Update kkt_errors!

[klamike]

Before this, the code was actually computing -1 * dual objective, it is why the gap was primal + dual instead of primal - dual. This PR updates the code to compute the dual objective with proper sign, which also cleans up the kernels (no more negation inside the positive_part/negative_part)

[gdalle]

The reason for my choice of notations was to stick to the published papers as much as possible. Is there a stronger reason to deviate?

[klamike]

Which paper? https://arxiv.org/pdf/2106.04756 equation 1, https://arxiv.org/pdf/2311.12180 equation 5 uses the form I am suggesting. I do believe that this form is more popular in the literature more broadly, definitely it is the one that I prefer.

[gdalle]

I think I used the notations from later papers such as

• The journal paper for PDLP: https://arxiv.org/abs/2501.07018
• The description of cuPDLPx: https://arxiv.org/abs/2507.14051

The team around Haihao Lu changed their notations sometime between the first cuPDLP.jl paper and these

[klamike]

I'm not sure I follow. How is it more general? It is just switching/optimizing the computation of $|x|^+$ and $|x|^-$, and computing dobj rather than -dobj. It does look like cuPDLPx is computing dobj since they have primal - dual (coolpdlp had primal + dual before this PR):
https://github.com/MIT-Lu-Lab/cuPDLPx/blob/c28d2c9032f42c7cd0d463f9da4f4c480bbad2e3/src/utils.cu#L883

[klamike]

To me, min(x, 0) and max(x, 0) is much clearer than -max(-x, 0) and -min(-x, 0). Furthermore, the current computation is relying on the fact that we can delay the outer negations all the way until we finally compute the gap. It just seems overly convoluted to me, and I don't understand why it helps to write it this way.

[klamike]

@gdalle friendliest of pings :) have you had a chance to take a look at this?