Update matrices

Consider the following QP

\[\begin{split}\begin{array}{ll} \mbox{minimize} & \frac{1}{2} x^T \begin{bmatrix}4 & 1\\ 1 & 2 \end{bmatrix} x + \begin{bmatrix}1 \\ 1\end{bmatrix}^T x \\ \mbox{subject to} & \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix} \leq \begin{bmatrix} 1 & 1\\ 1 & 0\\ 0 & 1\end{bmatrix} x \leq \begin{bmatrix}1 \\ 0.7 \\ 0.7\end{bmatrix} \end{array}\end{split}\]

We show below how to setup and solve the problem. Then we update the matrices \(P\) and \(A\) and solve the updated problem

\[\begin{split}\begin{array}{ll} \mbox{minimize} & \frac{1}{2} x^T \begin{bmatrix}5 & 1.5\\ 1.5 & 1 \end{bmatrix} x + \begin{bmatrix}1 \\ 1\end{bmatrix}^T x \\ \mbox{subject to} & \begin{bmatrix}1 \\ 0 \\ 0\end{bmatrix} \leq \begin{bmatrix} 1.2 & 1.1\\ 1.5 & 0\\ 0 & 0.8\end{bmatrix} x \leq \begin{bmatrix}1 \\ 0.7 \\ 0.7\end{bmatrix} \end{array}\end{split}\]

Python

import osqp
import numpy as np
from scipy import sparse

# Define problem data
P = sparse.csc_matrix([[4, 1], [1, 2]])
q = np.array([1, 1])
A = sparse.csc_matrix([[1, 1], [1, 0], [0, 1]])
l = np.array([1, 0, 0])
u = np.array([1, 0.7, 0.7])

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

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

# Solve problem
res = prob.solve()

# Update problem
# NB: Update only upper triangular part of P
P_new = sparse.csc_matrix([[5, 1.5], [1.5, 1]])
A_new = sparse.csc_matrix([[1.2, 1.1], [1.5, 0], [0, 0.8]])
prob.update(Px=sparse.triu(P_new).data, Ax=A_new.data)

# Solve updated problem
res = prob.solve()

Matlab

% Define problem data
P = sparse([4, 1; 1, 2]);
q = [1; 1];
A = sparse([1, 1; 1, 0; 0, 1]);
l = [1; 0; 0];
u = [1; 0.7; 0.7];

% Create an OSQP object
prob = osqp;

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

% Solve problem
res = prob.solve();

% Update problem
% NB: Update only upper triangular part of P
P_new = sparse([5, 1.5; 1.5, 1]);
A_new = sparse([1.2, 1.1; 1.5, 0; 0, 0.8]);
prob.update('Px', nonzeros(triu(P_new)), 'Ax', nonzeros(A_new));

% Solve updated problem
res = prob.solve();

Julia

using OSQP
using Compat.SparseArrays, Compat.LinearAlgebra

# Define problem data
P = sparse([4. 1.; 1. 2.])
q = [1.; 1.]
A = sparse([1. 1.; 1. 0.; 0. 1.])
l = [1.; 0.; 0.]
u = [1.; 0.7; 0.7]

# Crate OSQP object
prob = OSQP.Model()

# Setup workspace
OSQP.setup!(prob; P=P, q=q, A=A, l=l, u=u)

# Solve problem
results = OSQP.solve!(prob)

# Update problem
# NB: Update only upper triangular part of P
P_new = sparse([5. 1.5; 1.5 1.])
A_new = sparse([1.2 1.1; 1.5 0.; 0. 0.8])
OSQP.update!(prob, Px=triu(P_new).nzval, Ax=A_new.nzval)

# Solve updated problem
results = OSQP.solve!(prob)

R

library(osqp)
library(Matrix)

# Define problem data
P <- Matrix(c(4., 1.,
              1., 2.), 2, 2, sparse = TRUE)
q <- c(1., 1.)
A <- Matrix(c(1., 1., 0.,
              1., 0., 1.), 3, 2, sparse = TRUE)
l <- c(1., 0., 0.)
u <- c(1., 0.7, 0.7)

# Setup workspace
model <- osqp(P, q, A, l, u)

# Solve problem
res <- model$Solve()

# Update problem
# NB: Update only upper triangular part of P
P_new <- Matrix(c(5., 1.5,
                  1.5, 1.), 2, 2, sparse = TRUE)
A_new <- Matrix(c(1.2, 1.5, 0.,
                  1.1, 0., 0.8), 3, 2, sparse = TRUE)
model$Update(Px = P_new@x, Ax = A_new@x)

# Solve updated problem
res <- model$Solve()

C

#include "osqp.h"

int main(int argc, char **argv) {
    // Load problem data
    c_float P_x[3] = {4.0, 1.0, 2.0, };
    c_float P_x_new[3] = {5.0, 1.5, 1.0, };
    c_int P_nnz = 3;
    c_int P_i[3] = {0, 0, 1, };
    c_int P_p[3] = {0, 1, 3, };
    c_float q[2] = {1.0, 1.0, };
    c_float q_new[2] = {2.0, 3.0, };
    c_float A_x[4] = {1.0, 1.0, 1.0, 1.0, };
    c_float A_x_new[4] = {1.2, 1.5, 1.1, 0.8, };
    c_int A_nnz = 4;
    c_int A_i[4] = {0, 1, 0, 2, };
    c_int A_p[3] = {0, 2, 4, };
    c_float l[3] = {1.0, 0.0, 0.0, };
    c_float l_new[3] = {2.0, -1.0, -1.0, };
    c_float u[3] = {1.0, 0.7, 0.7, };
    c_float u_new[3] = {2.0, 2.5, 2.5, };
    c_int n = 2;
    c_int m = 3;

    // Exitflag
    c_int exitflag = 0;

    // Workspace structures
    OSQPWorkspace *work;
    OSQPSettings  *settings = (OSQPSettings *)c_malloc(sizeof(OSQPSettings));
    OSQPData      *data     = (OSQPData *)c_malloc(sizeof(OSQPData));

    // Populate data
    if (data) {
        data = (OSQPData *)c_malloc(sizeof(OSQPData));
        data->n = n;
        data->m = m;
        data->P = csc_matrix(data->n, data->n, P_nnz, P_x, P_i, P_p);
        data->q = q;
        data->A = csc_matrix(data->m, data->n, A_nnz, A_x, A_i, A_p);
        data->l = l;
        data->u = u;
    }

    // Define Solver settings as default
    if (settings) osqp_set_default_settings(settings);

    // Setup workspace
    exitflag = osqp_setup(&work, data, settings);

    // Solve problem
    osqp_solve(work);

    // Update problem
    // NB: Update only upper triangular part of P
    osqp_update_P(work, P_x_new, OSQP_NULL, 3);
    osqp_update_A(work, A_x_new, OSQP_NULL, 4);

    // Solve updated problem
    osqp_solve(work);

    // Cleanup
    osqp_cleanup(work);
    if (data) {
        if (data->A) c_free(data->A);
        if (data->P) c_free(data->P);
        c_free(data);
    }
    if (settings) c_free(settings);

    return exitflag;
};