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A proper, lower semicontinuous, convex function with no subgradient?
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Let $X$ be a Banach space and $f:Xto Bbb R cup{infty}$ is a proper, lower semicontinuous and convex function.
Is it possible that $partial f(x)=emptyset$ for all $xin text{dom} f$?
If $text{ int dom} fne emptyset$ then the above situation is not possible. However, I couldn't think of a counterexample for the case $text{ int dom} f = emptyset$. Does anyone know if the above statement is true or false?
real-analysis functional-analysis convex-analysis convex-optimization
asked Jan 31 at 16:44
BigbearZzzBigbearZzz
9,01521652
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add a comment |
$begingroup$
Let $X$ be a Banach space and $f:Xto Bbb R cup{infty}$ is a proper, lower semicontinuous and convex function.
Is it possible that $partial f(x)=emptyset$ for all $xin text{dom} f$?
If $text{ int dom} fne emptyset$ then the above situation is not possible. However, I couldn't think of a counterexample for the case $text{ int dom} f = emptyset$. Does anyone know if the above statement is true or false?
real-analysis functional-analysis convex-analysis convex-optimization
asked Jan 31 at 16:44
BigbearZzzBigbearZzz
9,01521652
$endgroup$
add a comment |
$begingroup$
Let $X$ be a Banach space and $f:Xto Bbb R cup{infty}$ is a proper, lower semicontinuous and convex function.
Is it possible that $partial f(x)=emptyset$ for all $xin text{dom} f$?
If $text{ int dom} fne emptyset$ then the above situation is not possible. However, I couldn't think of a counterexample for the case $text{ int dom} f = emptyset$. Does anyone know if the above statement is true or false?
real-analysis functional-analysis convex-analysis convex-optimization
asked Jan 31 at 16:44
BigbearZzzBigbearZzz
9,01521652
$endgroup$
Let $X$ be a Banach space and $f:Xto Bbb R cup{infty}$ is a proper, lower semicontinuous and convex function.
Is it possible that $partial f(x)=emptyset$ for all $xin text{dom} f$?
If $text{ int dom} fne emptyset$ then the above situation is not possible. However, I couldn't think of a counterexample for the case $text{ int dom} f = emptyset$. Does anyone know if the above statement is true or false?
real-analysis functional-analysis convex-analysis convex-optimization
real-analysis functional-analysis convex-analysis convex-optimization
asked Jan 31 at 16:44
BigbearZzzBigbearZzz
9,01521652
asked Jan 31 at 16:44
BigbearZzzBigbearZzz
9,01521652
asked Jan 31 at 16:44
BigbearZzzBigbearZzz
9,01521652
asked Jan 31 at 16:44
BigbearZzzBigbearZzz
9,01521652
asked Jan 31 at 16:44
BigbearZzzBigbearZzz
9,01521652
9,01521652
add a comment |
add a comment |
1 Answer
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The domain of $partial f$ is a dense subset of $text{dom} f$, so it cannot be empty.
(See e.g. Barbu-Precupanu, Convexity and optimization in Banach spaces, Corollary 2.44.)
answered Jan 31 at 18:28
RigelRigel
11.4k11320
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add a comment |
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1 Answer
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$begingroup$
The domain of $partial f$ is a dense subset of $text{dom} f$, so it cannot be empty.
(See e.g. Barbu-Precupanu, Convexity and optimization in Banach spaces, Corollary 2.44.)
answered Jan 31 at 18:28
RigelRigel
11.4k11320
$endgroup$
add a comment |
$begingroup$
The domain of $partial f$ is a dense subset of $text{dom} f$, so it cannot be empty.
(See e.g. Barbu-Precupanu, Convexity and optimization in Banach spaces, Corollary 2.44.)
answered Jan 31 at 18:28
RigelRigel
11.4k11320
$endgroup$
add a comment |
$begingroup$
The domain of $partial f$ is a dense subset of $text{dom} f$, so it cannot be empty.
(See e.g. Barbu-Precupanu, Convexity and optimization in Banach spaces, Corollary 2.44.)
answered Jan 31 at 18:28
RigelRigel
11.4k11320
$endgroup$
The domain of $partial f$ is a dense subset of $text{dom} f$, so it cannot be empty.
(See e.g. Barbu-Precupanu, Convexity and optimization in Banach spaces, Corollary 2.44.)
answered Jan 31 at 18:28
RigelRigel
11.4k11320
answered Jan 31 at 18:28
RigelRigel
11.4k11320
answered Jan 31 at 18:28
RigelRigel
11.4k11320
answered Jan 31 at 18:28
RigelRigel
11.4k11320
11.4k11320
add a comment |
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