How to show that $min_{x in mathbb{R}^4} x^Tx$ subject to $x^TAx geq 1$ has a global minimizer?
$begingroup$
Consider the following problem:
$$min_{x in mathbb{R}^4} x^Tx$$
over $C={x in mathbb{R}^4 mid x^TAx geq 1}$ where $A in mathbb{R}^{4 times 4}$ is a symmetric matrix with two distinct positive eigenvalues and other eigenvalues of $A$ are nonpositive.
How to show that this problem has a global minimizer?
I need to show that there exist $x_*$ in $C$ for which I have
$$
x_*^Tx_* leq x^Tx ,,, forall x in mathbb{R}^4
$$
or to come up with something like the following
$$
|x|^2= |x_*|^2 + alpha ,,, forall x in mathbb{R}^4 ,,,, 0<alpha in mathbb{R}
$$
My try:
$A$ is symmetric, so it can be written as $A=u Lambda u^T$. Hence,
$$
x^TAx=x^Tu Lambda u^Tx geq 1
$$
So
$$
z^T Lambda z geq 1
$$
where $u^Tx=z in mathbb{R}^4$. So the optimization problem would be
$$min_{x in mathbb{R}^4} z^Tz$$
over $z^T Lambda z geq 1$.
How can we proceed?
linear-algebra optimization symmetric-matrices
$endgroup$
add a comment |
$begingroup$
Consider the following problem:
$$min_{x in mathbb{R}^4} x^Tx$$
over $C={x in mathbb{R}^4 mid x^TAx geq 1}$ where $A in mathbb{R}^{4 times 4}$ is a symmetric matrix with two distinct positive eigenvalues and other eigenvalues of $A$ are nonpositive.
How to show that this problem has a global minimizer?
I need to show that there exist $x_*$ in $C$ for which I have
$$
x_*^Tx_* leq x^Tx ,,, forall x in mathbb{R}^4
$$
or to come up with something like the following
$$
|x|^2= |x_*|^2 + alpha ,,, forall x in mathbb{R}^4 ,,,, 0<alpha in mathbb{R}
$$
My try:
$A$ is symmetric, so it can be written as $A=u Lambda u^T$. Hence,
$$
x^TAx=x^Tu Lambda u^Tx geq 1
$$
So
$$
z^T Lambda z geq 1
$$
where $u^Tx=z in mathbb{R}^4$. So the optimization problem would be
$$min_{x in mathbb{R}^4} z^Tz$$
over $z^T Lambda z geq 1$.
How can we proceed?
linear-algebra optimization symmetric-matrices
$endgroup$
add a comment |
$begingroup$
Consider the following problem:
$$min_{x in mathbb{R}^4} x^Tx$$
over $C={x in mathbb{R}^4 mid x^TAx geq 1}$ where $A in mathbb{R}^{4 times 4}$ is a symmetric matrix with two distinct positive eigenvalues and other eigenvalues of $A$ are nonpositive.
How to show that this problem has a global minimizer?
I need to show that there exist $x_*$ in $C$ for which I have
$$
x_*^Tx_* leq x^Tx ,,, forall x in mathbb{R}^4
$$
or to come up with something like the following
$$
|x|^2= |x_*|^2 + alpha ,,, forall x in mathbb{R}^4 ,,,, 0<alpha in mathbb{R}
$$
My try:
$A$ is symmetric, so it can be written as $A=u Lambda u^T$. Hence,
$$
x^TAx=x^Tu Lambda u^Tx geq 1
$$
So
$$
z^T Lambda z geq 1
$$
where $u^Tx=z in mathbb{R}^4$. So the optimization problem would be
$$min_{x in mathbb{R}^4} z^Tz$$
over $z^T Lambda z geq 1$.
How can we proceed?
linear-algebra optimization symmetric-matrices
$endgroup$
Consider the following problem:
$$min_{x in mathbb{R}^4} x^Tx$$
over $C={x in mathbb{R}^4 mid x^TAx geq 1}$ where $A in mathbb{R}^{4 times 4}$ is a symmetric matrix with two distinct positive eigenvalues and other eigenvalues of $A$ are nonpositive.
How to show that this problem has a global minimizer?
I need to show that there exist $x_*$ in $C$ for which I have
$$
x_*^Tx_* leq x^Tx ,,, forall x in mathbb{R}^4
$$
or to come up with something like the following
$$
|x|^2= |x_*|^2 + alpha ,,, forall x in mathbb{R}^4 ,,,, 0<alpha in mathbb{R}
$$
My try:
$A$ is symmetric, so it can be written as $A=u Lambda u^T$. Hence,
$$
x^TAx=x^Tu Lambda u^Tx geq 1
$$
So
$$
z^T Lambda z geq 1
$$
where $u^Tx=z in mathbb{R}^4$. So the optimization problem would be
$$min_{x in mathbb{R}^4} z^Tz$$
over $z^T Lambda z geq 1$.
How can we proceed?
linear-algebra optimization symmetric-matrices
linear-algebra optimization symmetric-matrices
asked Dec 17 '18 at 1:40
SepideSepide
4938
4938
add a comment |
add a comment |
1 Answer
1
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$begingroup$
Choose some $R>0$ such that $B_R:={z^Tzle R^2}$ has nonempty intersection with $F:={z^TAzge1}$. Now $B_Rcap F$ is compact and has a global minimiser, which must also be a global minimiser for the original problem (why?).
Hint: minimisation over $F$ is minimisation over $(Fcap B_R)cup (Fcaptext{cl}( B_R^c))$. But minimising $|x|^2$ over $(Fcap B_R)$ must yield an optimal value that is smaller than or equal to minimising over $(Fcaptext{cl}( B_R^c))$.
$endgroup$
$begingroup$
I do not know, can you explain it to me?
$endgroup$
– Sepide
Dec 17 '18 at 2:08
$begingroup$
@Sepide because any objective value outside B_R is surely >= inside B_R.
$endgroup$
– Vim
Dec 17 '18 at 2:10
$begingroup$
Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
$endgroup$
– Sepide
Dec 17 '18 at 2:22
$begingroup$
@Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
$endgroup$
– Vim
Dec 17 '18 at 3:17
add a comment |
Your Answer
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1 Answer
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1 Answer
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active
oldest
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active
oldest
votes
active
oldest
votes
$begingroup$
Choose some $R>0$ such that $B_R:={z^Tzle R^2}$ has nonempty intersection with $F:={z^TAzge1}$. Now $B_Rcap F$ is compact and has a global minimiser, which must also be a global minimiser for the original problem (why?).
Hint: minimisation over $F$ is minimisation over $(Fcap B_R)cup (Fcaptext{cl}( B_R^c))$. But minimising $|x|^2$ over $(Fcap B_R)$ must yield an optimal value that is smaller than or equal to minimising over $(Fcaptext{cl}( B_R^c))$.
$endgroup$
$begingroup$
I do not know, can you explain it to me?
$endgroup$
– Sepide
Dec 17 '18 at 2:08
$begingroup$
@Sepide because any objective value outside B_R is surely >= inside B_R.
$endgroup$
– Vim
Dec 17 '18 at 2:10
$begingroup$
Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
$endgroup$
– Sepide
Dec 17 '18 at 2:22
$begingroup$
@Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
$endgroup$
– Vim
Dec 17 '18 at 3:17
add a comment |
$begingroup$
Choose some $R>0$ such that $B_R:={z^Tzle R^2}$ has nonempty intersection with $F:={z^TAzge1}$. Now $B_Rcap F$ is compact and has a global minimiser, which must also be a global minimiser for the original problem (why?).
Hint: minimisation over $F$ is minimisation over $(Fcap B_R)cup (Fcaptext{cl}( B_R^c))$. But minimising $|x|^2$ over $(Fcap B_R)$ must yield an optimal value that is smaller than or equal to minimising over $(Fcaptext{cl}( B_R^c))$.
$endgroup$
$begingroup$
I do not know, can you explain it to me?
$endgroup$
– Sepide
Dec 17 '18 at 2:08
$begingroup$
@Sepide because any objective value outside B_R is surely >= inside B_R.
$endgroup$
– Vim
Dec 17 '18 at 2:10
$begingroup$
Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
$endgroup$
– Sepide
Dec 17 '18 at 2:22
$begingroup$
@Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
$endgroup$
– Vim
Dec 17 '18 at 3:17
add a comment |
$begingroup$
Choose some $R>0$ such that $B_R:={z^Tzle R^2}$ has nonempty intersection with $F:={z^TAzge1}$. Now $B_Rcap F$ is compact and has a global minimiser, which must also be a global minimiser for the original problem (why?).
Hint: minimisation over $F$ is minimisation over $(Fcap B_R)cup (Fcaptext{cl}( B_R^c))$. But minimising $|x|^2$ over $(Fcap B_R)$ must yield an optimal value that is smaller than or equal to minimising over $(Fcaptext{cl}( B_R^c))$.
$endgroup$
Choose some $R>0$ such that $B_R:={z^Tzle R^2}$ has nonempty intersection with $F:={z^TAzge1}$. Now $B_Rcap F$ is compact and has a global minimiser, which must also be a global minimiser for the original problem (why?).
Hint: minimisation over $F$ is minimisation over $(Fcap B_R)cup (Fcaptext{cl}( B_R^c))$. But minimising $|x|^2$ over $(Fcap B_R)$ must yield an optimal value that is smaller than or equal to minimising over $(Fcaptext{cl}( B_R^c))$.
edited Dec 17 '18 at 3:21
answered Dec 17 '18 at 1:47
VimVim
8,13631348
8,13631348
$begingroup$
I do not know, can you explain it to me?
$endgroup$
– Sepide
Dec 17 '18 at 2:08
$begingroup$
@Sepide because any objective value outside B_R is surely >= inside B_R.
$endgroup$
– Vim
Dec 17 '18 at 2:10
$begingroup$
Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
$endgroup$
– Sepide
Dec 17 '18 at 2:22
$begingroup$
@Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
$endgroup$
– Vim
Dec 17 '18 at 3:17
add a comment |
$begingroup$
I do not know, can you explain it to me?
$endgroup$
– Sepide
Dec 17 '18 at 2:08
$begingroup$
@Sepide because any objective value outside B_R is surely >= inside B_R.
$endgroup$
– Vim
Dec 17 '18 at 2:10
$begingroup$
Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
$endgroup$
– Sepide
Dec 17 '18 at 2:22
$begingroup$
@Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
$endgroup$
– Vim
Dec 17 '18 at 3:17
$begingroup$
I do not know, can you explain it to me?
$endgroup$
– Sepide
Dec 17 '18 at 2:08
$begingroup$
I do not know, can you explain it to me?
$endgroup$
– Sepide
Dec 17 '18 at 2:08
$begingroup$
@Sepide because any objective value outside B_R is surely >= inside B_R.
$endgroup$
– Vim
Dec 17 '18 at 2:10
$begingroup$
@Sepide because any objective value outside B_R is surely >= inside B_R.
$endgroup$
– Vim
Dec 17 '18 at 2:10
$begingroup$
Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
$endgroup$
– Sepide
Dec 17 '18 at 2:22
$begingroup$
Is the ball contained in the feasible set? If so, for the points that are not in the ball but are in the feasible set, how we know that we cannot find a point that has a value less that what we get from the point that is inside the ball?
$endgroup$
– Sepide
Dec 17 '18 at 2:22
$begingroup$
@Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
$endgroup$
– Vim
Dec 17 '18 at 3:17
$begingroup$
@Sepide the ball doesn't have to be contained in the feasible region because we only care about the intersection of the ball and the feasibility. Please see my edit.
$endgroup$
– Vim
Dec 17 '18 at 3:17
add a comment |
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