Is the S-procedure or S-lemma valid for nonquadratic inequalities?

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The s-procedure or s-lemma tries to solve a system of quadratic inequalities via a linear matrix inequality (LMI) relaxation. There are many different enunciations and if you are not sure what i mean you can check here.

I have seen it be used in papers where the inequalities are not quadratic, for example here, but I cannot find a form of the theorem that seems to apply. Can anyone help me?

Thanks in advance.