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In optimization, a self-concordant function is a function f : R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } for which
or, equivalently, a function f : R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } that, wherever f ″ > 0 {\displaystyle f''>0} , satisfies
and which satisfies f ‴ = 0 {\displaystyle f'''=0} elsewhere.
More generally, a multivariate function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is self-concordant if
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