Mixing
Subgradient calculus: Understanding weighted sums property proof
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I'm reading page 72 of this pdf, and I can't understand the proof of the weighted sums property of the subdifferential $partial f$. Weighted sums satisfies:
Given two convex functions $f$ and $g$ on $mathbb{R}^n$ and $lambda, mu > 0$, and a function $h(x) = lambda f(x) + mu g(x)$, then:
$$partial h(x) = lambda partial f(x) + mu partial g(x) $$
For any $x in text{int Dom } h$
For the proof, author uses Corollary 1.2.1 and propositions 3.4.1:
Proposition 3.4.1 (page 69):
$$f′_h(x) = max{h^T d ; d in partial f(x)}$$
with $h in mathbb{R}^n$ and $f$ is a convex function. In short, the directional derivative is the support function of the set $partial f$.
Corollary 1.2.1 in this pdf:
Let $psi_M(x) = sup{y^Tx; y in M}$.
$M$ is a convex set. Function $psi_M(x)$ is called the support function of the set $M$.
Let $M1$ and $M2$ be two closed convex sets.
If for any $x in text{Dom }psi_{M2}$ we have $psi_{M1}(x) leq psi_{M2}(x)$ then $M1 subset M2$
Let $text{Dom }psi_{M1} = text{Dom }psi_{M2}$ and for any $x in text{Dom }psi_{M1}$ we have $psi_{M1}(x) = psi_{M2}(x)$. Then $M1 equiv M2$
Proof of weighted sums:
Let $x in text{int Dom } f cap text{int Dom } g$. Then for any vector $h in mathbb{R}^n$ (notation in this pdf is weird because author uses the same letter for vectors and functions):
$$f'_h(x) = lambda f'_h(x) + mu g'_h(x)$$
I think here's a typo: the term to the left should be $h'_h(x)$:
$$h'_h(x) = lambda f'_h(x) + mu g'_h(x) $$
$$h'_h(x) = max{lambda h^Td_1 ; d_1 in partial f (x) } + max { mu h^Td_2 ; d_2 in partial g(x) } $$
$$ = max{h^T (lambda d_1 + mu d_2) ; d_1 in partial f (x) ; d_2 in partial g(x) } $$
$$ = max{h^T d ; d in lambda partial f (x) + mu partial g(x) }$$
from this point, author uses Corollary 1.2.1 to obtain the property of weighted sums, but doesn't say how. Was the author saying that function $f$ is equal to function $h$ because of this corollary? How do I obtain the equivalence with $partial h(x)$?
convex-optimization subgradient
asked Feb 1 at 20:15
Broken_WindowBroken_Window
267211
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add a comment |
$begingroup$
I'm reading page 72 of this pdf, and I can't understand the proof of the weighted sums property of the subdifferential $partial f$. Weighted sums satisfies:
Given two convex functions $f$ and $g$ on $mathbb{R}^n$ and $lambda, mu > 0$, and a function $h(x) = lambda f(x) + mu g(x)$, then:
$$partial h(x) = lambda partial f(x) + mu partial g(x) $$
For any $x in text{int Dom } h$
For the proof, author uses Corollary 1.2.1 and propositions 3.4.1:
Proposition 3.4.1 (page 69):
$$f′_h(x) = max{h^T d ; d in partial f(x)}$$
with $h in mathbb{R}^n$ and $f$ is a convex function. In short, the directional derivative is the support function of the set $partial f$.
Corollary 1.2.1 in this pdf:
Let $psi_M(x) = sup{y^Tx; y in M}$.
$M$ is a convex set. Function $psi_M(x)$ is called the support function of the set $M$.
Let $M1$ and $M2$ be two closed convex sets.
If for any $x in text{Dom }psi_{M2}$ we have $psi_{M1}(x) leq psi_{M2}(x)$ then $M1 subset M2$
Let $text{Dom }psi_{M1} = text{Dom }psi_{M2}$ and for any $x in text{Dom }psi_{M1}$ we have $psi_{M1}(x) = psi_{M2}(x)$. Then $M1 equiv M2$
Proof of weighted sums:
Let $x in text{int Dom } f cap text{int Dom } g$. Then for any vector $h in mathbb{R}^n$ (notation in this pdf is weird because author uses the same letter for vectors and functions):
$$f'_h(x) = lambda f'_h(x) + mu g'_h(x)$$
I think here's a typo: the term to the left should be $h'_h(x)$:
$$h'_h(x) = lambda f'_h(x) + mu g'_h(x) $$
$$h'_h(x) = max{lambda h^Td_1 ; d_1 in partial f (x) } + max { mu h^Td_2 ; d_2 in partial g(x) } $$
$$ = max{h^T (lambda d_1 + mu d_2) ; d_1 in partial f (x) ; d_2 in partial g(x) } $$
$$ = max{h^T d ; d in lambda partial f (x) + mu partial g(x) }$$
from this point, author uses Corollary 1.2.1 to obtain the property of weighted sums, but doesn't say how. Was the author saying that function $f$ is equal to function $h$ because of this corollary? How do I obtain the equivalence with $partial h(x)$?
convex-optimization subgradient
asked Feb 1 at 20:15
Broken_WindowBroken_Window
267211
$endgroup$
add a comment |
$begingroup$
I'm reading page 72 of this pdf, and I can't understand the proof of the weighted sums property of the subdifferential $partial f$. Weighted sums satisfies:
Given two convex functions $f$ and $g$ on $mathbb{R}^n$ and $lambda, mu > 0$, and a function $h(x) = lambda f(x) + mu g(x)$, then:
$$partial h(x) = lambda partial f(x) + mu partial g(x) $$
For any $x in text{int Dom } h$
For the proof, author uses Corollary 1.2.1 and propositions 3.4.1:
Proposition 3.4.1 (page 69):
$$f′_h(x) = max{h^T d ; d in partial f(x)}$$
with $h in mathbb{R}^n$ and $f$ is a convex function. In short, the directional derivative is the support function of the set $partial f$.
Corollary 1.2.1 in this pdf:
Let $psi_M(x) = sup{y^Tx; y in M}$.
$M$ is a convex set. Function $psi_M(x)$ is called the support function of the set $M$.
Let $M1$ and $M2$ be two closed convex sets.
If for any $x in text{Dom }psi_{M2}$ we have $psi_{M1}(x) leq psi_{M2}(x)$ then $M1 subset M2$
Let $text{Dom }psi_{M1} = text{Dom }psi_{M2}$ and for any $x in text{Dom }psi_{M1}$ we have $psi_{M1}(x) = psi_{M2}(x)$. Then $M1 equiv M2$
Proof of weighted sums:
Let $x in text{int Dom } f cap text{int Dom } g$. Then for any vector $h in mathbb{R}^n$ (notation in this pdf is weird because author uses the same letter for vectors and functions):
$$f'_h(x) = lambda f'_h(x) + mu g'_h(x)$$
I think here's a typo: the term to the left should be $h'_h(x)$:
$$h'_h(x) = lambda f'_h(x) + mu g'_h(x) $$
$$h'_h(x) = max{lambda h^Td_1 ; d_1 in partial f (x) } + max { mu h^Td_2 ; d_2 in partial g(x) } $$
$$ = max{h^T (lambda d_1 + mu d_2) ; d_1 in partial f (x) ; d_2 in partial g(x) } $$
$$ = max{h^T d ; d in lambda partial f (x) + mu partial g(x) }$$
from this point, author uses Corollary 1.2.1 to obtain the property of weighted sums, but doesn't say how. Was the author saying that function $f$ is equal to function $h$ because of this corollary? How do I obtain the equivalence with $partial h(x)$?
convex-optimization subgradient
asked Feb 1 at 20:15
Broken_WindowBroken_Window
267211
$endgroup$
I'm reading page 72 of this pdf, and I can't understand the proof of the weighted sums property of the subdifferential $partial f$. Weighted sums satisfies:
Given two convex functions $f$ and $g$ on $mathbb{R}^n$ and $lambda, mu > 0$, and a function $h(x) = lambda f(x) + mu g(x)$, then:
$$partial h(x) = lambda partial f(x) + mu partial g(x) $$
For any $x in text{int Dom } h$
For the proof, author uses Corollary 1.2.1 and propositions 3.4.1:
Proposition 3.4.1 (page 69):
$$f′_h(x) = max{h^T d ; d in partial f(x)}$$
with $h in mathbb{R}^n$ and $f$ is a convex function. In short, the directional derivative is the support function of the set $partial f$.
Corollary 1.2.1 in this pdf:
Let $psi_M(x) = sup{y^Tx; y in M}$.
$M$ is a convex set. Function $psi_M(x)$ is called the support function of the set $M$.
Let $M1$ and $M2$ be two closed convex sets.
If for any $x in text{Dom }psi_{M2}$ we have $psi_{M1}(x) leq psi_{M2}(x)$ then $M1 subset M2$
Let $text{Dom }psi_{M1} = text{Dom }psi_{M2}$ and for any $x in text{Dom }psi_{M1}$ we have $psi_{M1}(x) = psi_{M2}(x)$. Then $M1 equiv M2$
Proof of weighted sums:
Let $x in text{int Dom } f cap text{int Dom } g$. Then for any vector $h in mathbb{R}^n$ (notation in this pdf is weird because author uses the same letter for vectors and functions):
$$f'_h(x) = lambda f'_h(x) + mu g'_h(x)$$
I think here's a typo: the term to the left should be $h'_h(x)$:
$$h'_h(x) = lambda f'_h(x) + mu g'_h(x) $$
$$h'_h(x) = max{lambda h^Td_1 ; d_1 in partial f (x) } + max { mu h^Td_2 ; d_2 in partial g(x) } $$
$$ = max{h^T (lambda d_1 + mu d_2) ; d_1 in partial f (x) ; d_2 in partial g(x) } $$
$$ = max{h^T d ; d in lambda partial f (x) + mu partial g(x) }$$
from this point, author uses Corollary 1.2.1 to obtain the property of weighted sums, but doesn't say how. Was the author saying that function $f$ is equal to function $h$ because of this corollary? How do I obtain the equivalence with $partial h(x)$?
convex-optimization subgradient
convex-optimization subgradient
asked Feb 1 at 20:15
Broken_WindowBroken_Window
267211
asked Feb 1 at 20:15
Broken_WindowBroken_Window
267211
asked Feb 1 at 20:15
Broken_WindowBroken_Window
267211
asked Feb 1 at 20:15
Broken_WindowBroken_Window
267211
asked Feb 1 at 20:15
Broken_WindowBroken_Window
267211
267211
add a comment |
add a comment |
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