Mixing
Positivity constraints in optimization
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How do you enforce positivity constraints in non-linear optimization (e.g. a constraint $x > 0$)? I remember there being a good reason for why most models use non-negativity constraints.
optimization
edited Jan 19 '12 at 11:38
Dirk
8,9672447
asked Jan 18 '12 at 0:16
JacobJacob
1,92811225
$endgroup$
add a comment |
$begingroup$
How do you enforce positivity constraints in non-linear optimization (e.g. a constraint $x > 0$)? I remember there being a good reason for why most models use non-negativity constraints.
optimization
edited Jan 19 '12 at 11:38
Dirk
8,9672447
asked Jan 18 '12 at 0:16
JacobJacob
1,92811225
$endgroup$
$begingroup$
How about the KKT conditions? en.wikipedia.org/wiki/…
$endgroup$
– matt
Jan 18 '12 at 11:43
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@matt:They handle non-negative and equality constraints.
$endgroup$
– Jacob
Jan 18 '12 at 12:03
$begingroup$
:You could use "Log Barrier methods". I will post an answer with more detail.
$endgroup$
– matt
Jan 19 '12 at 10:12
add a comment |
$begingroup$
How do you enforce positivity constraints in non-linear optimization (e.g. a constraint $x > 0$)? I remember there being a good reason for why most models use non-negativity constraints.
optimization
edited Jan 19 '12 at 11:38
Dirk
8,9672447
asked Jan 18 '12 at 0:16
JacobJacob
1,92811225
$endgroup$
How do you enforce positivity constraints in non-linear optimization (e.g. a constraint $x > 0$)? I remember there being a good reason for why most models use non-negativity constraints.
optimization
optimization
edited Jan 19 '12 at 11:38
Dirk
8,9672447
asked Jan 18 '12 at 0:16
JacobJacob
1,92811225
edited Jan 19 '12 at 11:38
Dirk
8,9672447
asked Jan 18 '12 at 0:16
JacobJacob
1,92811225
edited Jan 19 '12 at 11:38
Dirk
8,9672447
edited Jan 19 '12 at 11:38
Dirk
8,9672447
edited Jan 19 '12 at 11:38
Dirk
8,9672447
8,9672447
asked Jan 18 '12 at 0:16
JacobJacob
1,92811225
asked Jan 18 '12 at 0:16
JacobJacob
1,92811225
asked Jan 18 '12 at 0:16
JacobJacob
1,92811225
1,92811225
$begingroup$
How about the KKT conditions? en.wikipedia.org/wiki/…
$endgroup$
– matt
Jan 18 '12 at 11:43
$begingroup$
@matt:They handle non-negative and equality constraints.
$endgroup$
– Jacob
Jan 18 '12 at 12:03
$begingroup$
:You could use "Log Barrier methods". I will post an answer with more detail.
$endgroup$
– matt
Jan 19 '12 at 10:12
add a comment |
$begingroup$
How about the KKT conditions? en.wikipedia.org/wiki/…
$endgroup$
– matt
Jan 18 '12 at 11:43
$begingroup$
@matt:They handle non-negative and equality constraints.
$endgroup$
– Jacob
Jan 18 '12 at 12:03
$begingroup$
:You could use "Log Barrier methods". I will post an answer with more detail.
$endgroup$
– matt
Jan 19 '12 at 10:12
$begingroup$
How about the KKT conditions? en.wikipedia.org/wiki/…
$endgroup$
– matt
Jan 18 '12 at 11:43
$begingroup$
How about the KKT conditions? en.wikipedia.org/wiki/…
$endgroup$
– matt
Jan 18 '12 at 11:43
$begingroup$
@matt:They handle non-negative and equality constraints.
$endgroup$
– Jacob
Jan 18 '12 at 12:03
$begingroup$
@matt:They handle non-negative and equality constraints.
$endgroup$
– Jacob
Jan 18 '12 at 12:03
$begingroup$
:You could use "Log Barrier methods". I will post an answer with more detail.
$endgroup$
– matt
Jan 19 '12 at 10:12
$begingroup$
:You could use "Log Barrier methods". I will post an answer with more detail.
$endgroup$
– matt
Jan 19 '12 at 10:12
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Consider the following:
$$begin{array}{rll}
min_x& f(x)&\
text{subject to}& g_i(x)leq0 &text{for each }i\
& h_i(x)=0 &text{for each }j \
end{array}$$
For $alpha>0$ we define the log barrier penalty function, $P_alpha$, to be:
$$ P_alpha(x)=f(x)-frac1alphasum_ilog(-g_i(x))+alphasum_jh_j(x)^2 $$
where $x$ must be strictly feasible, i.e. $g_i(x)<0$ for each $i$, in order for the log term to be defined.
We seek to minimise $P_alpha(x)$. The idea is that the boundary of the feasible region (i.e $g_i(x)=0$) acts a a barrier for $x$ close to $0$.
edited Feb 3 at 9:02
Appliqué
3,98631839
answered Jan 19 '12 at 10:29
mattmatt
1,3161118
$endgroup$
1
$begingroup$
Should that be $log(-g_i(x))$? I think you have an extra left parenthesis.
$endgroup$
– rhombidodecahedron
Nov 17 '14 at 19:18
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
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active
oldest
votes
$begingroup$
Consider the following:
$$begin{array}{rll}
min_x& f(x)&\
text{subject to}& g_i(x)leq0 &text{for each }i\
& h_i(x)=0 &text{for each }j \
end{array}$$
For $alpha>0$ we define the log barrier penalty function, $P_alpha$, to be:
$$ P_alpha(x)=f(x)-frac1alphasum_ilog(-g_i(x))+alphasum_jh_j(x)^2 $$
where $x$ must be strictly feasible, i.e. $g_i(x)<0$ for each $i$, in order for the log term to be defined.
We seek to minimise $P_alpha(x)$. The idea is that the boundary of the feasible region (i.e $g_i(x)=0$) acts a a barrier for $x$ close to $0$.
edited Feb 3 at 9:02
Appliqué
3,98631839
answered Jan 19 '12 at 10:29
mattmatt
1,3161118
$endgroup$
1
$begingroup$
Should that be $log(-g_i(x))$? I think you have an extra left parenthesis.
$endgroup$
– rhombidodecahedron
Nov 17 '14 at 19:18
add a comment |
$begingroup$
Consider the following:
$$begin{array}{rll}
min_x& f(x)&\
text{subject to}& g_i(x)leq0 &text{for each }i\
& h_i(x)=0 &text{for each }j \
end{array}$$
For $alpha>0$ we define the log barrier penalty function, $P_alpha$, to be:
$$ P_alpha(x)=f(x)-frac1alphasum_ilog(-g_i(x))+alphasum_jh_j(x)^2 $$
where $x$ must be strictly feasible, i.e. $g_i(x)<0$ for each $i$, in order for the log term to be defined.
We seek to minimise $P_alpha(x)$. The idea is that the boundary of the feasible region (i.e $g_i(x)=0$) acts a a barrier for $x$ close to $0$.
edited Feb 3 at 9:02
Appliqué
3,98631839
answered Jan 19 '12 at 10:29
mattmatt
1,3161118
$endgroup$
1
$begingroup$
Should that be $log(-g_i(x))$? I think you have an extra left parenthesis.
$endgroup$
– rhombidodecahedron
Nov 17 '14 at 19:18
add a comment |
$begingroup$
Consider the following:
$$begin{array}{rll}
min_x& f(x)&\
text{subject to}& g_i(x)leq0 &text{for each }i\
& h_i(x)=0 &text{for each }j \
end{array}$$
For $alpha>0$ we define the log barrier penalty function, $P_alpha$, to be:
$$ P_alpha(x)=f(x)-frac1alphasum_ilog(-g_i(x))+alphasum_jh_j(x)^2 $$
where $x$ must be strictly feasible, i.e. $g_i(x)<0$ for each $i$, in order for the log term to be defined.
We seek to minimise $P_alpha(x)$. The idea is that the boundary of the feasible region (i.e $g_i(x)=0$) acts a a barrier for $x$ close to $0$.
edited Feb 3 at 9:02
Appliqué
3,98631839
answered Jan 19 '12 at 10:29
mattmatt
1,3161118
$endgroup$
Consider the following:
$$begin{array}{rll}
min_x& f(x)&\
text{subject to}& g_i(x)leq0 &text{for each }i\
& h_i(x)=0 &text{for each }j \
end{array}$$
For $alpha>0$ we define the log barrier penalty function, $P_alpha$, to be:
$$ P_alpha(x)=f(x)-frac1alphasum_ilog(-g_i(x))+alphasum_jh_j(x)^2 $$
where $x$ must be strictly feasible, i.e. $g_i(x)<0$ for each $i$, in order for the log term to be defined.
We seek to minimise $P_alpha(x)$. The idea is that the boundary of the feasible region (i.e $g_i(x)=0$) acts a a barrier for $x$ close to $0$.
edited Feb 3 at 9:02
Appliqué
3,98631839
answered Jan 19 '12 at 10:29
mattmatt
1,3161118
edited Feb 3 at 9:02
Appliqué
3,98631839
edited Feb 3 at 9:02
Appliqué
3,98631839
edited Feb 3 at 9:02
Appliqué
3,98631839
3,98631839
answered Jan 19 '12 at 10:29
mattmatt
1,3161118
answered Jan 19 '12 at 10:29
mattmatt
1,3161118
answered Jan 19 '12 at 10:29
mattmatt
1,3161118
1,3161118
1
$begingroup$
Should that be $log(-g_i(x))$? I think you have an extra left parenthesis.
$endgroup$
– rhombidodecahedron
Nov 17 '14 at 19:18
add a comment |
1
$begingroup$
Should that be $log(-g_i(x))$? I think you have an extra left parenthesis.
$endgroup$
– rhombidodecahedron
Nov 17 '14 at 19:18
1
1
$begingroup$
Should that be $log(-g_i(x))$? I think you have an extra left parenthesis.
$endgroup$
– rhombidodecahedron
Nov 17 '14 at 19:18
$begingroup$
Should that be $log(-g_i(x))$? I think you have an extra left parenthesis.
$endgroup$
– rhombidodecahedron
Nov 17 '14 at 19:18
add a comment |
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$begingroup$
How about the KKT conditions? en.wikipedia.org/wiki/…
$endgroup$
– matt
Jan 18 '12 at 11:43
$begingroup$
@matt:They handle non-negative and equality constraints.
$endgroup$
– Jacob
Jan 18 '12 at 12:03
$begingroup$
:You could use "Log Barrier methods". I will post an answer with more detail.
$endgroup$
– matt
Jan 19 '12 at 10:12