Logarithmically concave function: Difference between revisions

Revision as of 21:01, 27 March 2010

Revision as of 00:20, 28 June 2010

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* A twice differentiable function with convex ___domain is log-concave if and only if for all <math>x\in\operatorname{dom} f</math> <math>f(x)>0</math> and

:<math>f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T</math>, <ref> Stephen Boyd and Lieven Vandenberghe, [http://www.stanford.edu/~boyd/cvxbook/ Convex Optimization] (PDF) p.105</ref>

i.e. <math>f(x)\nabla^2f(x) - \nabla f(x)\nabla f(x)^T</math> is [[positive-definite matrix|negative semi-definite]]. If <math>f:\mathbb{R}\to\mathbb{R}^+</math>, this condition simplifies to

If <math>f:\mathbb{R}\to\mathbb{R}^+</math>, this condition simplifies to

:<math>f(x)f''(x) \leq (f'(x))^2</math>