Least-squares

Consider the following constrained least-squares problem

\[\begin{split}\begin{array}{ll} \mbox{minimize} & \frac{1}{2} \|Ax - b\|_2^2 \\ \mbox{subject to} & 0 \leq x \leq 1 \end{array}\end{split}\]

The problem has the following equivalent form

\[\begin{split}\begin{array}{ll} \mbox{minimize} & \frac{1}{2} y^T y \\ \mbox{subject to} & y = A x - b \\ & 0 \le x \le 1 \end{array}\end{split}\]

Python

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

# Generate problem data
sp.random.seed(1)
m = 30
n = 20
Ad = sparse.random(m, n, density=0.7, format='csc')
b = np.random.randn(m)

# OSQP data
P = sparse.block_diag((sparse.csc_matrix((n, n)), sparse.eye(m)), format='csc')
q = np.zeros(n+m);
A = sparse.vstack([
        sparse.hstack([Ad, -sparse.eye(m)]),
        sparse.hstack((sparse.eye(n), sparse.csc_matrix((n, m))))
    ]).tocsc()
l = np.hstack([b, np.zeros(n)])
u = np.hstack([b, np.ones(n)])

# 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)
m = 30;
n = 20;
Ad = sprandn(m, n, 0.7);
b = randn(m, 1);

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

% Create an OSQP object
prob = osqp;

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

% Solve problem
res = prob.solve();

YALMIP

% Generate data
rng(1)
m = 30;
n = 20;
A = sprandn(m, n, 0.7);
b = randn(m, 1);

% Define problem
x = sdpvar(n, 1);
objective = 0.5*norm(A*x - b)^2;
constraints = [ 0 <= x <= 1];

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

% Get optimal primal and dual solution
x_opt = value(x);
y_opt = dual(constraints(1));