Lagrangian method for a silmple linear programming
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I'm trying to solve a simple LP using Lagrangian method. But I don't know how to use the soloution of the dual problem to find the solution of the main LP.
I consider the following simple problem:
$$ min_{x} sum_{i=1}^n -f_ix_i $$
$$ text{s.t.}quad sum_{i=0}^n x_i=1, xge 0 $$
The general form is:
$$ min c^Tx $$
$$ text{s.t.}quad Ax=b, xge 0 $$
In my problem: $A=[1 , 1,1 cdots ,1]$,
$c=-f=[-f_1,-f_2,cdots,-f_n]^T$ and $b=1$.
The dual problem will be:
$$max ,-v$$
$$text{s.t. }A^Tv-fge 0$$
The constraint means:
$$forall ile n, , vge f_iRightarrow -vle -f_i$$
$$Rightarrow -vle min{-f_i} Rightarrow ,-v= min{-f_i}= -max f_iRightarrow v=max f_i$$
Therefore I've solved the dual problem. but what is the solution of the original problem? My question is:
In general how can I find the solution $x$ after solving the dual problem for $v$?
optimization convex-optimization linear-programming lagrange-multiplier
asked Dec 20 '18 at 5:10
SMA.DSMA.D
452420
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add a comment |
$begingroup$
I'm trying to solve a simple LP using Lagrangian method. But I don't know how to use the soloution of the dual problem to find the solution of the main LP.
I consider the following simple problem:
$$ min_{x} sum_{i=1}^n -f_ix_i $$
$$ text{s.t.}quad sum_{i=0}^n x_i=1, xge 0 $$
The general form is:
$$ min c^Tx $$
$$ text{s.t.}quad Ax=b, xge 0 $$
In my problem: $A=[1 , 1,1 cdots ,1]$,
$c=-f=[-f_1,-f_2,cdots,-f_n]^T$ and $b=1$.
The dual problem will be:
$$max ,-v$$
$$text{s.t. }A^Tv-fge 0$$
The constraint means:
$$forall ile n, , vge f_iRightarrow -vle -f_i$$
$$Rightarrow -vle min{-f_i} Rightarrow ,-v= min{-f_i}= -max f_iRightarrow v=max f_i$$
Therefore I've solved the dual problem. but what is the solution of the original problem? My question is:
In general how can I find the solution $x$ after solving the dual problem for $v$?
optimization convex-optimization linear-programming lagrange-multiplier
asked Dec 20 '18 at 5:10
SMA.DSMA.D
452420
$endgroup$
add a comment |
$begingroup$
I'm trying to solve a simple LP using Lagrangian method. But I don't know how to use the soloution of the dual problem to find the solution of the main LP.
I consider the following simple problem:
$$ min_{x} sum_{i=1}^n -f_ix_i $$
$$ text{s.t.}quad sum_{i=0}^n x_i=1, xge 0 $$
The general form is:
$$ min c^Tx $$
$$ text{s.t.}quad Ax=b, xge 0 $$
In my problem: $A=[1 , 1,1 cdots ,1]$,
$c=-f=[-f_1,-f_2,cdots,-f_n]^T$ and $b=1$.
The dual problem will be:
$$max ,-v$$
$$text{s.t. }A^Tv-fge 0$$
The constraint means:
$$forall ile n, , vge f_iRightarrow -vle -f_i$$
$$Rightarrow -vle min{-f_i} Rightarrow ,-v= min{-f_i}= -max f_iRightarrow v=max f_i$$
Therefore I've solved the dual problem. but what is the solution of the original problem? My question is:
In general how can I find the solution $x$ after solving the dual problem for $v$?
optimization convex-optimization linear-programming lagrange-multiplier
asked Dec 20 '18 at 5:10
SMA.DSMA.D
452420
$endgroup$
I'm trying to solve a simple LP using Lagrangian method. But I don't know how to use the soloution of the dual problem to find the solution of the main LP.
I consider the following simple problem:
$$ min_{x} sum_{i=1}^n -f_ix_i $$
$$ text{s.t.}quad sum_{i=0}^n x_i=1, xge 0 $$
The general form is:
$$ min c^Tx $$
$$ text{s.t.}quad Ax=b, xge 0 $$
In my problem: $A=[1 , 1,1 cdots ,1]$,
$c=-f=[-f_1,-f_2,cdots,-f_n]^T$ and $b=1$.
The dual problem will be:
$$max ,-v$$
$$text{s.t. }A^Tv-fge 0$$
The constraint means:
$$forall ile n, , vge f_iRightarrow -vle -f_i$$
$$Rightarrow -vle min{-f_i} Rightarrow ,-v= min{-f_i}= -max f_iRightarrow v=max f_i$$
Therefore I've solved the dual problem. but what is the solution of the original problem? My question is:
In general how can I find the solution $x$ after solving the dual problem for $v$?
optimization convex-optimization linear-programming lagrange-multiplier
optimization convex-optimization linear-programming lagrange-multiplier
asked Dec 20 '18 at 5:10
SMA.DSMA.D
452420
asked Dec 20 '18 at 5:10
SMA.DSMA.D
452420
asked Dec 20 '18 at 5:10
SMA.DSMA.D
452420
asked Dec 20 '18 at 5:10
SMA.DSMA.D
452420
asked Dec 20 '18 at 5:10
SMA.DSMA.D
452420
452420
add a comment |
add a comment |
1 Answer
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Complementary Slackness Principle tells us that either the inequality $i$ in $A^Tv_{opt}-fge 0$ becomes equality or $x_i=0$. Assuming all the numbers $f_i$ are distinct and $nu_{opt}=max f_i=f_k$. Then $nu_{opt}>f_i$ for $ine k$, hence, $x_i=0$ for $ine k$. Clearly then $x_k=1$.
answered Dec 20 '18 at 7:48
A.Γ.A.Γ.
22.8k32656
$endgroup$
$begingroup$
Thanks for answering. Is it similar to KKT? I didn't understand the descriptions of Wikipedia for complementary slackness.
$endgroup$
– SMA.D
Dec 20 '18 at 8:50
$begingroup$
@SMA.D It is a part of KKT. For inequalities $g(x)le 0$ the Lagrange function is $f(x)+u^Tg(x)$, then in KKT there is a condition that $u_i g_i(x)=0$. It is CSP.
$endgroup$
– A.Γ.
Dec 20 '18 at 8:57
add a comment |
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1 Answer
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1 Answer
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active
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votes
$begingroup$
Complementary Slackness Principle tells us that either the inequality $i$ in $A^Tv_{opt}-fge 0$ becomes equality or $x_i=0$. Assuming all the numbers $f_i$ are distinct and $nu_{opt}=max f_i=f_k$. Then $nu_{opt}>f_i$ for $ine k$, hence, $x_i=0$ for $ine k$. Clearly then $x_k=1$.
answered Dec 20 '18 at 7:48
A.Γ.A.Γ.
22.8k32656
$endgroup$
$begingroup$
Thanks for answering. Is it similar to KKT? I didn't understand the descriptions of Wikipedia for complementary slackness.
$endgroup$
– SMA.D
Dec 20 '18 at 8:50
$begingroup$
@SMA.D It is a part of KKT. For inequalities $g(x)le 0$ the Lagrange function is $f(x)+u^Tg(x)$, then in KKT there is a condition that $u_i g_i(x)=0$. It is CSP.
$endgroup$
– A.Γ.
Dec 20 '18 at 8:57
add a comment |
$begingroup$
Complementary Slackness Principle tells us that either the inequality $i$ in $A^Tv_{opt}-fge 0$ becomes equality or $x_i=0$. Assuming all the numbers $f_i$ are distinct and $nu_{opt}=max f_i=f_k$. Then $nu_{opt}>f_i$ for $ine k$, hence, $x_i=0$ for $ine k$. Clearly then $x_k=1$.
answered Dec 20 '18 at 7:48
A.Γ.A.Γ.
22.8k32656
$endgroup$
$begingroup$
Thanks for answering. Is it similar to KKT? I didn't understand the descriptions of Wikipedia for complementary slackness.
$endgroup$
– SMA.D
Dec 20 '18 at 8:50
$begingroup$
@SMA.D It is a part of KKT. For inequalities $g(x)le 0$ the Lagrange function is $f(x)+u^Tg(x)$, then in KKT there is a condition that $u_i g_i(x)=0$. It is CSP.
$endgroup$
– A.Γ.
Dec 20 '18 at 8:57
add a comment |
$begingroup$
Complementary Slackness Principle tells us that either the inequality $i$ in $A^Tv_{opt}-fge 0$ becomes equality or $x_i=0$. Assuming all the numbers $f_i$ are distinct and $nu_{opt}=max f_i=f_k$. Then $nu_{opt}>f_i$ for $ine k$, hence, $x_i=0$ for $ine k$. Clearly then $x_k=1$.
answered Dec 20 '18 at 7:48
A.Γ.A.Γ.
22.8k32656
$endgroup$
Complementary Slackness Principle tells us that either the inequality $i$ in $A^Tv_{opt}-fge 0$ becomes equality or $x_i=0$. Assuming all the numbers $f_i$ are distinct and $nu_{opt}=max f_i=f_k$. Then $nu_{opt}>f_i$ for $ine k$, hence, $x_i=0$ for $ine k$. Clearly then $x_k=1$.
answered Dec 20 '18 at 7:48
A.Γ.A.Γ.
22.8k32656
answered Dec 20 '18 at 7:48
A.Γ.A.Γ.
22.8k32656
answered Dec 20 '18 at 7:48
A.Γ.A.Γ.
22.8k32656
answered Dec 20 '18 at 7:48
A.Γ.A.Γ.
22.8k32656
22.8k32656
$begingroup$
Thanks for answering. Is it similar to KKT? I didn't understand the descriptions of Wikipedia for complementary slackness.
$endgroup$
– SMA.D
Dec 20 '18 at 8:50
$begingroup$
@SMA.D It is a part of KKT. For inequalities $g(x)le 0$ the Lagrange function is $f(x)+u^Tg(x)$, then in KKT there is a condition that $u_i g_i(x)=0$. It is CSP.
$endgroup$
– A.Γ.
Dec 20 '18 at 8:57
add a comment |
$begingroup$
Thanks for answering. Is it similar to KKT? I didn't understand the descriptions of Wikipedia for complementary slackness.
$endgroup$
– SMA.D
Dec 20 '18 at 8:50
$begingroup$
@SMA.D It is a part of KKT. For inequalities $g(x)le 0$ the Lagrange function is $f(x)+u^Tg(x)$, then in KKT there is a condition that $u_i g_i(x)=0$. It is CSP.
$endgroup$
– A.Γ.
Dec 20 '18 at 8:57
$begingroup$
Thanks for answering. Is it similar to KKT? I didn't understand the descriptions of Wikipedia for complementary slackness.
$endgroup$
– SMA.D
Dec 20 '18 at 8:50
$begingroup$
Thanks for answering. Is it similar to KKT? I didn't understand the descriptions of Wikipedia for complementary slackness.
$endgroup$
– SMA.D
Dec 20 '18 at 8:50
$begingroup$
@SMA.D It is a part of KKT. For inequalities $g(x)le 0$ the Lagrange function is $f(x)+u^Tg(x)$, then in KKT there is a condition that $u_i g_i(x)=0$. It is CSP.
$endgroup$
– A.Γ.
Dec 20 '18 at 8:57
$begingroup$
@SMA.D It is a part of KKT. For inequalities $g(x)le 0$ the Lagrange function is $f(x)+u^Tg(x)$, then in KKT there is a condition that $u_i g_i(x)=0$. It is CSP.
$endgroup$
– A.Γ.
Dec 20 '18 at 8:57
add a comment |
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