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I am trying to understand and remember the conditions here but I cannot yet understand them. You could take any example like here to explain or more theretical slant with proper references. My lecturer is just stating that they are the necessary conditions but I cannot understand why, please, explain. I can see that when $m=0$, it is the Lagrange -multiplier case.

From the Wikipedia:

enter image description here

where you can see the $m$.

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Chapter 5 of Stephen Boyd's book on convex optimization explains this really well. – Nick Alger May 21 '12 at 10:39
@NickAlger: thank you, will take some time to read that. – hhh May 21 '12 at 14:30

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