Consider the equality-form Linear Program

\begin{aligned} &\text{minimize} & & c^T x \\ & \text{subject to} & & A x = b \\ & & & x \geq 0, \end{aligned}

which has the graph form representation

\begin{aligned} &\text{minimize} & & I(y_{1..m} = b) + y_{m+1} + I(x \geq 0) \\ & \text{subject to} & & y = \begin{bmatrix}A \\\ c^T \end{bmatrix} x \end{aligned}

or equivalently

\begin{aligned} &\text{minimize} & & f(y) + g(x) \\ & \text{subject to} & & y = A x, \end{aligned}

where

f_i(y_i) = \left\{\begin{aligned} & I(y_i = b_i) & i = 1\ldots m \\ & y_i & i = m+1 \end{aligned} \right., ~~\text{ and } ~~g_j(x_j) = I(x_j \geq 0).

### MATLAB Code

 1 2 3 4 5 6 7 8 9 10 11 12 % Generate Data A = rand(100, 10); b = A * rand(10, 1); c = rand(10, 1); % Populate f and g f.h = [kIndEq0(100); kIdentity]; f.b = [b; 0]; g.h = kIndGe0; % Solve. x = pogs([A; c'], f, g); 

This example can be found in the file

 1 /examples/matlab/lp_eq.m 
 .

### R Code

 1 2 3 4 5 6 7 8 9 10 11 # Generate Data A = matrix(runif(100 * 10), 100, 10) b = A %*% runif(10) c = runif(10); # Populate f and g f = list(h = c(kIndEq0(100), kIdentity()), b = c(b, 0)) g = list(h = kIndGe0()) # Solve. solution = pogs(rbind(A, c), f, g) 

This example can be found in the file

 1 /examples/r/lp_eq.m 
 .

### C++ Code

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 #include #include #include "pogs.h" int main() { // Generate Data size_t m = 100, n = 10; std::vector A((m + 1) * n); std::vector b(m); std::vector x(n); std::vector y(m + 1); std::default_random_engine generator; std::uniform_real_distribution u_dist(0.0, 1.0); for (unsigned int i = 0; i < (m + 1) * n; ++i) A[i] = u_dist(generator); for (unsigned int i = 0; i < n; ++i) v[i] = u_dist(generator); for (unsigned int i = 0; i < m; ++i) for (unsigned int j = 0; j < n; ++j) b[i] += A[i * n + j] * v[j]; // Populate f and g PogsData pogs_data(A.data(), m + 1, n); pogs_data.x = x.data(); pogs_data.y = y.data(); pogs_data.f.reserve(m + 1); for (unsigned int i = 0; i < m; ++i) pogs_data.f.emplace_back(kIndEq0, 1.0, b[i]); pogs_data.f.emplace_back(kIdentity); pogs_data.g.reserve(n); for (unsigned int i = 0; i < n; ++i) pogs_data.g.emplace_back(kIndGe0); // Solve Pogs(&pogs_data); } 

This example can be found in the file

 1 /examples/cpp/lp_eq.m 
 .