Why is positive definiteness required for a global minimum to exist? The 2019 Stack Overflow Developer Survey Results Are Inhessian matrix not positive definite at a minimum?Positive definiteness of matrix?Does existence of global minimum imply coercivity?Coerciveness and Positive definiteness relation?Why (x'Ay)^2 <=(x'Ax)(y'Ay), when A is positive definite?minimum eigenvalue of the Hadamard product of positive semi-definite matricesPositive definiteness of matrix A.Minimum eigenvalue of sums of positive semi-definite matricesGlobal minimum of $f(x)=langle Ax,x rangle +2 langle x,b rangle+c$Positive definiteness of “sub-matrices”
What does "rabbited" mean/imply in this sentence?
What do the Banks children have against barley water?
Which Sci-Fi work first showed weapon of galactic-scale mass destruction?
How to change the limits of integration
Extreme, unacceptable situation and I can't attend work tomorrow morning
How come people say “Would of”?
What does Linus Torvalds mean when he says that Git "never ever" tracks a file?
Is "plugging out" electronic devices an American expression?
How was Skylab's orbit inclination chosen?
Unbreakable Formation vs. Cry of the Carnarium
How to create dashed lines/arrows in Illustrator
Is domain driven design an anti-SQL pattern?
Could JWST stay at L2 "forever"?
"To split hairs" vs "To be pedantic"
What could be the right powersource for 15 seconds lifespan disposable giant chainsaw?
Limit the amount of RAM Mathematica may access?
Idiomatic way to prevent slicing?
Inline version of a function returns different value than non-inline version
JSON.serialize: is it possible to suppress null values of a map?
How to make payment on the internet without leaving a money trail?
Is it possible for the two major parties in the UK to form a coalition with each other instead of a much smaller party?
Why is the maximum length of OpenWrt’s root password 8 characters?
How can I create a character who can assume the widest possible range of creature sizes?
What do hard-Brexiteers want with respect to the Irish border?
Why is positive definiteness required for a global minimum to exist?
The 2019 Stack Overflow Developer Survey Results Are Inhessian matrix not positive definite at a minimum?Positive definiteness of matrix?Does existence of global minimum imply coercivity?Coerciveness and Positive definiteness relation?Why (x'Ay)^2 <=(x'Ax)(y'Ay), when A is positive definite?minimum eigenvalue of the Hadamard product of positive semi-definite matricesPositive definiteness of matrix A.Minimum eigenvalue of sums of positive semi-definite matricesGlobal minimum of $f(x)=langle Ax,x rangle +2 langle x,b rangle+c$Positive definiteness of “sub-matrices”
$begingroup$
When we optimize
$$min x'Ax$$
why does matrix $A$ need to be positive definite in order to have a global minimum?
matrices optimization convex-optimization positive-definite quadratic-programming
edited Mar 23 at 9:22
Rodrigo de Azevedo
13.2k41962
asked Mar 23 at 3:13
JOHNJOHN
183
$endgroup$
|
show 4 more comments
$begingroup$
When we optimize
$$min x'Ax$$
why does matrix $A$ need to be positive definite in order to have a global minimum?
matrices optimization convex-optimization positive-definite quadratic-programming
edited Mar 23 at 9:22
Rodrigo de Azevedo
13.2k41962
asked Mar 23 at 3:13
JOHNJOHN
183
$endgroup$
1
$begingroup$
It is not required. If $A=0$ then there is a global minimum. You need positive semi definite. Otherwise there is a direction in which you can produce values that are unbounded below.
$endgroup$
– copper.hat
Mar 23 at 3:15
$begingroup$
@copper.hat what is the requirement to have a unique solution?
$endgroup$
– JOHN
Mar 23 at 3:19
$begingroup$
Can you guess? Where is the derivative (of a convex function) zero?
$endgroup$
– copper.hat
Mar 23 at 3:19
$begingroup$
@copper.hat you would want $Ax=0$ has a solution, so the det𝐀 != 0. and second derivative to be negative. I don't know how could A be expressed to meet this.
$endgroup$
– JOHN
Mar 23 at 3:28
$begingroup$
What is the second derivative of $x mapsto x'Ax$??
$endgroup$
– copper.hat
Mar 23 at 3:29
|
show 4 more comments
$begingroup$
When we optimize
$$min x'Ax$$
why does matrix $A$ need to be positive definite in order to have a global minimum?
matrices optimization convex-optimization positive-definite quadratic-programming
edited Mar 23 at 9:22
Rodrigo de Azevedo
13.2k41962
asked Mar 23 at 3:13
JOHNJOHN
183
$endgroup$
When we optimize
$$min x'Ax$$
why does matrix $A$ need to be positive definite in order to have a global minimum?
matrices optimization convex-optimization positive-definite quadratic-programming
matrices optimization convex-optimization positive-definite quadratic-programming
edited Mar 23 at 9:22
Rodrigo de Azevedo
13.2k41962
asked Mar 23 at 3:13
JOHNJOHN
183
edited Mar 23 at 9:22
Rodrigo de Azevedo
13.2k41962
asked Mar 23 at 3:13
JOHNJOHN
183
edited Mar 23 at 9:22
Rodrigo de Azevedo
13.2k41962
edited Mar 23 at 9:22
Rodrigo de Azevedo
13.2k41962
edited Mar 23 at 9:22
Rodrigo de Azevedo
13.2k41962
13.2k41962
asked Mar 23 at 3:13
JOHNJOHN
183
asked Mar 23 at 3:13
JOHNJOHN
183
asked Mar 23 at 3:13
JOHNJOHN
183
183
1
$begingroup$
It is not required. If $A=0$ then there is a global minimum. You need positive semi definite. Otherwise there is a direction in which you can produce values that are unbounded below.
$endgroup$
– copper.hat
Mar 23 at 3:15
$begingroup$
@copper.hat what is the requirement to have a unique solution?
$endgroup$
– JOHN
Mar 23 at 3:19
$begingroup$
Can you guess? Where is the derivative (of a convex function) zero?
$endgroup$
– copper.hat
Mar 23 at 3:19
$begingroup$
@copper.hat you would want $Ax=0$ has a solution, so the det𝐀 != 0. and second derivative to be negative. I don't know how could A be expressed to meet this.
$endgroup$
– JOHN
Mar 23 at 3:28
$begingroup$
What is the second derivative of $x mapsto x'Ax$??
$endgroup$
– copper.hat
Mar 23 at 3:29
|
show 4 more comments
1
$begingroup$
It is not required. If $A=0$ then there is a global minimum. You need positive semi definite. Otherwise there is a direction in which you can produce values that are unbounded below.
$endgroup$
– copper.hat
Mar 23 at 3:15
$begingroup$
@copper.hat what is the requirement to have a unique solution?
$endgroup$
– JOHN
Mar 23 at 3:19
$begingroup$
Can you guess? Where is the derivative (of a convex function) zero?
$endgroup$
– copper.hat
Mar 23 at 3:19
$begingroup$
@copper.hat you would want $Ax=0$ has a solution, so the det𝐀 != 0. and second derivative to be negative. I don't know how could A be expressed to meet this.
$endgroup$
– JOHN
Mar 23 at 3:28
$begingroup$
What is the second derivative of $x mapsto x'Ax$??
$endgroup$
– copper.hat
Mar 23 at 3:29
1
1
$begingroup$
It is not required. If $A=0$ then there is a global minimum. You need positive semi definite. Otherwise there is a direction in which you can produce values that are unbounded below.
$endgroup$
– copper.hat
Mar 23 at 3:15
$begingroup$
It is not required. If $A=0$ then there is a global minimum. You need positive semi definite. Otherwise there is a direction in which you can produce values that are unbounded below.
$endgroup$
– copper.hat
Mar 23 at 3:15
$begingroup$
@copper.hat what is the requirement to have a unique solution?
$endgroup$
– JOHN
Mar 23 at 3:19
$begingroup$
@copper.hat what is the requirement to have a unique solution?
$endgroup$
– JOHN
Mar 23 at 3:19
$begingroup$
Can you guess? Where is the derivative (of a convex function) zero?
$endgroup$
– copper.hat
Mar 23 at 3:19
$begingroup$
Can you guess? Where is the derivative (of a convex function) zero?
$endgroup$
– copper.hat
Mar 23 at 3:19
$begingroup$
@copper.hat you would want $Ax=0$ has a solution, so the det𝐀 != 0. and second derivative to be negative. I don't know how could A be expressed to meet this.
$endgroup$
– JOHN
Mar 23 at 3:28
$begingroup$
@copper.hat you would want $Ax=0$ has a solution, so the det𝐀 != 0. and second derivative to be negative. I don't know how could A be expressed to meet this.
$endgroup$
– JOHN
Mar 23 at 3:28
$begingroup$
What is the second derivative of $x mapsto x'Ax$??
$endgroup$
– copper.hat
Mar 23 at 3:29
$begingroup$
What is the second derivative of $x mapsto x'Ax$??
$endgroup$
– copper.hat
Mar 23 at 3:29
|
show 4 more comments
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%2f3158912%2fwhy-is-positive-definiteness-required-for-a-global-minimum-to-exist%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%2f3158912%2fwhy-is-positive-definiteness-required-for-a-global-minimum-to-exist%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
1
$begingroup$
It is not required. If $A=0$ then there is a global minimum. You need positive semi definite. Otherwise there is a direction in which you can produce values that are unbounded below.
$endgroup$
– copper.hat
Mar 23 at 3:15
$begingroup$
@copper.hat what is the requirement to have a unique solution?
$endgroup$
– JOHN
Mar 23 at 3:19
$begingroup$
Can you guess? Where is the derivative (of a convex function) zero?
$endgroup$
– copper.hat
Mar 23 at 3:19
$begingroup$
@copper.hat you would want $Ax=0$ has a solution, so the det𝐀 != 0. and second derivative to be negative. I don't know how could A be expressed to meet this.
$endgroup$
– JOHN
Mar 23 at 3:28
$begingroup$
What is the second derivative of $x mapsto x'Ax$??
$endgroup$
– copper.hat
Mar 23 at 3:29