential least squares programming
opt = minimize(objective,x0 = x0,
                        constraints=cons,method='SLSQP',options={'disp': True})

First, what can I use for x0? x is unknown, and I need an x0 which satisfies the constraint M*x0 = y: How can I find an initial guess which satisfies the constraint? M is a matrix of independent Gaussian variables (~N(0,1)) if that helps.

Second, Is there a problem with the way I've set this up? When I use the true x (which I happen to know in the development phase) for x0, I expect it to return x = x0 quickly. Instead, it returns a zero vector x = [0,0,0...,0]. This behavior is unexpected.

Edit:

Here is a solution using cvxpy** solving min(L_1(x)) subject to Mx=y:

import cvxpy as cvx

x = cvx.Variable(b) #b is dim x  
objective = cvx.Minimize(cvx.norm(x,1)) #L_1 norm objective function
constraints = [M*x == y] #y is dim a and M is dim a by b
prob = cvx.Problem(objective,constraints)
result = prob.solve(verbose=False)

#then clean up and chop the 1e-12 vals out of the solution
x = np.array(x.value) #extract array from variable 
x = np.array([a for b in x for a in b]) #unpack the extra brackets
x[np.abs(x)<1e-9]=0 #chop small numbers to 0

来源：https://stackoverflow.com/questions/49787068/l1-convex-optimization-with-equality-constraints-in-python

标签

python

scipy

convex-optimization