# Chemical equilibrium problem

$LaTeX: \min \left\{ c^Tx + \sum_j \left(x_j \ln(x_j) \right) : x > 0, \; \sum\limits_j x_j = M, \; Ax=b \right\},$
where the objective is the Gibbs free energy function with $LaTeX: x_j$ being the number of moles of species $LaTeX: j$, $LaTeX: b_i$ being the atomic weight of atom of type $LaTeX: i$, and $LaTeX: A_{i,j}$ being the number of atoms of type $LaTeX: i$ in one mole of species $LaTeX: j$. The equation, $LaTeX: Ax=b$, is mass balance. (Its significance is that it was an early application to show how an LP approach can apply to a nonlinear mathematical program.)