Equalization of inequality constraints

The nullspace_optimizer package includes the class EqualizedOptimizable that converts inequality constraints into equality constraints by adding slack variables. More precisely, it converts an optimization problem of the form

\[\begin{aligned} \min_{x\in \mathcal{X}}& \quad J(x)\\ \textrm{s.t.} & \left\{\begin{aligned} g_i(x)&=0, \text{ for all } 1\leqslant i\leqslant p,\\ h_j(x) &\leqslant 0, \text{ for all }1\leqslant j \leqslant q,\\ \end{aligned}\right. \end{aligned}\]

into

\[\begin{aligned} \min_{x\in \mathcal{X}}& \quad J(x)\\ \textrm{s.t.} & \left\{\begin{aligned} g_i(x)&=0, \text{ for all } 1\leqslant i\leqslant p,\\ h_j(x)+\frac{1}{2}z_j^2 &= 0, \text{ for all }1\leqslant j \leqslant q,\\ \end{aligned}\right. \end{aligned}\]

The class is used as follows:

python

from nullspace_optimizer import Optimizable, EqualizedOptimizable, \
                                nlspace_solve

class Problem(Optimizable):
     def x0(self):
         #x0
     #...

     def J(self, x):
     # J

     # etc...

 # Equalized version of the optimization problem Problem
 equalized_problem = EqualizedOptimizable(Problem())
 # Solve the equalized version of the problem
 results = nlspace_solve(equalized_problem)