Add a square slack model

[MaxenceGollier]

I propose to add a new slack-model: square-slack-model.
It would transform the problem

$$\begin{aligned} \min_x \quad & f(x)\\\ \mathrm{s.t.} \quad & g(x) \leq 0, \\\ \end{aligned}$$

into

$$\begin{aligned} \min_{x, y} \quad & f(x)\\\ \mathrm{s.t.} \quad & g(x) + y^2 = 0,\\\ \end{aligned}$$

see

• https://publications.polymtl.ca/4755/
• https://epubs.siam.org/doi/abs/10.1137/23M1608343

I have a few questions @dpo, is this the formulation you had in mind last week ? Do we apply this procedure only on nonlinear inequality constraints or to box constraints as well ? For my solver, I'd need to apply it to both but i am not sure whether this is a good thing to do in general.

[dpo]

@MaxenceGollier The squared slacks transformation is useful in theory (for example, to derive the augmented Lagrangian method for inequality constraints or the second-order stationarity conditions for problems with inequality constraints), but in practice, it has serious disadvantages: https://publications.polymtl.ca/4755/1/2012_Armand_Squared_slacks_transformation_nonlinear_programming.pdf

[MaxenceGollier]

@dpo, I am confused, you suggested a few weeks ago for me to use this approach to allow my equality-constraint solver to handle inequalities ?

[dpo]

I said it would be an easy way to perform quick tests on inequality-constrained problems, but I don't think it's a viable long-term approach.

[MaxenceGollier]

Ha, ok, closing this then.