Mixing
How to fix this dual cone?
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Consider the following cone:
$$mathbbG_n=Bigg,(xoplusthetaopluskappa) inmathbbR^noplusmathbbR_+oplusmathbbR_+,colon thetasum_iin [n]expbigg(frac-x_ithetabigg)leqkappa Bigg.$$
Where $0exp(fracalpha0)=0$ for each $alphainmathbbR$.
I am trying to calculate the dual of $mathbbG_n$. A similar result is presented in https://tel.archives-ouvertes.fr/tel-00006861/document Sec 6.3 and my attempt is to adapt it to my case. However, I found the following problems:
1- While considering the case where $theta=0$, we still have that $(x,0,kappa)inmathbbG_n$ for each $(x,kappa)inmathbbR^noplusmathbbR_+$. Because $x$ is not necessairly nonnegative, I couldn't obtain any corresponding sign constraints on $x^ast$ and $kappa^ast$ (besides $x^ast=0$ and $kappa^astgeq0$).
2- While minimizing a single term of the summation, I would obtain that $t_i=−log(frackappa^ast)$ because there is no $log$ for negative numbers, is this right?
3- Why is the minimum equal to zero when $kappa^ast=0$?
4- The two final cases colapse and then we obtain that the minimum is $−x_i^astlog(fracx_i^astkappa^ast)$ either way?
5- Finally, we conclude that $x^astin$?, $kappa^astin$? and $theta^astgeqsum_i x_i^astlog(fracx_i^astkappa^ast)−x_i^ast$. Right?
Can anyone help me to fix these?
proof-verification convex-analysis convex-cone dual-cone
asked Jul 22 at 20:11
Ariel Serranoni
7913
add a comment |Â
up vote
0
down vote
favorite
Consider the following cone:
$$mathbbG_n=Bigg,(xoplusthetaopluskappa) inmathbbR^noplusmathbbR_+oplusmathbbR_+,colon thetasum_iin [n]expbigg(frac-x_ithetabigg)leqkappa Bigg.$$
Where $0exp(fracalpha0)=0$ for each $alphainmathbbR$.
I am trying to calculate the dual of $mathbbG_n$. A similar result is presented in https://tel.archives-ouvertes.fr/tel-00006861/document Sec 6.3 and my attempt is to adapt it to my case. However, I found the following problems:
1- While considering the case where $theta=0$, we still have that $(x,0,kappa)inmathbbG_n$ for each $(x,kappa)inmathbbR^noplusmathbbR_+$. Because $x$ is not necessairly nonnegative, I couldn't obtain any corresponding sign constraints on $x^ast$ and $kappa^ast$ (besides $x^ast=0$ and $kappa^astgeq0$).
2- While minimizing a single term of the summation, I would obtain that $t_i=−log(frackappa^ast)$ because there is no $log$ for negative numbers, is this right?
3- Why is the minimum equal to zero when $kappa^ast=0$?
4- The two final cases colapse and then we obtain that the minimum is $−x_i^astlog(fracx_i^astkappa^ast)$ either way?
5- Finally, we conclude that $x^astin$?, $kappa^astin$? and $theta^astgeqsum_i x_i^astlog(fracx_i^astkappa^ast)−x_i^ast$. Right?
Can anyone help me to fix these?
proof-verification convex-analysis convex-cone dual-cone
asked Jul 22 at 20:11
Ariel Serranoni
7913
What does $oplus$ mean?
– copper.hat
Jul 22 at 21:28
1
It is a cartesian product of vector spaces that carries their structure. More formally, if $V$ and $W$ are Euclidean spaces we define the direct sum of $V$ and $W$ as the vector space $Voplus W=voplus w,colon vin Vtext and win W$ where we consider, for each $v_1oplus w_1, v_2oplus w_2in Voplus W$ and $alphainmathbbR$, $v_1oplus w_1 + v_2oplus w_2= (v_1+_Vv_2)oplus (w_1+_Ww_2) $ and $alpha(v_1oplus w_2)=alpha v_1oplusalpha w_1.$ Also, we consider $langle v_1oplus w_1,v_2oplus w_2rangle=langle v_1,v_2rangle_V +langle w_1,w_2rangle_W$.
– Ariel Serranoni
Jul 22 at 21:58
How do you interpret the constraining equation when $theta = 0$? I am assuming that by cone you mean that $t x in C$ for all $t ge 0, x in C$.
– copper.hat
Jul 22 at 23:03
@copper.hat, thanks for pointing this out, I will edit my question to make it clearer. And you are right about the definition of a cone (indeed your definition is for closed cones but it is the case).
– Ariel Serranoni
Jul 23 at 5:15
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider the following cone:
$$mathbbG_n=Bigg,(xoplusthetaopluskappa) inmathbbR^noplusmathbbR_+oplusmathbbR_+,colon thetasum_iin [n]expbigg(frac-x_ithetabigg)leqkappa Bigg.$$
Where $0exp(fracalpha0)=0$ for each $alphainmathbbR$.
I am trying to calculate the dual of $mathbbG_n$. A similar result is presented in https://tel.archives-ouvertes.fr/tel-00006861/document Sec 6.3 and my attempt is to adapt it to my case. However, I found the following problems:
1- While considering the case where $theta=0$, we still have that $(x,0,kappa)inmathbbG_n$ for each $(x,kappa)inmathbbR^noplusmathbbR_+$. Because $x$ is not necessairly nonnegative, I couldn't obtain any corresponding sign constraints on $x^ast$ and $kappa^ast$ (besides $x^ast=0$ and $kappa^astgeq0$).
2- While minimizing a single term of the summation, I would obtain that $t_i=−log(frackappa^ast)$ because there is no $log$ for negative numbers, is this right?
3- Why is the minimum equal to zero when $kappa^ast=0$?
4- The two final cases colapse and then we obtain that the minimum is $−x_i^astlog(fracx_i^astkappa^ast)$ either way?
5- Finally, we conclude that $x^astin$?, $kappa^astin$? and $theta^astgeqsum_i x_i^astlog(fracx_i^astkappa^ast)−x_i^ast$. Right?
Can anyone help me to fix these?
proof-verification convex-analysis convex-cone dual-cone
asked Jul 22 at 20:11
Ariel Serranoni
7913
Consider the following cone:
$$mathbbG_n=Bigg,(xoplusthetaopluskappa) inmathbbR^noplusmathbbR_+oplusmathbbR_+,colon thetasum_iin [n]expbigg(frac-x_ithetabigg)leqkappa Bigg.$$
Where $0exp(fracalpha0)=0$ for each $alphainmathbbR$.
I am trying to calculate the dual of $mathbbG_n$. A similar result is presented in https://tel.archives-ouvertes.fr/tel-00006861/document Sec 6.3 and my attempt is to adapt it to my case. However, I found the following problems:
1- While considering the case where $theta=0$, we still have that $(x,0,kappa)inmathbbG_n$ for each $(x,kappa)inmathbbR^noplusmathbbR_+$. Because $x$ is not necessairly nonnegative, I couldn't obtain any corresponding sign constraints on $x^ast$ and $kappa^ast$ (besides $x^ast=0$ and $kappa^astgeq0$).
2- While minimizing a single term of the summation, I would obtain that $t_i=−log(frackappa^ast)$ because there is no $log$ for negative numbers, is this right?
3- Why is the minimum equal to zero when $kappa^ast=0$?
4- The two final cases colapse and then we obtain that the minimum is $−x_i^astlog(fracx_i^astkappa^ast)$ either way?
5- Finally, we conclude that $x^astin$?, $kappa^astin$? and $theta^astgeqsum_i x_i^astlog(fracx_i^astkappa^ast)−x_i^ast$. Right?
Can anyone help me to fix these?
proof-verification convex-analysis convex-cone dual-cone
asked Jul 22 at 20:11
Ariel Serranoni
7913
edited Jul 23 at 5:16
asked Jul 22 at 20:11
Ariel Serranoni
7913
asked Jul 22 at 20:11
Ariel Serranoni
7913
asked Jul 22 at 20:11
Ariel Serranoni
7913
7913
What does $oplus$ mean?
– copper.hat
Jul 22 at 21:28
1
It is a cartesian product of vector spaces that carries their structure. More formally, if $V$ and $W$ are Euclidean spaces we define the direct sum of $V$ and $W$ as the vector space $Voplus W=voplus w,colon vin Vtext and win W$ where we consider, for each $v_1oplus w_1, v_2oplus w_2in Voplus W$ and $alphainmathbbR$, $v_1oplus w_1 + v_2oplus w_2= (v_1+_Vv_2)oplus (w_1+_Ww_2) $ and $alpha(v_1oplus w_2)=alpha v_1oplusalpha w_1.$ Also, we consider $langle v_1oplus w_1,v_2oplus w_2rangle=langle v_1,v_2rangle_V +langle w_1,w_2rangle_W$.
– Ariel Serranoni
Jul 22 at 21:58
How do you interpret the constraining equation when $theta = 0$? I am assuming that by cone you mean that $t x in C$ for all $t ge 0, x in C$.
– copper.hat
Jul 22 at 23:03
@copper.hat, thanks for pointing this out, I will edit my question to make it clearer. And you are right about the definition of a cone (indeed your definition is for closed cones but it is the case).
– Ariel Serranoni
Jul 23 at 5:15
add a comment |Â
What does $oplus$ mean?
– copper.hat
Jul 22 at 21:28
1
It is a cartesian product of vector spaces that carries their structure. More formally, if $V$ and $W$ are Euclidean spaces we define the direct sum of $V$ and $W$ as the vector space $Voplus W=voplus w,colon vin Vtext and win W$ where we consider, for each $v_1oplus w_1, v_2oplus w_2in Voplus W$ and $alphainmathbbR$, $v_1oplus w_1 + v_2oplus w_2= (v_1+_Vv_2)oplus (w_1+_Ww_2) $ and $alpha(v_1oplus w_2)=alpha v_1oplusalpha w_1.$ Also, we consider $langle v_1oplus w_1,v_2oplus w_2rangle=langle v_1,v_2rangle_V +langle w_1,w_2rangle_W$.
– Ariel Serranoni
Jul 22 at 21:58
How do you interpret the constraining equation when $theta = 0$? I am assuming that by cone you mean that $t x in C$ for all $t ge 0, x in C$.
– copper.hat
Jul 22 at 23:03
@copper.hat, thanks for pointing this out, I will edit my question to make it clearer. And you are right about the definition of a cone (indeed your definition is for closed cones but it is the case).
– Ariel Serranoni
Jul 23 at 5:15
What does $oplus$ mean?
– copper.hat
Jul 22 at 21:28
What does $oplus$ mean?
– copper.hat
Jul 22 at 21:28
1
1
It is a cartesian product of vector spaces that carries their structure. More formally, if $V$ and $W$ are Euclidean spaces we define the direct sum of $V$ and $W$ as the vector space $Voplus W=voplus w,colon vin Vtext and win W$ where we consider, for each $v_1oplus w_1, v_2oplus w_2in Voplus W$ and $alphainmathbbR$, $v_1oplus w_1 + v_2oplus w_2= (v_1+_Vv_2)oplus (w_1+_Ww_2) $ and $alpha(v_1oplus w_2)=alpha v_1oplusalpha w_1.$ Also, we consider $langle v_1oplus w_1,v_2oplus w_2rangle=langle v_1,v_2rangle_V +langle w_1,w_2rangle_W$.
– Ariel Serranoni
Jul 22 at 21:58
It is a cartesian product of vector spaces that carries their structure. More formally, if $V$ and $W$ are Euclidean spaces we define the direct sum of $V$ and $W$ as the vector space $Voplus W=voplus w,colon vin Vtext and win W$ where we consider, for each $v_1oplus w_1, v_2oplus w_2in Voplus W$ and $alphainmathbbR$, $v_1oplus w_1 + v_2oplus w_2= (v_1+_Vv_2)oplus (w_1+_Ww_2) $ and $alpha(v_1oplus w_2)=alpha v_1oplusalpha w_1.$ Also, we consider $langle v_1oplus w_1,v_2oplus w_2rangle=langle v_1,v_2rangle_V +langle w_1,w_2rangle_W$.
– Ariel Serranoni
Jul 22 at 21:58
How do you interpret the constraining equation when $theta = 0$? I am assuming that by cone you mean that $t x in C$ for all $t ge 0, x in C$.
– copper.hat
Jul 22 at 23:03
How do you interpret the constraining equation when $theta = 0$? I am assuming that by cone you mean that $t x in C$ for all $t ge 0, x in C$.
– copper.hat
Jul 22 at 23:03
@copper.hat, thanks for pointing this out, I will edit my question to make it clearer. And you are right about the definition of a cone (indeed your definition is for closed cones but it is the case).
– Ariel Serranoni
Jul 23 at 5:15
@copper.hat, thanks for pointing this out, I will edit my question to make it clearer. And you are right about the definition of a cone (indeed your definition is for closed cones but it is the case).
– Ariel Serranoni
Jul 23 at 5:15
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2859727%2fhow-to-fix-this-dual-cone%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
What does $oplus$ mean?
– copper.hat
Jul 22 at 21:28
1
It is a cartesian product of vector spaces that carries their structure. More formally, if $V$ and $W$ are Euclidean spaces we define the direct sum of $V$ and $W$ as the vector space $Voplus W=voplus w,colon vin Vtext and win W$ where we consider, for each $v_1oplus w_1, v_2oplus w_2in Voplus W$ and $alphainmathbbR$, $v_1oplus w_1 + v_2oplus w_2= (v_1+_Vv_2)oplus (w_1+_Ww_2) $ and $alpha(v_1oplus w_2)=alpha v_1oplusalpha w_1.$ Also, we consider $langle v_1oplus w_1,v_2oplus w_2rangle=langle v_1,v_2rangle_V +langle w_1,w_2rangle_W$.
– Ariel Serranoni
Jul 22 at 21:58
How do you interpret the constraining equation when $theta = 0$? I am assuming that by cone you mean that $t x in C$ for all $t ge 0, x in C$.
– copper.hat
Jul 22 at 23:03
@copper.hat, thanks for pointing this out, I will edit my question to make it clearer. And you are right about the definition of a cone (indeed your definition is for closed cones but it is the case).
– Ariel Serranoni
Jul 23 at 5:15