Mixing
How to write a convex combination in terms of optimization problem?
$begingroup$
Let $w$, $s$ and $y$ be vectors in $C in mathbb{R}^n$ and $t in [0,1]$. The convex combination of $w$ and $s$ are as follows
$$
y = (1-t)w + ts tag{1}
$$
Consider the following optimization problem:
$$
y_* = argmin_{y in C} -tlangle y,s -w rangle + frac{1}{2} |y - w|_2^2 tag{2}
$$
which can be solved to get $y_*$ because the objective, i.e., $f(y) = -tlangle y,s -wrangle + frac{1}{2} |y - w|_2^2$ is convex, so the necessary and sufficient condition in order $y_*$ be the minimizer of $(2)$ is the following:
$$
langle -t (s -w) + (y-w), y- y_* rangle geq 0 ,,,, forall y in C
$$
$$
langle y - ((1-t)w +t s) , y - y_* rangle geq 0 ,,,, forall y in C
tag{3}$$
Evidently, for $y_* = (1-t)w +t s$ $(3)$ holds true not only for all $y in C$ but also for every $y$ in $mathbb{R}^n$. However, the cost function is strongly convex so it is strictly convex, therefore, it has one global minimizer.
Question:
Using $(3)$ how can we prove that, $y_*$ is indeed $(1-t)w +t s$?
linear-algebra inequality convex-analysis convex-optimization
asked Feb 1 at 22:10
SaeedSaeed
1,149310
$endgroup$
|
show 2 more comments
$begingroup$
Let $w$, $s$ and $y$ be vectors in $C in mathbb{R}^n$ and $t in [0,1]$. The convex combination of $w$ and $s$ are as follows
$$
y = (1-t)w + ts tag{1}
$$
Consider the following optimization problem:
$$
y_* = argmin_{y in C} -tlangle y,s -w rangle + frac{1}{2} |y - w|_2^2 tag{2}
$$
which can be solved to get $y_*$ because the objective, i.e., $f(y) = -tlangle y,s -wrangle + frac{1}{2} |y - w|_2^2$ is convex, so the necessary and sufficient condition in order $y_*$ be the minimizer of $(2)$ is the following:
$$
langle -t (s -w) + (y-w), y- y_* rangle geq 0 ,,,, forall y in C
$$
$$
langle y - ((1-t)w +t s) , y - y_* rangle geq 0 ,,,, forall y in C
tag{3}$$
Evidently, for $y_* = (1-t)w +t s$ $(3)$ holds true not only for all $y in C$ but also for every $y$ in $mathbb{R}^n$. However, the cost function is strongly convex so it is strictly convex, therefore, it has one global minimizer.
Question:
Using $(3)$ how can we prove that, $y_*$ is indeed $(1-t)w +t s$?
linear-algebra inequality convex-analysis convex-optimization
asked Feb 1 at 22:10
SaeedSaeed
1,149310
$endgroup$
$begingroup$
Do you know that $y_* in C$? If not, it isn't feasible to (2), so your claim doesn't hold. If so, then $y_*$ is an unconstrained minimizer of the objective of (2) and therefore must be a (constrained) minimizer of (2) as well.
$endgroup$
– madnessweasley
Feb 1 at 23:46
$begingroup$
@madnessweasley: $y_*$ is the minimizer of $(2)$ and has to be in $C$ because the optimization is over $C$. My questions is why choosing $y_*=(1-t)w+ts$ which is a point in $C$ is the minimizer of constraint problem. In other words, assume we just had $(3)$, can we think of other $y in C$ for which $(3)$ holds?
$endgroup$
– Saeed
Feb 2 at 2:52
$begingroup$
@LinAlg: I have marked all the ones that has been answered. If I have forgotten one and you aware of that, please let me know.
$endgroup$
– Saeed
Feb 4 at 17:07
$begingroup$
@LinAlg: Dear LinAlg, there are several questions on the statement that are left unanswered :-). Your answer addressed just 2 of them but it was helpful and gave me insight. I marked it off.
$endgroup$
– Saeed
Feb 4 at 18:48
$begingroup$
Do you need to use (3)? Why not use the first order criterion?
$endgroup$
– LinAlg
Feb 4 at 18:49
|
show 2 more comments
$begingroup$
Let $w$, $s$ and $y$ be vectors in $C in mathbb{R}^n$ and $t in [0,1]$. The convex combination of $w$ and $s$ are as follows
$$
y = (1-t)w + ts tag{1}
$$
Consider the following optimization problem:
$$
y_* = argmin_{y in C} -tlangle y,s -w rangle + frac{1}{2} |y - w|_2^2 tag{2}
$$
which can be solved to get $y_*$ because the objective, i.e., $f(y) = -tlangle y,s -wrangle + frac{1}{2} |y - w|_2^2$ is convex, so the necessary and sufficient condition in order $y_*$ be the minimizer of $(2)$ is the following:
$$
langle -t (s -w) + (y-w), y- y_* rangle geq 0 ,,,, forall y in C
$$
$$
langle y - ((1-t)w +t s) , y - y_* rangle geq 0 ,,,, forall y in C
tag{3}$$
Evidently, for $y_* = (1-t)w +t s$ $(3)$ holds true not only for all $y in C$ but also for every $y$ in $mathbb{R}^n$. However, the cost function is strongly convex so it is strictly convex, therefore, it has one global minimizer.
Question:
Using $(3)$ how can we prove that, $y_*$ is indeed $(1-t)w +t s$?
linear-algebra inequality convex-analysis convex-optimization
asked Feb 1 at 22:10
SaeedSaeed
1,149310
$endgroup$
Let $w$, $s$ and $y$ be vectors in $C in mathbb{R}^n$ and $t in [0,1]$. The convex combination of $w$ and $s$ are as follows
$$
y = (1-t)w + ts tag{1}
$$
Consider the following optimization problem:
$$
y_* = argmin_{y in C} -tlangle y,s -w rangle + frac{1}{2} |y - w|_2^2 tag{2}
$$
which can be solved to get $y_*$ because the objective, i.e., $f(y) = -tlangle y,s -wrangle + frac{1}{2} |y - w|_2^2$ is convex, so the necessary and sufficient condition in order $y_*$ be the minimizer of $(2)$ is the following:
$$
langle -t (s -w) + (y-w), y- y_* rangle geq 0 ,,,, forall y in C
$$
$$
langle y - ((1-t)w +t s) , y - y_* rangle geq 0 ,,,, forall y in C
tag{3}$$
Evidently, for $y_* = (1-t)w +t s$ $(3)$ holds true not only for all $y in C$ but also for every $y$ in $mathbb{R}^n$. However, the cost function is strongly convex so it is strictly convex, therefore, it has one global minimizer.
Question:
Using $(3)$ how can we prove that, $y_*$ is indeed $(1-t)w +t s$?
linear-algebra inequality convex-analysis convex-optimization
linear-algebra inequality convex-analysis convex-optimization
asked Feb 1 at 22:10
SaeedSaeed
1,149310
asked Feb 1 at 22:10
SaeedSaeed
1,149310
asked Feb 1 at 22:10
SaeedSaeed
1,149310
asked Feb 1 at 22:10
SaeedSaeed
1,149310
asked Feb 1 at 22:10
SaeedSaeed
1,149310
1,149310
$begingroup$
Do you know that $y_* in C$? If not, it isn't feasible to (2), so your claim doesn't hold. If so, then $y_*$ is an unconstrained minimizer of the objective of (2) and therefore must be a (constrained) minimizer of (2) as well.
$endgroup$
– madnessweasley
Feb 1 at 23:46
$begingroup$
@madnessweasley: $y_*$ is the minimizer of $(2)$ and has to be in $C$ because the optimization is over $C$. My questions is why choosing $y_*=(1-t)w+ts$ which is a point in $C$ is the minimizer of constraint problem. In other words, assume we just had $(3)$, can we think of other $y in C$ for which $(3)$ holds?
$endgroup$
– Saeed
Feb 2 at 2:52
$begingroup$
@LinAlg: I have marked all the ones that has been answered. If I have forgotten one and you aware of that, please let me know.
$endgroup$
– Saeed
Feb 4 at 17:07
$begingroup$
@LinAlg: Dear LinAlg, there are several questions on the statement that are left unanswered :-). Your answer addressed just 2 of them but it was helpful and gave me insight. I marked it off.
$endgroup$
– Saeed
Feb 4 at 18:48
$begingroup$
Do you need to use (3)? Why not use the first order criterion?
$endgroup$
– LinAlg
Feb 4 at 18:49
|
show 2 more comments
$begingroup$
Do you know that $y_* in C$? If not, it isn't feasible to (2), so your claim doesn't hold. If so, then $y_*$ is an unconstrained minimizer of the objective of (2) and therefore must be a (constrained) minimizer of (2) as well.
$endgroup$
– madnessweasley
Feb 1 at 23:46
$begingroup$
@madnessweasley: $y_*$ is the minimizer of $(2)$ and has to be in $C$ because the optimization is over $C$. My questions is why choosing $y_*=(1-t)w+ts$ which is a point in $C$ is the minimizer of constraint problem. In other words, assume we just had $(3)$, can we think of other $y in C$ for which $(3)$ holds?
$endgroup$
– Saeed
Feb 2 at 2:52
$begingroup$
@LinAlg: I have marked all the ones that has been answered. If I have forgotten one and you aware of that, please let me know.
$endgroup$
– Saeed
Feb 4 at 17:07
$begingroup$
@LinAlg: Dear LinAlg, there are several questions on the statement that are left unanswered :-). Your answer addressed just 2 of them but it was helpful and gave me insight. I marked it off.
$endgroup$
– Saeed
Feb 4 at 18:48
$begingroup$
Do you need to use (3)? Why not use the first order criterion?
$endgroup$
– LinAlg
Feb 4 at 18:49
$begingroup$
Do you know that $y_* in C$? If not, it isn't feasible to (2), so your claim doesn't hold. If so, then $y_*$ is an unconstrained minimizer of the objective of (2) and therefore must be a (constrained) minimizer of (2) as well.
$endgroup$
– madnessweasley
Feb 1 at 23:46
$begingroup$
Do you know that $y_* in C$? If not, it isn't feasible to (2), so your claim doesn't hold. If so, then $y_*$ is an unconstrained minimizer of the objective of (2) and therefore must be a (constrained) minimizer of (2) as well.
$endgroup$
– madnessweasley
Feb 1 at 23:46
$begingroup$
@madnessweasley: $y_*$ is the minimizer of $(2)$ and has to be in $C$ because the optimization is over $C$. My questions is why choosing $y_*=(1-t)w+ts$ which is a point in $C$ is the minimizer of constraint problem. In other words, assume we just had $(3)$, can we think of other $y in C$ for which $(3)$ holds?
$endgroup$
– Saeed
Feb 2 at 2:52
$begingroup$
@madnessweasley: $y_*$ is the minimizer of $(2)$ and has to be in $C$ because the optimization is over $C$. My questions is why choosing $y_*=(1-t)w+ts$ which is a point in $C$ is the minimizer of constraint problem. In other words, assume we just had $(3)$, can we think of other $y in C$ for which $(3)$ holds?
$endgroup$
– Saeed
Feb 2 at 2:52
$begingroup$
@LinAlg: I have marked all the ones that has been answered. If I have forgotten one and you aware of that, please let me know.
$endgroup$
– Saeed
Feb 4 at 17:07
$begingroup$
@LinAlg: I have marked all the ones that has been answered. If I have forgotten one and you aware of that, please let me know.
$endgroup$
– Saeed
Feb 4 at 17:07
$begingroup$
@LinAlg: Dear LinAlg, there are several questions on the statement that are left unanswered :-). Your answer addressed just 2 of them but it was helpful and gave me insight. I marked it off.
$endgroup$
– Saeed
Feb 4 at 18:48
$begingroup$
@LinAlg: Dear LinAlg, there are several questions on the statement that are left unanswered :-). Your answer addressed just 2 of them but it was helpful and gave me insight. I marked it off.
$endgroup$
– Saeed
Feb 4 at 18:48
$begingroup$
Do you need to use (3)? Why not use the first order criterion?
$endgroup$
– LinAlg
Feb 4 at 18:49
$begingroup$
Do you need to use (3)? Why not use the first order criterion?
$endgroup$
– LinAlg
Feb 4 at 18:49
|
show 2 more comments
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$begingroup$
Do you know that $y_* in C$? If not, it isn't feasible to (2), so your claim doesn't hold. If so, then $y_*$ is an unconstrained minimizer of the objective of (2) and therefore must be a (constrained) minimizer of (2) as well.
$endgroup$
– madnessweasley
Feb 1 at 23:46
$begingroup$
@madnessweasley: $y_*$ is the minimizer of $(2)$ and has to be in $C$ because the optimization is over $C$. My questions is why choosing $y_*=(1-t)w+ts$ which is a point in $C$ is the minimizer of constraint problem. In other words, assume we just had $(3)$, can we think of other $y in C$ for which $(3)$ holds?
$endgroup$
– Saeed
Feb 2 at 2:52
$begingroup$
@LinAlg: I have marked all the ones that has been answered. If I have forgotten one and you aware of that, please let me know.
$endgroup$
– Saeed
Feb 4 at 17:07
$begingroup$
@LinAlg: Dear LinAlg, there are several questions on the statement that are left unanswered :-). Your answer addressed just 2 of them but it was helpful and gave me insight. I marked it off.
$endgroup$
– Saeed
Feb 4 at 18:48
$begingroup$
Do you need to use (3)? Why not use the first order criterion?
$endgroup$
– LinAlg
Feb 4 at 18:49