Where am I making a mistake in my calculation ( deriving the dual form from the inequality )
I'm trying to do the derivation for the second problem on the page 13(slide 12) of this pdf : http://www.seas.ucla.edu/~vandenbe/ee236a/lectures/duality.pdf
I'm using Fenchel-Rockafellar duality. If we let $A$ from $X$ to $Y$, ( $(X,W)$ and $(Y,Z)$ are pairs hence $((X,Y),(W,Z))$ is a dual pair) and
$$f(x)=c^{T}x+delta_{R_{+}^{n}}(x)$$
$$g(y)=delta_{R_{+}^{n}}(y)$$
then $f^star(w)=delta_{R_{+}^{n}}(c-w)$ and $g^star(z)=b^{T}z$ then the duality will be $sup-{delta_{R_{+}^{n}}(c+A^star z)+b^{T}z}$ according to the Fenchel-Rockafellar duality. But how I read this is maximimize $-b^{T}z$ subject to $c+A^star zgeq 0$ which is not similar to the given duality. What am I doing wrong?
I'm using this pdf as a source https://people.math.ethz.ch/~patrickc/CA2013.pdf . You can find the definition I'm using from this pdf or Ican write them if needed.
convex-optimization linear-programming duality-theorems
asked Dec 26 at 16:10
dankmemer
149112
add a comment |
I'm trying to do the derivation for the second problem on the page 13(slide 12) of this pdf : http://www.seas.ucla.edu/~vandenbe/ee236a/lectures/duality.pdf
I'm using Fenchel-Rockafellar duality. If we let $A$ from $X$ to $Y$, ( $(X,W)$ and $(Y,Z)$ are pairs hence $((X,Y),(W,Z))$ is a dual pair) and
$$f(x)=c^{T}x+delta_{R_{+}^{n}}(x)$$
$$g(y)=delta_{R_{+}^{n}}(y)$$
then $f^star(w)=delta_{R_{+}^{n}}(c-w)$ and $g^star(z)=b^{T}z$ then the duality will be $sup-{delta_{R_{+}^{n}}(c+A^star z)+b^{T}z}$ according to the Fenchel-Rockafellar duality. But how I read this is maximimize $-b^{T}z$ subject to $c+A^star zgeq 0$ which is not similar to the given duality. What am I doing wrong?
I'm using this pdf as a source https://people.math.ethz.ch/~patrickc/CA2013.pdf . You can find the definition I'm using from this pdf or Ican write them if needed.
convex-optimization linear-programming duality-theorems
asked Dec 26 at 16:10
dankmemer
149112
Does the dual problem given on slide 2 not directly apply to the example on slide 13?
– littleO
Dec 26 at 16:22
I just started to study this stuff i don't know how it exactly works. When i use the same method on the dual problem given on slide 2 i get the same exact thing when i take the conjugates and stuff. But for the problem on slide 13 if what you said is the approach then why do i need to apply the same stuff twice? @littleO
– dankmemer
Dec 26 at 16:28
On slide 13 one of the primal variables is $t$, but I don't see $t$ in the work you've shown. Where's $t$?
– littleO
Dec 26 at 16:36
oops i when i said page 13 i meant page 13 of the pdf, it corresponds to slide 12
– dankmemer
Dec 26 at 16:40
add a comment |
I'm trying to do the derivation for the second problem on the page 13(slide 12) of this pdf : http://www.seas.ucla.edu/~vandenbe/ee236a/lectures/duality.pdf
I'm using Fenchel-Rockafellar duality. If we let $A$ from $X$ to $Y$, ( $(X,W)$ and $(Y,Z)$ are pairs hence $((X,Y),(W,Z))$ is a dual pair) and
$$f(x)=c^{T}x+delta_{R_{+}^{n}}(x)$$
$$g(y)=delta_{R_{+}^{n}}(y)$$
then $f^star(w)=delta_{R_{+}^{n}}(c-w)$ and $g^star(z)=b^{T}z$ then the duality will be $sup-{delta_{R_{+}^{n}}(c+A^star z)+b^{T}z}$ according to the Fenchel-Rockafellar duality. But how I read this is maximimize $-b^{T}z$ subject to $c+A^star zgeq 0$ which is not similar to the given duality. What am I doing wrong?
I'm using this pdf as a source https://people.math.ethz.ch/~patrickc/CA2013.pdf . You can find the definition I'm using from this pdf or Ican write them if needed.
convex-optimization linear-programming duality-theorems
asked Dec 26 at 16:10
dankmemer
149112
I'm trying to do the derivation for the second problem on the page 13(slide 12) of this pdf : http://www.seas.ucla.edu/~vandenbe/ee236a/lectures/duality.pdf
I'm using Fenchel-Rockafellar duality. If we let $A$ from $X$ to $Y$, ( $(X,W)$ and $(Y,Z)$ are pairs hence $((X,Y),(W,Z))$ is a dual pair) and
$$f(x)=c^{T}x+delta_{R_{+}^{n}}(x)$$
$$g(y)=delta_{R_{+}^{n}}(y)$$
then $f^star(w)=delta_{R_{+}^{n}}(c-w)$ and $g^star(z)=b^{T}z$ then the duality will be $sup-{delta_{R_{+}^{n}}(c+A^star z)+b^{T}z}$ according to the Fenchel-Rockafellar duality. But how I read this is maximimize $-b^{T}z$ subject to $c+A^star zgeq 0$ which is not similar to the given duality. What am I doing wrong?
I'm using this pdf as a source https://people.math.ethz.ch/~patrickc/CA2013.pdf . You can find the definition I'm using from this pdf or Ican write them if needed.
convex-optimization linear-programming duality-theorems
convex-optimization linear-programming duality-theorems
asked Dec 26 at 16:10
dankmemer
149112
asked Dec 26 at 16:10
dankmemer
149112
edited Dec 26 at 16:40
asked Dec 26 at 16:10
dankmemer
149112
asked Dec 26 at 16:10
dankmemer
149112
asked Dec 26 at 16:10
dankmemer
149112
149112
Does the dual problem given on slide 2 not directly apply to the example on slide 13?
– littleO
Dec 26 at 16:22
I just started to study this stuff i don't know how it exactly works. When i use the same method on the dual problem given on slide 2 i get the same exact thing when i take the conjugates and stuff. But for the problem on slide 13 if what you said is the approach then why do i need to apply the same stuff twice? @littleO
– dankmemer
Dec 26 at 16:28
On slide 13 one of the primal variables is $t$, but I don't see $t$ in the work you've shown. Where's $t$?
– littleO
Dec 26 at 16:36
oops i when i said page 13 i meant page 13 of the pdf, it corresponds to slide 12
– dankmemer
Dec 26 at 16:40
add a comment |
Does the dual problem given on slide 2 not directly apply to the example on slide 13?
– littleO
Dec 26 at 16:22
I just started to study this stuff i don't know how it exactly works. When i use the same method on the dual problem given on slide 2 i get the same exact thing when i take the conjugates and stuff. But for the problem on slide 13 if what you said is the approach then why do i need to apply the same stuff twice? @littleO
– dankmemer
Dec 26 at 16:28
On slide 13 one of the primal variables is $t$, but I don't see $t$ in the work you've shown. Where's $t$?
– littleO
Dec 26 at 16:36
oops i when i said page 13 i meant page 13 of the pdf, it corresponds to slide 12
– dankmemer
Dec 26 at 16:40
Does the dual problem given on slide 2 not directly apply to the example on slide 13?
– littleO
Dec 26 at 16:22
Does the dual problem given on slide 2 not directly apply to the example on slide 13?
– littleO
Dec 26 at 16:22
I just started to study this stuff i don't know how it exactly works. When i use the same method on the dual problem given on slide 2 i get the same exact thing when i take the conjugates and stuff. But for the problem on slide 13 if what you said is the approach then why do i need to apply the same stuff twice? @littleO
– dankmemer
Dec 26 at 16:28
I just started to study this stuff i don't know how it exactly works. When i use the same method on the dual problem given on slide 2 i get the same exact thing when i take the conjugates and stuff. But for the problem on slide 13 if what you said is the approach then why do i need to apply the same stuff twice? @littleO
– dankmemer
Dec 26 at 16:28
On slide 13 one of the primal variables is $t$, but I don't see $t$ in the work you've shown. Where's $t$?
– littleO
Dec 26 at 16:36
On slide 13 one of the primal variables is $t$, but I don't see $t$ in the work you've shown. Where's $t$?
– littleO
Dec 26 at 16:36
oops i when i said page 13 i meant page 13 of the pdf, it corresponds to slide 12
– dankmemer
Dec 26 at 16:40
oops i when i said page 13 i meant page 13 of the pdf, it corresponds to slide 12
– dankmemer
Dec 26 at 16:40
add a comment |
1 Answer
1
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oldest
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Let $y = -z$. Then the dual problem you derived can be written as maximizing $b^T y$ subject to $A^T y leq c$. This agrees with the dual problem shown on slide 12.
By the way, the Cheridito lecture notes you linked to look quite good, but Vandenberghe's lecture notes are self-contained. In this case you can derive the dual problem by first converting the primal problem to inequality form and then applying the result on slide 2.
You might also consider reading chapter 5 of the Boyd and Vandenberghe Convex Optimization textbook (which is free online). It gives a simple explanation of Lagrange duality, which is an easy way to derive the dual problem shown on slide 2. (Not that there is anything wrong with the Fenchel duality approach; it's also very useful.)
answered Dec 26 at 16:46
littleO
29.2k644108
add a comment |
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1 Answer
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1 Answer
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Let $y = -z$. Then the dual problem you derived can be written as maximizing $b^T y$ subject to $A^T y leq c$. This agrees with the dual problem shown on slide 12.
By the way, the Cheridito lecture notes you linked to look quite good, but Vandenberghe's lecture notes are self-contained. In this case you can derive the dual problem by first converting the primal problem to inequality form and then applying the result on slide 2.
You might also consider reading chapter 5 of the Boyd and Vandenberghe Convex Optimization textbook (which is free online). It gives a simple explanation of Lagrange duality, which is an easy way to derive the dual problem shown on slide 2. (Not that there is anything wrong with the Fenchel duality approach; it's also very useful.)
answered Dec 26 at 16:46
littleO
29.2k644108
add a comment |
Let $y = -z$. Then the dual problem you derived can be written as maximizing $b^T y$ subject to $A^T y leq c$. This agrees with the dual problem shown on slide 12.
By the way, the Cheridito lecture notes you linked to look quite good, but Vandenberghe's lecture notes are self-contained. In this case you can derive the dual problem by first converting the primal problem to inequality form and then applying the result on slide 2.
You might also consider reading chapter 5 of the Boyd and Vandenberghe Convex Optimization textbook (which is free online). It gives a simple explanation of Lagrange duality, which is an easy way to derive the dual problem shown on slide 2. (Not that there is anything wrong with the Fenchel duality approach; it's also very useful.)
answered Dec 26 at 16:46
littleO
29.2k644108
add a comment |
Let $y = -z$. Then the dual problem you derived can be written as maximizing $b^T y$ subject to $A^T y leq c$. This agrees with the dual problem shown on slide 12.
By the way, the Cheridito lecture notes you linked to look quite good, but Vandenberghe's lecture notes are self-contained. In this case you can derive the dual problem by first converting the primal problem to inequality form and then applying the result on slide 2.
You might also consider reading chapter 5 of the Boyd and Vandenberghe Convex Optimization textbook (which is free online). It gives a simple explanation of Lagrange duality, which is an easy way to derive the dual problem shown on slide 2. (Not that there is anything wrong with the Fenchel duality approach; it's also very useful.)
answered Dec 26 at 16:46
littleO
29.2k644108
Let $y = -z$. Then the dual problem you derived can be written as maximizing $b^T y$ subject to $A^T y leq c$. This agrees with the dual problem shown on slide 12.
By the way, the Cheridito lecture notes you linked to look quite good, but Vandenberghe's lecture notes are self-contained. In this case you can derive the dual problem by first converting the primal problem to inequality form and then applying the result on slide 2.
You might also consider reading chapter 5 of the Boyd and Vandenberghe Convex Optimization textbook (which is free online). It gives a simple explanation of Lagrange duality, which is an easy way to derive the dual problem shown on slide 2. (Not that there is anything wrong with the Fenchel duality approach; it's also very useful.)
answered Dec 26 at 16:46
littleO
29.2k644108
edited Dec 27 at 2:31
answered Dec 26 at 16:46
littleO
29.2k644108
answered Dec 26 at 16:46
littleO
29.2k644108
answered Dec 26 at 16:46
littleO
29.2k644108
29.2k644108
add a comment |
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Does the dual problem given on slide 2 not directly apply to the example on slide 13?
– littleO
Dec 26 at 16:22
I just started to study this stuff i don't know how it exactly works. When i use the same method on the dual problem given on slide 2 i get the same exact thing when i take the conjugates and stuff. But for the problem on slide 13 if what you said is the approach then why do i need to apply the same stuff twice? @littleO
– dankmemer
Dec 26 at 16:28
On slide 13 one of the primal variables is $t$, but I don't see $t$ in the work you've shown. Where's $t$?
– littleO
Dec 26 at 16:36
oops i when i said page 13 i meant page 13 of the pdf, it corresponds to slide 12
– dankmemer
Dec 26 at 16:40