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Nov 28, 2021 · $g(x)=e^{-x}, f(x)=x^{2}$ is a counter-example. If add the condition that $g$ is also increasing then the result is true. [ $g\circ f$ is concave on $(0,\frac 1 {\sqrt 2})$ and convex on $(\frac 1 {\sqrt 2}, \infty)$ ].
Free Functions Concavity Calculator - find function concavity intervlas step-by-step.
- Examples on Rn and Rm×n
- f (y) ≥ f (x) + ∇f (x)T (y − x)
- f (x) = ( n k=1 xk)1/n
- Pointwise supremum
- Quasiconvex functions
- Properties of log-concave functions
- f (x, y) dy
- • w ∈ Rn:
affine functions are convex and concave; all norms are convex examples on Rn affine function f (x) P = aT x + b norms: kxk n
for all x, y ∈ dom f (y) (x) + ∇f (x)T (y − x) (x, f (x)) first-order approximation of f is global underestimator
on Rn ++ is concave (similar proof as for log-sum-exp)
f (x, y) is convex in x for each y ∈ A, then g(x) = sup f (x, y) y∈A support function of a set C: SC (x) = sup y∈C yT x is convex distance to farthest point in a set C: (x) = sup kx − yk y∈C maximum eigenvalue of symmetric matrix: for X
Rn → R is quasiconvex if dom f is convex and the sublevel sets
twice differentiable f with convex domain is log-concave if and only if (x)∇2f (x) ∇ f (x)∇f (x)T for all x ∈ dom f product of log-concave functions is log-concave sum of log-concave functions is not always log-concave integration: if f : n m R × R → R is log-concave, then Z g(x) =
is log-concave (not easy to show) consequences of integration property convolution f ∗ g of log-concave functions Z (f ∗ g)(x) = f (x − , g is log-concave y)g(y)dy if C ⊆ R n convex and y is a random variable with log-concave pdf then is log-concave proof: write f (x) Z
random variations of parameters in manufactured product • S: set of acceptable values S is convex and w has a log-concave pdf, then • Y is log-concave • yield regions {x | Y (x) ≥ α} are convex
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above gives g(x) = g(P n x ne n) = n g(x nen) = n x ng(en) = P n x na n = Ax. Finally, f(x) = g(x) + y = Ax+ y. For N = 1, the next result says that a function is concave i , informally, its slope is weakly decreasing. If the function is di erentiable then the implication is that the derivative is weakly decreasing. Theorem 3. Let C R be an ...
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Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is $g(f)$ strictly convex.
dom f = {x ∈ dom g| g(x) ∈ dom h} Simple case: h:R R and g:Rn R, general domain and non-differentiable. Example of sufficient (but not necessary) conditions: f convex if h is convex and ~h nondecreasing and g is convex ~. f convex if h is convex and h nonincreasing and g is concave.
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g(t) = f(x + tv), dom g = {t | x + tv ∈ dom f} is convex (in t) for any x ∈ dom f, v ∈ Rn can check convexity of f by checking convexity of functions of one variable example. f : Sn R with f(X) = logdetX, dom f = S ++ → n g(t) = logdet(X + tV ) = logdetX +logdet(I + tX−1/2VX−1/2) n = logdetX + log(1+tλ i) i=1 where λ
