Squashed 'third_party/osqp/' content from commit 33454b3e23
Change-Id: I056df0582ca06664e86554c341a94c47ab932001
git-subtree-dir: third_party/osqp
git-subtree-split: 33454b3e236f1f44193bfbbb6b8c8e71f8f04e9a
Signed-off-by: Austin Schuh <austin.linux@gmail.com>
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+Least-squares
+=============
+
+Consider the following constrained least-squares problem
+
+.. math::
+ \begin{array}{ll}
+ \mbox{minimize} & \frac{1}{2} \|Ax - b\|_2^2 \\
+ \mbox{subject to} & 0 \leq x \leq 1
+ \end{array}
+
+The problem has the following equivalent form
+
+.. math::
+ \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}
+
+
+
+Python
+------
+
+.. code:: python
+
+ import osqp
+ import numpy as np
+ import scipy as sp
+ from scipy import 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))])], format='csc')
+ 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
+------
+
+.. code:: 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();
+
+
+
+CVXPY
+-----
+
+.. code:: python
+
+ from cvxpy import *
+ import numpy as np
+ import scipy as sp
+ from scipy import sparse
+
+ # Generate problem data
+ sp.random.seed(1)
+ m = 30
+ n = 20
+ A = sparse.random(m, n, density=0.7, format='csc')
+ b = np.random.randn(m)
+
+ # Define problem
+ x = Variable(n)
+ objective = 0.5*sum_squares(A*x - b)
+ constraints = [x >= 0, x <= 1]
+
+ # Solve with OSQP
+ Problem(Minimize(objective), constraints).solve(solver=OSQP)
+
+
+
+YALMIP
+------
+
+.. code:: matlab
+
+ % 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);
+