Famous theorems that are special cases of linear programming (or convex) duality
The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any other examples of famous theorems that are also corollaries of LP duality, or duality of convex optimization? The Farkas lemma and the hyperplane separation theorem would be other candidates, although they look more like equivalent statements to me.
oc.optimization-and-control convex-optimization linear-programming
add a comment |
The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any other examples of famous theorems that are also corollaries of LP duality, or duality of convex optimization? The Farkas lemma and the hyperplane separation theorem would be other candidates, although they look more like equivalent statements to me.
oc.optimization-and-control convex-optimization linear-programming
The existence of Nash equilibria for matrix games with probabilistic strategies came to my mind, but I don't think there is a famous theorem behind this.
– M. Winter
8 hours ago
1
mathoverflow.net/q/252206/12674 looks relevant.
– Thomas Kalinowski
3 hours ago
add a comment |
The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any other examples of famous theorems that are also corollaries of LP duality, or duality of convex optimization? The Farkas lemma and the hyperplane separation theorem would be other candidates, although they look more like equivalent statements to me.
oc.optimization-and-control convex-optimization linear-programming
The max flow-min cut theorem is one of the most famous theorems of discrete optimization, although it is very straightforward to prove using duality theory from linear programming. Are there any other examples of famous theorems that are also corollaries of LP duality, or duality of convex optimization? The Farkas lemma and the hyperplane separation theorem would be other candidates, although they look more like equivalent statements to me.
oc.optimization-and-control convex-optimization linear-programming
oc.optimization-and-control convex-optimization linear-programming
asked 9 hours ago
community wiki
Tom Solberg
The existence of Nash equilibria for matrix games with probabilistic strategies came to my mind, but I don't think there is a famous theorem behind this.
– M. Winter
8 hours ago
1
mathoverflow.net/q/252206/12674 looks relevant.
– Thomas Kalinowski
3 hours ago
add a comment |
The existence of Nash equilibria for matrix games with probabilistic strategies came to my mind, but I don't think there is a famous theorem behind this.
– M. Winter
8 hours ago
1
mathoverflow.net/q/252206/12674 looks relevant.
– Thomas Kalinowski
3 hours ago
The existence of Nash equilibria for matrix games with probabilistic strategies came to my mind, but I don't think there is a famous theorem behind this.
– M. Winter
8 hours ago
The existence of Nash equilibria for matrix games with probabilistic strategies came to my mind, but I don't think there is a famous theorem behind this.
– M. Winter
8 hours ago
1
1
mathoverflow.net/q/252206/12674 looks relevant.
– Thomas Kalinowski
3 hours ago
mathoverflow.net/q/252206/12674 looks relevant.
– Thomas Kalinowski
3 hours ago
add a comment |
2 Answers
2
active
oldest
votes
To elaborate on M. Winter's comment: Von Neumann's minimax theorem for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem did not make this connection explicit.
add a comment |
Kőnig's theorem and Hall's marriage theorem are famous and follow from LP duality (together with an integrality argument).
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: "504"
};
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%2fmathoverflow.net%2fquestions%2f320589%2ffamous-theorems-that-are-special-cases-of-linear-programming-or-convex-duality%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
To elaborate on M. Winter's comment: Von Neumann's minimax theorem for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem did not make this connection explicit.
add a comment |
To elaborate on M. Winter's comment: Von Neumann's minimax theorem for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem did not make this connection explicit.
add a comment |
To elaborate on M. Winter's comment: Von Neumann's minimax theorem for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem did not make this connection explicit.
To elaborate on M. Winter's comment: Von Neumann's minimax theorem for two-person zero-sum games can be thought of as a consequence of LP duality, although his first proof of the theorem did not make this connection explicit.
answered 6 hours ago
community wiki
Timothy Chow
add a comment |
add a comment |
Kőnig's theorem and Hall's marriage theorem are famous and follow from LP duality (together with an integrality argument).
add a comment |
Kőnig's theorem and Hall's marriage theorem are famous and follow from LP duality (together with an integrality argument).
add a comment |
Kőnig's theorem and Hall's marriage theorem are famous and follow from LP duality (together with an integrality argument).
Kőnig's theorem and Hall's marriage theorem are famous and follow from LP duality (together with an integrality argument).
answered 2 hours ago
community wiki
Thomas Kalinowski
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- 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%2fmathoverflow.net%2fquestions%2f320589%2ffamous-theorems-that-are-special-cases-of-linear-programming-or-convex-duality%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
The existence of Nash equilibria for matrix games with probabilistic strategies came to my mind, but I don't think there is a famous theorem behind this.
– M. Winter
8 hours ago
1
mathoverflow.net/q/252206/12674 looks relevant.
– Thomas Kalinowski
3 hours ago