How to show directly $min frac{1}{2}|x-z|_2^2$ has a unique global minimizer over $Ax=b$?
$begingroup$
Consider the following problem:
$$
min_{Ax=b} frac{1}{2}|x-z|_2^2
$$
where $z in mathbb{R}^{n}$ is a given vector, $x in mathbb{R}^{n}$, $A in mathbb{R}^{m times n}$ and $b in mathbb{R}^{m}$.
Show that there exists a unique global minimizer?
My try:
It is clear that this problem is a convex optimization problem and the local minimizer is a global minimizer. Also, it is strogly convex so it has at most one global minimizer. However, can we show the uniqueness of the minimizer directly?
To that end, we need to need to assume $x^*, y^* in mathbb{R}^{n}$ be to global minimizer.
$$
frac{1}{2}|x^*-z|_2^2 leq frac{1}{2}|x-z|_2^2,,, forall x
$$
Also,
$$
frac{1}{2}|y^*-z|_2^2 leq frac{1}{2}|x-z|_2^2,,, forall x
$$
Using these two we need to come up with a contradiction. How?
optimization convex-analysis convex-optimization
asked Dec 14 '18 at 2:28
SepideSepide
3038
$endgroup$
add a comment |
$begingroup$
Consider the following problem:
$$
min_{Ax=b} frac{1}{2}|x-z|_2^2
$$
where $z in mathbb{R}^{n}$ is a given vector, $x in mathbb{R}^{n}$, $A in mathbb{R}^{m times n}$ and $b in mathbb{R}^{m}$.
Show that there exists a unique global minimizer?
My try:
It is clear that this problem is a convex optimization problem and the local minimizer is a global minimizer. Also, it is strogly convex so it has at most one global minimizer. However, can we show the uniqueness of the minimizer directly?
To that end, we need to need to assume $x^*, y^* in mathbb{R}^{n}$ be to global minimizer.
$$
frac{1}{2}|x^*-z|_2^2 leq frac{1}{2}|x-z|_2^2,,, forall x
$$
Also,
$$
frac{1}{2}|y^*-z|_2^2 leq frac{1}{2}|x-z|_2^2,,, forall x
$$
Using these two we need to come up with a contradiction. How?
optimization convex-analysis convex-optimization
asked Dec 14 '18 at 2:28
SepideSepide
3038
$endgroup$
$begingroup$
If you have strong convexity already for the objective function $a,$ then you know that $a((1 - t)x + ty) < (1 - t) a(x) + t a(y)$ for all $t in (0, 1),$ hence $x$ and $y$ cannot be global minimisers.
$endgroup$
– Will M.
Dec 14 '18 at 4:26
add a comment |
$begingroup$
Consider the following problem:
$$
min_{Ax=b} frac{1}{2}|x-z|_2^2
$$
where $z in mathbb{R}^{n}$ is a given vector, $x in mathbb{R}^{n}$, $A in mathbb{R}^{m times n}$ and $b in mathbb{R}^{m}$.
Show that there exists a unique global minimizer?
My try:
It is clear that this problem is a convex optimization problem and the local minimizer is a global minimizer. Also, it is strogly convex so it has at most one global minimizer. However, can we show the uniqueness of the minimizer directly?
To that end, we need to need to assume $x^*, y^* in mathbb{R}^{n}$ be to global minimizer.
$$
frac{1}{2}|x^*-z|_2^2 leq frac{1}{2}|x-z|_2^2,,, forall x
$$
Also,
$$
frac{1}{2}|y^*-z|_2^2 leq frac{1}{2}|x-z|_2^2,,, forall x
$$
Using these two we need to come up with a contradiction. How?
optimization convex-analysis convex-optimization
asked Dec 14 '18 at 2:28
SepideSepide
3038
$endgroup$
Consider the following problem:
$$
min_{Ax=b} frac{1}{2}|x-z|_2^2
$$
where $z in mathbb{R}^{n}$ is a given vector, $x in mathbb{R}^{n}$, $A in mathbb{R}^{m times n}$ and $b in mathbb{R}^{m}$.
Show that there exists a unique global minimizer?
My try:
It is clear that this problem is a convex optimization problem and the local minimizer is a global minimizer. Also, it is strogly convex so it has at most one global minimizer. However, can we show the uniqueness of the minimizer directly?
To that end, we need to need to assume $x^*, y^* in mathbb{R}^{n}$ be to global minimizer.
$$
frac{1}{2}|x^*-z|_2^2 leq frac{1}{2}|x-z|_2^2,,, forall x
$$
Also,
$$
frac{1}{2}|y^*-z|_2^2 leq frac{1}{2}|x-z|_2^2,,, forall x
$$
Using these two we need to come up with a contradiction. How?
optimization convex-analysis convex-optimization
optimization convex-analysis convex-optimization
asked Dec 14 '18 at 2:28
SepideSepide
3038
asked Dec 14 '18 at 2:28
SepideSepide
3038
asked Dec 14 '18 at 2:28
SepideSepide
3038
asked Dec 14 '18 at 2:28
SepideSepide
3038
asked Dec 14 '18 at 2:28
SepideSepide
3038
3038
$begingroup$
If you have strong convexity already for the objective function $a,$ then you know that $a((1 - t)x + ty) < (1 - t) a(x) + t a(y)$ for all $t in (0, 1),$ hence $x$ and $y$ cannot be global minimisers.
$endgroup$
– Will M.
Dec 14 '18 at 4:26
add a comment |
$begingroup$
If you have strong convexity already for the objective function $a,$ then you know that $a((1 - t)x + ty) < (1 - t) a(x) + t a(y)$ for all $t in (0, 1),$ hence $x$ and $y$ cannot be global minimisers.
$endgroup$
– Will M.
Dec 14 '18 at 4:26
$begingroup$
If you have strong convexity already for the objective function $a,$ then you know that $a((1 - t)x + ty) < (1 - t) a(x) + t a(y)$ for all $t in (0, 1),$ hence $x$ and $y$ cannot be global minimisers.
$endgroup$
– Will M.
Dec 14 '18 at 4:26
$begingroup$
If you have strong convexity already for the objective function $a,$ then you know that $a((1 - t)x + ty) < (1 - t) a(x) + t a(y)$ for all $t in (0, 1),$ hence $x$ and $y$ cannot be global minimisers.
$endgroup$
– Will M.
Dec 14 '18 at 4:26
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The set $C:={xinBbb R^n : Ax=b}$ is an affine subset, hence it is closed and convex. The projection of $z$ into $C$ has a unique solution i.e. $exists x_0in C$ such that
$$
||x_0 - z||_2 = min_{yin C} ||y-z||_2.
$$
It is easy to see that $x_0$ also solves your minimization problem. (You can prove the existence of $x_0$ by taking a minimizing sequence and use compactness argument to show that it converges. The statement is even true for a Hilbert space but the prove is a bit more involved.)
For the second part, $frac 12||cdot - z||^2_2$ is strictly convex. Assume for contradiction that there exist two distinct minimizers $x_1, x_2$ and show that $frac {x_1+x_2}2$ is an even better minimizer. (In fact, the uniqueness of the projection from the previous part already implies uniqueness of the minimizer.)
answered Dec 14 '18 at 4:07
BigbearZzzBigbearZzz
8,67621652
$endgroup$
add a comment |
$begingroup$
If we introduce $y=x-z$, the problem takes the form of a least squares problem
$$
min_{Ay=b-Az} frac12|y|_2^2,
$$
which has an unique global minimizer $y= A^{+}(b-Az)$. Thus, the global minimizer of the original problem is $x=A^{+}(b-Az)+z$.
answered Dec 14 '18 at 4:18
AVKAVK
2,0961517
$endgroup$
add a comment |
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2 Answers
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active
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2 Answers
2
active
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votes
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votes
$begingroup$
The set $C:={xinBbb R^n : Ax=b}$ is an affine subset, hence it is closed and convex. The projection of $z$ into $C$ has a unique solution i.e. $exists x_0in C$ such that
$$
||x_0 - z||_2 = min_{yin C} ||y-z||_2.
$$
It is easy to see that $x_0$ also solves your minimization problem. (You can prove the existence of $x_0$ by taking a minimizing sequence and use compactness argument to show that it converges. The statement is even true for a Hilbert space but the prove is a bit more involved.)
For the second part, $frac 12||cdot - z||^2_2$ is strictly convex. Assume for contradiction that there exist two distinct minimizers $x_1, x_2$ and show that $frac {x_1+x_2}2$ is an even better minimizer. (In fact, the uniqueness of the projection from the previous part already implies uniqueness of the minimizer.)
answered Dec 14 '18 at 4:07
BigbearZzzBigbearZzz
8,67621652
$endgroup$
add a comment |
$begingroup$
The set $C:={xinBbb R^n : Ax=b}$ is an affine subset, hence it is closed and convex. The projection of $z$ into $C$ has a unique solution i.e. $exists x_0in C$ such that
$$
||x_0 - z||_2 = min_{yin C} ||y-z||_2.
$$
It is easy to see that $x_0$ also solves your minimization problem. (You can prove the existence of $x_0$ by taking a minimizing sequence and use compactness argument to show that it converges. The statement is even true for a Hilbert space but the prove is a bit more involved.)
For the second part, $frac 12||cdot - z||^2_2$ is strictly convex. Assume for contradiction that there exist two distinct minimizers $x_1, x_2$ and show that $frac {x_1+x_2}2$ is an even better minimizer. (In fact, the uniqueness of the projection from the previous part already implies uniqueness of the minimizer.)
answered Dec 14 '18 at 4:07
BigbearZzzBigbearZzz
8,67621652
$endgroup$
add a comment |
$begingroup$
The set $C:={xinBbb R^n : Ax=b}$ is an affine subset, hence it is closed and convex. The projection of $z$ into $C$ has a unique solution i.e. $exists x_0in C$ such that
$$
||x_0 - z||_2 = min_{yin C} ||y-z||_2.
$$
It is easy to see that $x_0$ also solves your minimization problem. (You can prove the existence of $x_0$ by taking a minimizing sequence and use compactness argument to show that it converges. The statement is even true for a Hilbert space but the prove is a bit more involved.)
For the second part, $frac 12||cdot - z||^2_2$ is strictly convex. Assume for contradiction that there exist two distinct minimizers $x_1, x_2$ and show that $frac {x_1+x_2}2$ is an even better minimizer. (In fact, the uniqueness of the projection from the previous part already implies uniqueness of the minimizer.)
answered Dec 14 '18 at 4:07
BigbearZzzBigbearZzz
8,67621652
$endgroup$
The set $C:={xinBbb R^n : Ax=b}$ is an affine subset, hence it is closed and convex. The projection of $z$ into $C$ has a unique solution i.e. $exists x_0in C$ such that
$$
||x_0 - z||_2 = min_{yin C} ||y-z||_2.
$$
It is easy to see that $x_0$ also solves your minimization problem. (You can prove the existence of $x_0$ by taking a minimizing sequence and use compactness argument to show that it converges. The statement is even true for a Hilbert space but the prove is a bit more involved.)
For the second part, $frac 12||cdot - z||^2_2$ is strictly convex. Assume for contradiction that there exist two distinct minimizers $x_1, x_2$ and show that $frac {x_1+x_2}2$ is an even better minimizer. (In fact, the uniqueness of the projection from the previous part already implies uniqueness of the minimizer.)
answered Dec 14 '18 at 4:07
BigbearZzzBigbearZzz
8,67621652
answered Dec 14 '18 at 4:07
BigbearZzzBigbearZzz
8,67621652
answered Dec 14 '18 at 4:07
BigbearZzzBigbearZzz
8,67621652
answered Dec 14 '18 at 4:07
BigbearZzzBigbearZzz
8,67621652
8,67621652
add a comment |
add a comment |
$begingroup$
If we introduce $y=x-z$, the problem takes the form of a least squares problem
$$
min_{Ay=b-Az} frac12|y|_2^2,
$$
which has an unique global minimizer $y= A^{+}(b-Az)$. Thus, the global minimizer of the original problem is $x=A^{+}(b-Az)+z$.
answered Dec 14 '18 at 4:18
AVKAVK
2,0961517
$endgroup$
add a comment |
$begingroup$
If we introduce $y=x-z$, the problem takes the form of a least squares problem
$$
min_{Ay=b-Az} frac12|y|_2^2,
$$
which has an unique global minimizer $y= A^{+}(b-Az)$. Thus, the global minimizer of the original problem is $x=A^{+}(b-Az)+z$.
answered Dec 14 '18 at 4:18
AVKAVK
2,0961517
$endgroup$
add a comment |
$begingroup$
If we introduce $y=x-z$, the problem takes the form of a least squares problem
$$
min_{Ay=b-Az} frac12|y|_2^2,
$$
which has an unique global minimizer $y= A^{+}(b-Az)$. Thus, the global minimizer of the original problem is $x=A^{+}(b-Az)+z$.
answered Dec 14 '18 at 4:18
AVKAVK
2,0961517
$endgroup$
If we introduce $y=x-z$, the problem takes the form of a least squares problem
$$
min_{Ay=b-Az} frac12|y|_2^2,
$$
which has an unique global minimizer $y= A^{+}(b-Az)$. Thus, the global minimizer of the original problem is $x=A^{+}(b-Az)+z$.
answered Dec 14 '18 at 4:18
AVKAVK
2,0961517
edited Dec 14 '18 at 4:50
answered Dec 14 '18 at 4:18
AVKAVK
2,0961517
answered Dec 14 '18 at 4:18
AVKAVK
2,0961517
answered Dec 14 '18 at 4:18
AVKAVK
2,0961517
2,0961517
add a comment |
add a comment |
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$begingroup$
If you have strong convexity already for the objective function $a,$ then you know that $a((1 - t)x + ty) < (1 - t) a(x) + t a(y)$ for all $t in (0, 1),$ hence $x$ and $y$ cannot be global minimisers.
$endgroup$
– Will M.
Dec 14 '18 at 4:26