Show that intersection of a polyhedron and affine set is a polyhedron.
$begingroup$
Let $P={x in mathbb{R}^n mid Ax geq b}$ be a nonempty polyhedron for a matrix $A in mathbb{R}^{m times n}$ and $b in mathbb{R}^{m}$.
Show that a nonempty intersection of $P$ and an affine set in $mathbb{R}^{n}$ is a polyhedron.
To show this I just know that an affine set in $mathbb{R}^{n}$ can be written as a subspace $L$ in $mathbb{R}^{n}$ plus some $u$ in $mathbb{R}^{n}$, that is
$$
L= {v: cv=0}
$$
$$
u+L={u+v mid v in N(C)} ,,,text{or},,, u+L = {z mid Cz=d}
$$
where $d=Cu$.
To show the claim I need to assume $x$ is in $P$, so
$$
Ax geq b
$$
Also, $x in u+L$, i.e., $Cx=d$ where $d=Cu$.
Now I need to show there exist an $A'$ and $b'$ where $A'x geq b'$.
How can I proceed?
linear-algebra convex-analysis
asked Dec 8 '18 at 0:00
SaeedSaeed
1,024310
$endgroup$
add a comment |
$begingroup$
Let $P={x in mathbb{R}^n mid Ax geq b}$ be a nonempty polyhedron for a matrix $A in mathbb{R}^{m times n}$ and $b in mathbb{R}^{m}$.
Show that a nonempty intersection of $P$ and an affine set in $mathbb{R}^{n}$ is a polyhedron.
To show this I just know that an affine set in $mathbb{R}^{n}$ can be written as a subspace $L$ in $mathbb{R}^{n}$ plus some $u$ in $mathbb{R}^{n}$, that is
$$
L= {v: cv=0}
$$
$$
u+L={u+v mid v in N(C)} ,,,text{or},,, u+L = {z mid Cz=d}
$$
where $d=Cu$.
To show the claim I need to assume $x$ is in $P$, so
$$
Ax geq b
$$
Also, $x in u+L$, i.e., $Cx=d$ where $d=Cu$.
Now I need to show there exist an $A'$ and $b'$ where $A'x geq b'$.
How can I proceed?
linear-algebra convex-analysis
asked Dec 8 '18 at 0:00
SaeedSaeed
1,024310
$endgroup$
add a comment |
$begingroup$
Let $P={x in mathbb{R}^n mid Ax geq b}$ be a nonempty polyhedron for a matrix $A in mathbb{R}^{m times n}$ and $b in mathbb{R}^{m}$.
Show that a nonempty intersection of $P$ and an affine set in $mathbb{R}^{n}$ is a polyhedron.
To show this I just know that an affine set in $mathbb{R}^{n}$ can be written as a subspace $L$ in $mathbb{R}^{n}$ plus some $u$ in $mathbb{R}^{n}$, that is
$$
L= {v: cv=0}
$$
$$
u+L={u+v mid v in N(C)} ,,,text{or},,, u+L = {z mid Cz=d}
$$
where $d=Cu$.
To show the claim I need to assume $x$ is in $P$, so
$$
Ax geq b
$$
Also, $x in u+L$, i.e., $Cx=d$ where $d=Cu$.
Now I need to show there exist an $A'$ and $b'$ where $A'x geq b'$.
How can I proceed?
linear-algebra convex-analysis
asked Dec 8 '18 at 0:00
SaeedSaeed
1,024310
$endgroup$
Let $P={x in mathbb{R}^n mid Ax geq b}$ be a nonempty polyhedron for a matrix $A in mathbb{R}^{m times n}$ and $b in mathbb{R}^{m}$.
Show that a nonempty intersection of $P$ and an affine set in $mathbb{R}^{n}$ is a polyhedron.
To show this I just know that an affine set in $mathbb{R}^{n}$ can be written as a subspace $L$ in $mathbb{R}^{n}$ plus some $u$ in $mathbb{R}^{n}$, that is
$$
L= {v: cv=0}
$$
$$
u+L={u+v mid v in N(C)} ,,,text{or},,, u+L = {z mid Cz=d}
$$
where $d=Cu$.
To show the claim I need to assume $x$ is in $P$, so
$$
Ax geq b
$$
Also, $x in u+L$, i.e., $Cx=d$ where $d=Cu$.
Now I need to show there exist an $A'$ and $b'$ where $A'x geq b'$.
How can I proceed?
linear-algebra convex-analysis
linear-algebra convex-analysis
asked Dec 8 '18 at 0:00
SaeedSaeed
1,024310
asked Dec 8 '18 at 0:00
SaeedSaeed
1,024310
edited Dec 8 '18 at 4:49
Saeed
asked Dec 8 '18 at 0:00
SaeedSaeed
1,024310
asked Dec 8 '18 at 0:00
SaeedSaeed
1,024310
asked Dec 8 '18 at 0:00
SaeedSaeed
1,024310
1,024310
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: first express your affine space as a system of linear equations $Cx = d$. Then append the rows of $C$ and $-C$ to $A$ to form a taller matrix $A$, as well as append $d$ and $-d$ to $b$ to form a taller column vector $b'$. Then the inequality $A'x le b'$ describes the intersection.
answered Dec 8 '18 at 0:24
Theo BenditTheo Bendit
18.1k12152
$endgroup$
$begingroup$
I think I got what you mean, just change the last $leq$ to $geq$.
$endgroup$
– Saeed
Dec 8 '18 at 5:02
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3030528%2fshow-that-intersection-of-a-polyhedron-and-affine-set-is-a-polyhedron%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: first express your affine space as a system of linear equations $Cx = d$. Then append the rows of $C$ and $-C$ to $A$ to form a taller matrix $A$, as well as append $d$ and $-d$ to $b$ to form a taller column vector $b'$. Then the inequality $A'x le b'$ describes the intersection.
answered Dec 8 '18 at 0:24
Theo BenditTheo Bendit
18.1k12152
$endgroup$
$begingroup$
I think I got what you mean, just change the last $leq$ to $geq$.
$endgroup$
– Saeed
Dec 8 '18 at 5:02
add a comment |
$begingroup$
Hint: first express your affine space as a system of linear equations $Cx = d$. Then append the rows of $C$ and $-C$ to $A$ to form a taller matrix $A$, as well as append $d$ and $-d$ to $b$ to form a taller column vector $b'$. Then the inequality $A'x le b'$ describes the intersection.
answered Dec 8 '18 at 0:24
Theo BenditTheo Bendit
18.1k12152
$endgroup$
$begingroup$
I think I got what you mean, just change the last $leq$ to $geq$.
$endgroup$
– Saeed
Dec 8 '18 at 5:02
add a comment |
$begingroup$
Hint: first express your affine space as a system of linear equations $Cx = d$. Then append the rows of $C$ and $-C$ to $A$ to form a taller matrix $A$, as well as append $d$ and $-d$ to $b$ to form a taller column vector $b'$. Then the inequality $A'x le b'$ describes the intersection.
answered Dec 8 '18 at 0:24
Theo BenditTheo Bendit
18.1k12152
$endgroup$
Hint: first express your affine space as a system of linear equations $Cx = d$. Then append the rows of $C$ and $-C$ to $A$ to form a taller matrix $A$, as well as append $d$ and $-d$ to $b$ to form a taller column vector $b'$. Then the inequality $A'x le b'$ describes the intersection.
answered Dec 8 '18 at 0:24
Theo BenditTheo Bendit
18.1k12152
answered Dec 8 '18 at 0:24
Theo BenditTheo Bendit
18.1k12152
answered Dec 8 '18 at 0:24
Theo BenditTheo Bendit
18.1k12152
answered Dec 8 '18 at 0:24
Theo BenditTheo Bendit
18.1k12152
18.1k12152
$begingroup$
I think I got what you mean, just change the last $leq$ to $geq$.
$endgroup$
– Saeed
Dec 8 '18 at 5:02
add a comment |
$begingroup$
I think I got what you mean, just change the last $leq$ to $geq$.
$endgroup$
– Saeed
Dec 8 '18 at 5:02
$begingroup$
I think I got what you mean, just change the last $leq$ to $geq$.
$endgroup$
– Saeed
Dec 8 '18 at 5:02
$begingroup$
I think I got what you mean, just change the last $leq$ to $geq$.
$endgroup$
– Saeed
Dec 8 '18 at 5:02
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3030528%2fshow-that-intersection-of-a-polyhedron-and-affine-set-is-a-polyhedron%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
