Huber fitting

Huber fitting or the robust least-squares problem performs linear regression under the assumption that there are outliers in the data. The fitting problem is written as

\[\begin{array}{ll} \mbox{minimize} & \sum_{i=1}^{m} \phi_{\rm hub}(a_i^T x - b_i), \end{array}\]

with the Huber penalty function \(\phi_{\rm hub}:\mathbf{R}\to\mathbf{R}\) defined as

\[\begin{split}\phi_{\rm hub}(u) = \begin{cases} u^2 & |u| \le 1 \\ (2|u| - 1) & |u| > 1 \end{cases}\end{split}\]

The problem has the following equivalent form (see here for more details)

\[\begin{split}\begin{array}{ll} \mbox{minimize} & u^T u + 2\,\boldsymbol{1}^T (r+s) \\ \mbox{subject to} & Ax - b - u = r - s \\ & r \ge 0 \\ & s \ge 0 \end{array}\end{split}\]

Python

import osqp
import numpy as np
import scipy as sp
from scipy import sparse

# Generate problem data
sp.random.seed(1)
n = 10
m = 100
Ad = sparse.random(m, n, density=0.5, format='csc')
x_true = np.random.randn(n) / np.sqrt(n)
ind95 = (np.random.rand(m) < 0.95).astype(float)
b = Ad@x_true + 0.5*np.random.randn(m) * ind95 \
    + 10.*np.random.rand(m) * (1. - ind95)

# OSQP data
Im = sparse.eye(m)
P = sparse.block_diag([sparse.csc_matrix((n, n)), 2*Im,
                       sparse.csc_matrix((2*m, 2*m))],
                       format='csc')
q = np.append(np.zeros(m+n), 2*np.ones(2*m))
A = sparse.bmat([[Ad,   -Im,   -Im,   Im],
                 [None,  None,  Im,   None],
                 [None,  None,  None, Im]], format='csc')
l = np.hstack([b, np.zeros(2*m)])
u = np.hstack([b, np.inf*np.ones(2*m)])

# Create an OSQP object
prob = osqp.OSQP()

# Setup workspace
prob.setup(P, q, A, l, u)

# Solve problem
res = prob.solve()

Matlab

% Generate problem data
rng(1)
n = 10;
m = 100;
Ad = sprandn(m, n, 0.5);
x_true = randn(n, 1) / sqrt(n);
ind95 = rand(m, 1) > 0.95;
b = Ad*x_true + 10*rand(m, 1).*ind95 + 0.5*randn(m, 1).*(1-ind95);

% OSQP data
Im = speye(m);
Om = sparse(m, m);
Omn = sparse(m, n);
P = blkdiag(sparse(n, n), 2*Im, sparse(2*m, 2*m));
q = [zeros(m + n, 1); 2*ones(2*m, 1)];
A = [Ad,  -Im, -Im, Im;
     Omn,  Om,  Im, Om;
     Omn,  Om,  Om, Im];
l = [b; zeros(2*m, 1)];
u = [b; inf*ones(2*m, 1)];

% Create an OSQP object
prob = osqp;

% Setup workspace
prob.setup(P, q, A, l, u);

% Solve problem
res = prob.solve();

CVXPY

from cvxpy import *
import numpy as np
import scipy as sp
from scipy import sparse

# Generate problem data
sp.random.seed(1)
n = 10
m = 100
A = sparse.random(m, n, density=0.5, format='csc')
x_true = np.random.randn(n) / np.sqrt(n)
ind95 = (np.random.rand(m) < 0.95).astype(float)
b = A@x_true + 0.5*np.random.randn(m) * ind95 \
    + 10.*np.random.rand(m) * (1. - ind95)

# Define problem
x = Variable(n)
objective = sum(huber(A@x - b))

# Solve with OSQP
Problem(Minimize(objective)).solve(solver=OSQP)

YALMIP

% Generate problem data
rng(1)
n = 10;
m = 100;
A = sprandn(m, n, 0.5);
x_true = randn(n, 1) / sqrt(n);
ind95 = rand(m, 1) > 0.95;
b = A*x_true + 10*rand(m, 1).*ind95 + 0.5*randn(m, 1).*(1-ind95);

% Define problem
x = sdpvar(n, 1);
objective = huber(A*x - b);

% Solve with OSQP
options = sdpsettings('solver', 'osqp');
optimize([], objective, options);