EqualizedOptimizable

class EqualizedOptimizable(problem, coeffs_z=None)

An Optimizable object automatically converts all inequality constraints into equality constraints with the method of slack variables.

Usage

python

equalizedProblem = EqualizedOptimizable(problem)

Supposing that problem corresponds to the optimization problem

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

the equalizedProblem corresponds to the equalized program

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

Optimization points for the EqualizedOptimizable object are tuples of the form (x,z) with x an optimization point for problem and z a list of size \(q\) the number of inequality constraints.

The initial value for the slack variable is set to

\[z_j(0) = \sqrt{2 |h_i(x(0))|} \text{ for } 1\leqslant j \leqslant q\]

The inner product chosen on the z variable is the usual Euclidean inner product. Derivatives with respect to the z variable are appended at the end of the derivative arrays with respect to x.

Note

In practice, the equalization helps to find a central path going from the initialization to the optimum in between the constraints. However, equalization can be inefficient or require some tuning for problems with a large number of constraints.

Problem:

An instance of Optimizable implementing an arbitrary optimization program.

Parameters:

coeffs_z – some coefficients to equalize \(h_j(x)\leqslant 0\) into \(h_j(x)+\frac{1}{2}\texttt{coeffs\_z}[j] z_j^2=0\).