Logarithmically concave function: Difference between revisions

Revision as of 23:26, 16 March 2016

Revision as of 17:18, 2 June 2016

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==Properties==

* A positive log-concave function is also [[Quasi-concave function|quasi-concave]]. This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are concave.<ref name=":0" />

* Every concave function that is nonnegative on its ___domain is log-concave. However, the reverse does not necessarily hold. An example is the [[Gaussian function]] {{math|''f''(''x'')}}&nbsp;=&nbsp;{{math|exp(&minus;x²/2)}} which is log-concave since {{math|log ''f''(''x'')}}&nbsp;=&nbsp;{{math|&minus;''x''²/2}} is a concave function of {{math|''x''}}. But {{math|''f''}} is not concave since the second derivative is positive for |{{math|''x''}}|&nbsp;>&nbsp;1:

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* A twice differentiable, nonnegative function with a convex ___domain is log-concave if and only if for all {{math|''x''}} satisfying {{math|''f''(''x'')&nbsp;>&nbsp;0}},

::<math>f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T</math>,<ref name=":0">Stephen Boyd and Lieven Vandenberghe, [http://www.stanford.edu/~boyd/cvxbook/ Convex Optimization] (PDF) pSection 3.1055</ref>

:i.e.