How to show entropy function $R(x)=sum_{i=1}^nx_ilog(x_i)$ is strongly convex?
Let $x in mathbb{R}^n$ belongs to $S$ where
$$
S= {x in mathbb{R}^n mid x succ 0, |x|_1 leq M}
$$
where $succ$ is the generalized inequality which means all elements of $x$ are positive and $log$ is natural logarithm. Use the following theorem to show that $R(x)=sum_{i=1}^nx_ilog(x_i)$ is $frac{1}{M}$-strongly convex over $S$.
Theorem: f is $alpha$-strongly convex if and only if $nabla^2f(x) succeq frac{alpha}{2}I$ for all $x$.
Definition:$f$ is $alpha$-strongly convex if there exist a constant $alpha$ such that
$$ f(y) geq f(x)+left<f '(x),y-xright>+frac{alpha}{2}|y-x|^2$$
for all $x,y$.
linear-algebra convex-analysis hessian-matrix
asked Dec 28 '18 at 18:57
SaeedSaeed
724310
add a comment |
Let $x in mathbb{R}^n$ belongs to $S$ where
$$
S= {x in mathbb{R}^n mid x succ 0, |x|_1 leq M}
$$
where $succ$ is the generalized inequality which means all elements of $x$ are positive and $log$ is natural logarithm. Use the following theorem to show that $R(x)=sum_{i=1}^nx_ilog(x_i)$ is $frac{1}{M}$-strongly convex over $S$.
Theorem: f is $alpha$-strongly convex if and only if $nabla^2f(x) succeq frac{alpha}{2}I$ for all $x$.
Definition:$f$ is $alpha$-strongly convex if there exist a constant $alpha$ such that
$$ f(y) geq f(x)+left<f '(x),y-xright>+frac{alpha}{2}|y-x|^2$$
for all $x,y$.
linear-algebra convex-analysis hessian-matrix
asked Dec 28 '18 at 18:57
SaeedSaeed
724310
add a comment |
Let $x in mathbb{R}^n$ belongs to $S$ where
$$
S= {x in mathbb{R}^n mid x succ 0, |x|_1 leq M}
$$
where $succ$ is the generalized inequality which means all elements of $x$ are positive and $log$ is natural logarithm. Use the following theorem to show that $R(x)=sum_{i=1}^nx_ilog(x_i)$ is $frac{1}{M}$-strongly convex over $S$.
Theorem: f is $alpha$-strongly convex if and only if $nabla^2f(x) succeq frac{alpha}{2}I$ for all $x$.
Definition:$f$ is $alpha$-strongly convex if there exist a constant $alpha$ such that
$$ f(y) geq f(x)+left<f '(x),y-xright>+frac{alpha}{2}|y-x|^2$$
for all $x,y$.
linear-algebra convex-analysis hessian-matrix
asked Dec 28 '18 at 18:57
SaeedSaeed
724310
Let $x in mathbb{R}^n$ belongs to $S$ where
$$
S= {x in mathbb{R}^n mid x succ 0, |x|_1 leq M}
$$
where $succ$ is the generalized inequality which means all elements of $x$ are positive and $log$ is natural logarithm. Use the following theorem to show that $R(x)=sum_{i=1}^nx_ilog(x_i)$ is $frac{1}{M}$-strongly convex over $S$.
Theorem: f is $alpha$-strongly convex if and only if $nabla^2f(x) succeq frac{alpha}{2}I$ for all $x$.
Definition:$f$ is $alpha$-strongly convex if there exist a constant $alpha$ such that
$$ f(y) geq f(x)+left<f '(x),y-xright>+frac{alpha}{2}|y-x|^2$$
for all $x,y$.
linear-algebra convex-analysis hessian-matrix
linear-algebra convex-analysis hessian-matrix
asked Dec 28 '18 at 18:57
SaeedSaeed
724310
asked Dec 28 '18 at 18:57
SaeedSaeed
724310
asked Dec 28 '18 at 18:57
SaeedSaeed
724310
asked Dec 28 '18 at 18:57
SaeedSaeed
724310
asked Dec 28 '18 at 18:57
SaeedSaeed
724310
724310
add a comment |
add a comment |
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