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Wikipedia here explains it like this:

enter image description here

I understand it so that either $\mu_i=0$ or $g_i=0$ but this answer here:

"If μ1≠0 and μ2≠0, then x is one of the two points at the intersection of the two circles (complementary slackness)"

where I cannot see at all what it has to do with complementary-slackness. Could someone clarify this concept and this example? Please, choose an simpler example if the latter too detailed.

P.s. I am doing the non-linear optimization 2.3139.

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You may want to see the video here and at the very end something about complementary-slackness -condition.

enter image description here

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