Strongly convex function

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Arises for LaTeX: \textstyle f \in C^2: eigenvalues of hessian are bounded away from zero (from below): there exists LaTeX: \textstyle K > 0 such that LaTeX: \textstyle h' H_f(x) h \ge K||h||^2 for all LaTeX: \textstyle h \in \mathbb{R}^n. For example, the function LaTeX: \exp(-x) is strictly convex on LaTeX: \mathbb{R}, but its second derivative is LaTeX: \textstyle \exp(-x), which is not bounded away from zero. The minimum is not achieved because the function approaches its infimum of zero without achieving it for any (finite) LaTeX: x. Strong convexity rules out such asymptotes.


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