If x* is a local optimum of a function in all directions, could it be optimal in a neighborhood of x*?
In a Euclidean space whose base is denoted $(e_1,e_2,...e_N)$. suppose that $x*$ is the local minimum of a function $f$ in all direction. Could we say that $x*$ is an optimum of $f$ in a neighborhood of $x*$ ??
optimization nonlinear-optimization
edited Nov 24 at 2:35
Moo
5,53131020
asked Nov 23 at 20:05
yas are
556
add a comment |
In a Euclidean space whose base is denoted $(e_1,e_2,...e_N)$. suppose that $x*$ is the local minimum of a function $f$ in all direction. Could we say that $x*$ is an optimum of $f$ in a neighborhood of $x*$ ??
optimization nonlinear-optimization
edited Nov 24 at 2:35
Moo
5,53131020
asked Nov 23 at 20:05
yas are
556
add a comment |
In a Euclidean space whose base is denoted $(e_1,e_2,...e_N)$. suppose that $x*$ is the local minimum of a function $f$ in all direction. Could we say that $x*$ is an optimum of $f$ in a neighborhood of $x*$ ??
optimization nonlinear-optimization
edited Nov 24 at 2:35
Moo
5,53131020
asked Nov 23 at 20:05
yas are
556
In a Euclidean space whose base is denoted $(e_1,e_2,...e_N)$. suppose that $x*$ is the local minimum of a function $f$ in all direction. Could we say that $x*$ is an optimum of $f$ in a neighborhood of $x*$ ??
optimization nonlinear-optimization
optimization nonlinear-optimization
edited Nov 24 at 2:35
Moo
5,53131020
asked Nov 23 at 20:05
yas are
556
edited Nov 24 at 2:35
Moo
5,53131020
asked Nov 23 at 20:05
yas are
556
edited Nov 24 at 2:35
Moo
5,53131020
edited Nov 24 at 2:35
Moo
5,53131020
edited Nov 24 at 2:35
Moo
5,53131020
5,53131020
asked Nov 23 at 20:05
yas are
556
asked Nov 23 at 20:05
yas are
556
asked Nov 23 at 20:05
yas are
556
556
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Thats what a local minimum means. f has a local extremum at x$^*$ if for some $epsilon$-neighborhood of x$^*$, f(x) $ge$ f(x$^*$) for all x in the neighborhood.
answered Nov 23 at 20:14
Joel Pereira
65119
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
– yas are
Nov 23 at 20:52
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%2f3010775%2fif-x-is-a-local-optimum-of-a-function-in-all-directions-could-it-be-optimal-in%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
Thats what a local minimum means. f has a local extremum at x$^*$ if for some $epsilon$-neighborhood of x$^*$, f(x) $ge$ f(x$^*$) for all x in the neighborhood.
answered Nov 23 at 20:14
Joel Pereira
65119
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
– yas are
Nov 23 at 20:52
add a comment |
Thats what a local minimum means. f has a local extremum at x$^*$ if for some $epsilon$-neighborhood of x$^*$, f(x) $ge$ f(x$^*$) for all x in the neighborhood.
answered Nov 23 at 20:14
Joel Pereira
65119
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
– yas are
Nov 23 at 20:52
add a comment |
Thats what a local minimum means. f has a local extremum at x$^*$ if for some $epsilon$-neighborhood of x$^*$, f(x) $ge$ f(x$^*$) for all x in the neighborhood.
answered Nov 23 at 20:14
Joel Pereira
65119
Thats what a local minimum means. f has a local extremum at x$^*$ if for some $epsilon$-neighborhood of x$^*$, f(x) $ge$ f(x$^*$) for all x in the neighborhood.
answered Nov 23 at 20:14
Joel Pereira
65119
answered Nov 23 at 20:14
Joel Pereira
65119
answered Nov 23 at 20:14
Joel Pereira
65119
answered Nov 23 at 20:14
Joel Pereira
65119
65119
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
– yas are
Nov 23 at 20:52
add a comment |
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
– yas are
Nov 23 at 20:52
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
– yas are
Nov 23 at 20:52
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
– yas are
Nov 23 at 20:52
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3010775%2fif-x-is-a-local-optimum-of-a-function-in-all-directions-could-it-be-optimal-in%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
