convexity(concavity) of linear optimization w.r.t. constraint
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I have a class of linear optimization problems $f:Dto mathbb{R}$:
$$f(d)=max_x b'x$$
s.t.
$$ Ax=d$$
$$x_ige 0$$
where $xin mathbb{R}^n$, $D={d|exists xin mathbb{R}^n:Ax=d,x_ige 0}$ . I've prooved that $f$ is homogeneous and concave, which impies:
$$f(d_1)+f(d_2)le f(d_1+d_2)$$
Is there any condition over $d_1$ and $d_2$ such that the strict inequality hold?
optimization convex-analysis
asked Dec 2 at 5:50
di bao
62
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up vote
1
down vote
favorite
I have a class of linear optimization problems $f:Dto mathbb{R}$:
$$f(d)=max_x b'x$$
s.t.
$$ Ax=d$$
$$x_ige 0$$
where $xin mathbb{R}^n$, $D={d|exists xin mathbb{R}^n:Ax=d,x_ige 0}$ . I've prooved that $f$ is homogeneous and concave, which impies:
$$f(d_1)+f(d_2)le f(d_1+d_2)$$
Is there any condition over $d_1$ and $d_2$ such that the strict inequality hold?
optimization convex-analysis
asked Dec 2 at 5:50
di bao
62
New contributor
di bao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Is there a non strict equality? If $d_k$ are not in the range of $A$ then $f(d_k) = -infty$ :-).
– copper.hat
Dec 2 at 6:45
Hmmm.. another typo. And the domain of $f$ shall have non-empty feasible set.
– di bao
Dec 2 at 7:23
Not sure what kind of a condition you are looking for. The strict inequality holds is already a condition. Do you need another one? Besides, $A$ and $b$ would for sure affect any condition.
– A.Γ.
Dec 2 at 12:32
Yes, I'm searching for condition like $g(d_1,d_2,A,b)>0$, and $g$ is explicitly defined.
– di bao
Dec 2 at 19:11
the optimal basis should be different for $d_1$ and $d_2$; revised simplex is your friend
– LinAlg
Dec 2 at 22:51
|
show 1 more comment
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I have a class of linear optimization problems $f:Dto mathbb{R}$:
$$f(d)=max_x b'x$$
s.t.
$$ Ax=d$$
$$x_ige 0$$
where $xin mathbb{R}^n$, $D={d|exists xin mathbb{R}^n:Ax=d,x_ige 0}$ . I've prooved that $f$ is homogeneous and concave, which impies:
$$f(d_1)+f(d_2)le f(d_1+d_2)$$
Is there any condition over $d_1$ and $d_2$ such that the strict inequality hold?
optimization convex-analysis
asked Dec 2 at 5:50
di bao
62
New contributor
di bao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I have a class of linear optimization problems $f:Dto mathbb{R}$:
$$f(d)=max_x b'x$$
s.t.
$$ Ax=d$$
$$x_ige 0$$
where $xin mathbb{R}^n$, $D={d|exists xin mathbb{R}^n:Ax=d,x_ige 0}$ . I've prooved that $f$ is homogeneous and concave, which impies:
$$f(d_1)+f(d_2)le f(d_1+d_2)$$
Is there any condition over $d_1$ and $d_2$ such that the strict inequality hold?
optimization convex-analysis
optimization convex-analysis
asked Dec 2 at 5:50
di bao
62
New contributor
di bao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 2 at 5:50
di bao
62
New contributor
di bao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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edited Dec 2 at 7:17
asked Dec 2 at 5:50
di bao
62
New contributor
di bao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 2 at 5:50
di bao
62
asked Dec 2 at 5:50
di bao
62
62
New contributor
di bao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
di bao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
di bao is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Is there a non strict equality? If $d_k$ are not in the range of $A$ then $f(d_k) = -infty$ :-).
– copper.hat
Dec 2 at 6:45
Hmmm.. another typo. And the domain of $f$ shall have non-empty feasible set.
– di bao
Dec 2 at 7:23
Not sure what kind of a condition you are looking for. The strict inequality holds is already a condition. Do you need another one? Besides, $A$ and $b$ would for sure affect any condition.
– A.Γ.
Dec 2 at 12:32
Yes, I'm searching for condition like $g(d_1,d_2,A,b)>0$, and $g$ is explicitly defined.
– di bao
Dec 2 at 19:11
the optimal basis should be different for $d_1$ and $d_2$; revised simplex is your friend
– LinAlg
Dec 2 at 22:51
|
show 1 more comment
Is there a non strict equality? If $d_k$ are not in the range of $A$ then $f(d_k) = -infty$ :-).
– copper.hat
Dec 2 at 6:45
Hmmm.. another typo. And the domain of $f$ shall have non-empty feasible set.
– di bao
Dec 2 at 7:23
Not sure what kind of a condition you are looking for. The strict inequality holds is already a condition. Do you need another one? Besides, $A$ and $b$ would for sure affect any condition.
– A.Γ.
Dec 2 at 12:32
Yes, I'm searching for condition like $g(d_1,d_2,A,b)>0$, and $g$ is explicitly defined.
– di bao
Dec 2 at 19:11
the optimal basis should be different for $d_1$ and $d_2$; revised simplex is your friend
– LinAlg
Dec 2 at 22:51
Is there a non strict equality? If $d_k$ are not in the range of $A$ then $f(d_k) = -infty$ :-).
– copper.hat
Dec 2 at 6:45
Is there a non strict equality? If $d_k$ are not in the range of $A$ then $f(d_k) = -infty$ :-).
– copper.hat
Dec 2 at 6:45
Hmmm.. another typo. And the domain of $f$ shall have non-empty feasible set.
– di bao
Dec 2 at 7:23
Hmmm.. another typo. And the domain of $f$ shall have non-empty feasible set.
– di bao
Dec 2 at 7:23
Not sure what kind of a condition you are looking for. The strict inequality holds is already a condition. Do you need another one? Besides, $A$ and $b$ would for sure affect any condition.
– A.Γ.
Dec 2 at 12:32
Not sure what kind of a condition you are looking for. The strict inequality holds is already a condition. Do you need another one? Besides, $A$ and $b$ would for sure affect any condition.
– A.Γ.
Dec 2 at 12:32
Yes, I'm searching for condition like $g(d_1,d_2,A,b)>0$, and $g$ is explicitly defined.
– di bao
Dec 2 at 19:11
Yes, I'm searching for condition like $g(d_1,d_2,A,b)>0$, and $g$ is explicitly defined.
– di bao
Dec 2 at 19:11
the optimal basis should be different for $d_1$ and $d_2$; revised simplex is your friend
– LinAlg
Dec 2 at 22:51
the optimal basis should be different for $d_1$ and $d_2$; revised simplex is your friend
– LinAlg
Dec 2 at 22:51
|
show 1 more comment
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di bao is a new contributor. Be nice, and check out our Code of Conduct.
di bao is a new contributor. Be nice, and check out our Code of Conduct.
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Is there a non strict equality? If $d_k$ are not in the range of $A$ then $f(d_k) = -infty$ :-).
– copper.hat
Dec 2 at 6:45
Hmmm.. another typo. And the domain of $f$ shall have non-empty feasible set.
– di bao
Dec 2 at 7:23
Not sure what kind of a condition you are looking for. The strict inequality holds is already a condition. Do you need another one? Besides, $A$ and $b$ would for sure affect any condition.
– A.Γ.
Dec 2 at 12:32
Yes, I'm searching for condition like $g(d_1,d_2,A,b)>0$, and $g$ is explicitly defined.
– di bao
Dec 2 at 19:11
the optimal basis should be different for $d_1$ and $d_2$; revised simplex is your friend
– LinAlg
Dec 2 at 22:51