Is there a finite set $mathcal{D}$ such that $E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2$...
$begingroup$
Let $mathcal{D} subset mathbb{R}^2$ be a finite set. Define a function $E : mathbb{R}^3 rightarrow mathbb{R}$ by
$$large E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$
Does there exist a set $mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.
I proved $E$ is convex, and I know that if $mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $mathcal{D}$ has multiple elements are appreciated.
real-analysis multivariable-calculus convex-analysis convex-optimization
asked Nov 26 '18 at 2:50
Joe Man AnalysisJoe Man Analysis
33419
$endgroup$
add a comment |
$begingroup$
Let $mathcal{D} subset mathbb{R}^2$ be a finite set. Define a function $E : mathbb{R}^3 rightarrow mathbb{R}$ by
$$large E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$
Does there exist a set $mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.
I proved $E$ is convex, and I know that if $mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $mathcal{D}$ has multiple elements are appreciated.
real-analysis multivariable-calculus convex-analysis convex-optimization
asked Nov 26 '18 at 2:50
Joe Man AnalysisJoe Man Analysis
33419
$endgroup$
add a comment |
$begingroup$
Let $mathcal{D} subset mathbb{R}^2$ be a finite set. Define a function $E : mathbb{R}^3 rightarrow mathbb{R}$ by
$$large E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$
Does there exist a set $mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.
I proved $E$ is convex, and I know that if $mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $mathcal{D}$ has multiple elements are appreciated.
real-analysis multivariable-calculus convex-analysis convex-optimization
asked Nov 26 '18 at 2:50
Joe Man AnalysisJoe Man Analysis
33419
$endgroup$
Let $mathcal{D} subset mathbb{R}^2$ be a finite set. Define a function $E : mathbb{R}^3 rightarrow mathbb{R}$ by
$$large E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$
Does there exist a set $mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.
I proved $E$ is convex, and I know that if $mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $mathcal{D}$ has multiple elements are appreciated.
real-analysis multivariable-calculus convex-analysis convex-optimization
real-analysis multivariable-calculus convex-analysis convex-optimization
asked Nov 26 '18 at 2:50
Joe Man AnalysisJoe Man Analysis
33419
asked Nov 26 '18 at 2:50
Joe Man AnalysisJoe Man Analysis
33419
edited Nov 26 '18 at 9:54
Joe Man Analysis
asked Nov 26 '18 at 2:50
Joe Man AnalysisJoe Man Analysis
33419
asked Nov 26 '18 at 2:50
Joe Man AnalysisJoe Man Analysis
33419
asked Nov 26 '18 at 2:50
Joe Man AnalysisJoe Man Analysis
33419
33419
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.
answered Nov 26 '18 at 12:19
gerwgerw
19.2k11334
$endgroup$
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%2f3013751%2fis-there-a-finite-set-mathcald-such-that-ea-b-c-sum-x-in-mathcal%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: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.
answered Nov 26 '18 at 12:19
gerwgerw
19.2k11334
$endgroup$
add a comment |
$begingroup$
Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.
answered Nov 26 '18 at 12:19
gerwgerw
19.2k11334
$endgroup$
add a comment |
$begingroup$
Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.
answered Nov 26 '18 at 12:19
gerwgerw
19.2k11334
$endgroup$
Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.
answered Nov 26 '18 at 12:19
gerwgerw
19.2k11334
answered Nov 26 '18 at 12:19
gerwgerw
19.2k11334
answered Nov 26 '18 at 12:19
gerwgerw
19.2k11334
answered Nov 26 '18 at 12:19
gerwgerw
19.2k11334
19.2k11334
add a comment |
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%2f3013751%2fis-there-a-finite-set-mathcald-such-that-ea-b-c-sum-x-in-mathcal%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
