Logarithmically concave function

In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality

f(\theta x %2B (1 - \theta) y) \geq f(x)^{\theta} f(y)^{1 - \theta}

for all x,y ∈ dom f and 0 < θ < 1. If f is strictly positive, this is equivalent to saying that the logarithm of the function, log ∘ f, is concave; that is,

\log f(\theta x %2B (1 - \theta) y) \geq \theta \log f(x) %2B (1-\theta) \log f(y)

for all x,y ∈ dom f and 0 < θ < 1.

Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function.

Similarly, a function is log-convex if satisfies the reverse inequality

f(\theta x %2B (1 - \theta) y) \leq f(x)^{\theta} f(y)^{1 - \theta}

for all x,y ∈ dom f and 0 < θ < 1.

Contents

Properties

f''(x)=e^{-\frac{x^2}{2}} (x^2-1) \nleq 0
f(x)\nabla^2f(x) \preceq \nabla f(x)\nabla f(x)^T, [1]
i.e.
f(x)\nabla^2f(x) - \nabla f(x)\nabla f(x)^T is
negative semi-definite. For functions of one variable, this condition simplifies to
f(x)f''(x) \leq (f'(x))^2

Operations preserving log-concavity

\log\,f(x) %2B \log\,g(x) = \log(f(x)g(x))
is concave, and hence also f g is log-concave.
g(x)=\int f(x,y) dy
is log-concave (see Prékopa–Leindler inequality).
(f*g)(x)=\int f(x-y)g(y) dy = \int h(x,y) dy
is log-concave.

Notes

  1. ^ Stephen Boyd and Lieven Vandenberghe, Convex Optimization (PDF) p.105

References

See also