Changing domain of solution in Integer Programming
$begingroup$
I'm given an inequality $a_1x_1 + a_2x_2 geq c_1$ where $a, c, x_1, x_2$ are integers and $x_1, x_2 text{ are either } 1 text{ or } 0$.
I'd like to construct another inequality $b_1y_1 + b_2y_2 geq c_2$ where $b, c, y_1, y_2$ are integers in range $[-k, k]$ in a way s.t. first inequality is satisfiable iff second inequality is satisfiable.
I've tried to map $x_1, x_2$ to $y_1, y_2$ via affine transformation but it doesn't seem like the right approach.
How should I approach this problem?
linear-algebra inequality integer-programming
asked Dec 18 '18 at 3:25
SpiderRicoSpiderRico
259111
$endgroup$
add a comment |
$begingroup$
I'm given an inequality $a_1x_1 + a_2x_2 geq c_1$ where $a, c, x_1, x_2$ are integers and $x_1, x_2 text{ are either } 1 text{ or } 0$.
I'd like to construct another inequality $b_1y_1 + b_2y_2 geq c_2$ where $b, c, y_1, y_2$ are integers in range $[-k, k]$ in a way s.t. first inequality is satisfiable iff second inequality is satisfiable.
I've tried to map $x_1, x_2$ to $y_1, y_2$ via affine transformation but it doesn't seem like the right approach.
How should I approach this problem?
linear-algebra inequality integer-programming
asked Dec 18 '18 at 3:25
SpiderRicoSpiderRico
259111
$endgroup$
$begingroup$
I take it $k$ is fixed? Do you know anything about the signs of the symbols in the first constraint?
$endgroup$
– prubin
Dec 18 '18 at 22:17
$begingroup$
@prubin yes k is fixed. No assumption about the signs of coefficients.
$endgroup$
– SpiderRico
Dec 18 '18 at 22:25
$begingroup$
Also just to be clear, the domain requirement ${-k,dots,k}$ applies not just to the $y$ variables but also to the coefficients $b$ and $c_2$?
$endgroup$
– prubin
Dec 20 '18 at 1:03
$begingroup$
@prubin This is correct. In second inequality, range condition applies both to coefficients and variables.
$endgroup$
– SpiderRico
Dec 21 '18 at 3:36
add a comment |
$begingroup$
I'm given an inequality $a_1x_1 + a_2x_2 geq c_1$ where $a, c, x_1, x_2$ are integers and $x_1, x_2 text{ are either } 1 text{ or } 0$.
I'd like to construct another inequality $b_1y_1 + b_2y_2 geq c_2$ where $b, c, y_1, y_2$ are integers in range $[-k, k]$ in a way s.t. first inequality is satisfiable iff second inequality is satisfiable.
I've tried to map $x_1, x_2$ to $y_1, y_2$ via affine transformation but it doesn't seem like the right approach.
How should I approach this problem?
linear-algebra inequality integer-programming
asked Dec 18 '18 at 3:25
SpiderRicoSpiderRico
259111
$endgroup$
I'm given an inequality $a_1x_1 + a_2x_2 geq c_1$ where $a, c, x_1, x_2$ are integers and $x_1, x_2 text{ are either } 1 text{ or } 0$.
I'd like to construct another inequality $b_1y_1 + b_2y_2 geq c_2$ where $b, c, y_1, y_2$ are integers in range $[-k, k]$ in a way s.t. first inequality is satisfiable iff second inequality is satisfiable.
I've tried to map $x_1, x_2$ to $y_1, y_2$ via affine transformation but it doesn't seem like the right approach.
How should I approach this problem?
linear-algebra inequality integer-programming
linear-algebra inequality integer-programming
asked Dec 18 '18 at 3:25
SpiderRicoSpiderRico
259111
asked Dec 18 '18 at 3:25
SpiderRicoSpiderRico
259111
asked Dec 18 '18 at 3:25
SpiderRicoSpiderRico
259111
asked Dec 18 '18 at 3:25
SpiderRicoSpiderRico
259111
asked Dec 18 '18 at 3:25
SpiderRicoSpiderRico
259111
259111
$begingroup$
I take it $k$ is fixed? Do you know anything about the signs of the symbols in the first constraint?
$endgroup$
– prubin
Dec 18 '18 at 22:17
$begingroup$
@prubin yes k is fixed. No assumption about the signs of coefficients.
$endgroup$
– SpiderRico
Dec 18 '18 at 22:25
$begingroup$
Also just to be clear, the domain requirement ${-k,dots,k}$ applies not just to the $y$ variables but also to the coefficients $b$ and $c_2$?
$endgroup$
– prubin
Dec 20 '18 at 1:03
$begingroup$
@prubin This is correct. In second inequality, range condition applies both to coefficients and variables.
$endgroup$
– SpiderRico
Dec 21 '18 at 3:36
add a comment |
$begingroup$
I take it $k$ is fixed? Do you know anything about the signs of the symbols in the first constraint?
$endgroup$
– prubin
Dec 18 '18 at 22:17
$begingroup$
@prubin yes k is fixed. No assumption about the signs of coefficients.
$endgroup$
– SpiderRico
Dec 18 '18 at 22:25
$begingroup$
Also just to be clear, the domain requirement ${-k,dots,k}$ applies not just to the $y$ variables but also to the coefficients $b$ and $c_2$?
$endgroup$
– prubin
Dec 20 '18 at 1:03
$begingroup$
@prubin This is correct. In second inequality, range condition applies both to coefficients and variables.
$endgroup$
– SpiderRico
Dec 21 '18 at 3:36
$begingroup$
I take it $k$ is fixed? Do you know anything about the signs of the symbols in the first constraint?
$endgroup$
– prubin
Dec 18 '18 at 22:17
$begingroup$
I take it $k$ is fixed? Do you know anything about the signs of the symbols in the first constraint?
$endgroup$
– prubin
Dec 18 '18 at 22:17
$begingroup$
@prubin yes k is fixed. No assumption about the signs of coefficients.
$endgroup$
– SpiderRico
Dec 18 '18 at 22:25
$begingroup$
@prubin yes k is fixed. No assumption about the signs of coefficients.
$endgroup$
– SpiderRico
Dec 18 '18 at 22:25
$begingroup$
Also just to be clear, the domain requirement ${-k,dots,k}$ applies not just to the $y$ variables but also to the coefficients $b$ and $c_2$?
$endgroup$
– prubin
Dec 20 '18 at 1:03
$begingroup$
Also just to be clear, the domain requirement ${-k,dots,k}$ applies not just to the $y$ variables but also to the coefficients $b$ and $c_2$?
$endgroup$
– prubin
Dec 20 '18 at 1:03
$begingroup$
@prubin This is correct. In second inequality, range condition applies both to coefficients and variables.
$endgroup$
– SpiderRico
Dec 21 '18 at 3:36
$begingroup$
@prubin This is correct. In second inequality, range condition applies both to coefficients and variables.
$endgroup$
– SpiderRico
Dec 21 '18 at 3:36
add a comment |
0
active
oldest
votes
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%2f3044746%2fchanging-domain-of-solution-in-integer-programming%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3044746%2fchanging-domain-of-solution-in-integer-programming%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
$begingroup$
I take it $k$ is fixed? Do you know anything about the signs of the symbols in the first constraint?
$endgroup$
– prubin
Dec 18 '18 at 22:17
$begingroup$
@prubin yes k is fixed. No assumption about the signs of coefficients.
$endgroup$
– SpiderRico
Dec 18 '18 at 22:25
$begingroup$
Also just to be clear, the domain requirement ${-k,dots,k}$ applies not just to the $y$ variables but also to the coefficients $b$ and $c_2$?
$endgroup$
– prubin
Dec 20 '18 at 1:03
$begingroup$
@prubin This is correct. In second inequality, range condition applies both to coefficients and variables.
$endgroup$
– SpiderRico
Dec 21 '18 at 3:36