Convergence in multi-objective coordinate descent
$begingroup$
I have two functions $f_1(x;y)$ and $f_2(y;x)$, which are both optimized in terms of the same variables $x$ and $y$. The argument notation with semicolon (e.g. $x;y$ in $f_1(x;y)$) should demonstrate that when optimizing $f_1$, only $x$ can be adapted and $y$ is considered as fixed (symmetric for $f_2$).
The two functions $f_1$ and $f_2$ are:
- $f_1(x;y) = xcdot(x+y)^2 + (1-x)cdot(2-x-y)$
- $f_2(y;x) = ycdot(x+y)^2 + (1-y)cdot(2-x-y)$
An optimization round works as follows (a kind of coordinate descent with multiple objective functions):
- $x^{(k+1)} = argmin_{x} f_1(x;y^{(k)})$
- $y^{(k+1)} = argmin_{y} f_2(y;x^{(k+1)})$
My question: How can we prove (or disprove) the convergence of such an optimization algorithm, where convergence should yield $(x',y')$ s.t.
$x' = argmin_{x} f_1(x;y')$ and- $y' = argmin_{y} f_2(y;x')$
i.e. both functions $f_1$ and $f_2$ are at the minimum with respect to their adjustable variable, given the fixed variable.
(I am not interested in the global minima of $f_1$ or $f_2$.)
convergence optimization convex-optimization
asked Jan 3 at 8:17
simaschsimasch
13
$endgroup$
add a comment |
$begingroup$
I have two functions $f_1(x;y)$ and $f_2(y;x)$, which are both optimized in terms of the same variables $x$ and $y$. The argument notation with semicolon (e.g. $x;y$ in $f_1(x;y)$) should demonstrate that when optimizing $f_1$, only $x$ can be adapted and $y$ is considered as fixed (symmetric for $f_2$).
The two functions $f_1$ and $f_2$ are:
- $f_1(x;y) = xcdot(x+y)^2 + (1-x)cdot(2-x-y)$
- $f_2(y;x) = ycdot(x+y)^2 + (1-y)cdot(2-x-y)$
An optimization round works as follows (a kind of coordinate descent with multiple objective functions):
- $x^{(k+1)} = argmin_{x} f_1(x;y^{(k)})$
- $y^{(k+1)} = argmin_{y} f_2(y;x^{(k+1)})$
My question: How can we prove (or disprove) the convergence of such an optimization algorithm, where convergence should yield $(x',y')$ s.t.
$x' = argmin_{x} f_1(x;y')$ and- $y' = argmin_{y} f_2(y;x')$
i.e. both functions $f_1$ and $f_2$ are at the minimum with respect to their adjustable variable, given the fixed variable.
(I am not interested in the global minima of $f_1$ or $f_2$.)
convergence optimization convex-optimization
asked Jan 3 at 8:17
simaschsimasch
13
$endgroup$
add a comment |
$begingroup$
I have two functions $f_1(x;y)$ and $f_2(y;x)$, which are both optimized in terms of the same variables $x$ and $y$. The argument notation with semicolon (e.g. $x;y$ in $f_1(x;y)$) should demonstrate that when optimizing $f_1$, only $x$ can be adapted and $y$ is considered as fixed (symmetric for $f_2$).
The two functions $f_1$ and $f_2$ are:
- $f_1(x;y) = xcdot(x+y)^2 + (1-x)cdot(2-x-y)$
- $f_2(y;x) = ycdot(x+y)^2 + (1-y)cdot(2-x-y)$
An optimization round works as follows (a kind of coordinate descent with multiple objective functions):
- $x^{(k+1)} = argmin_{x} f_1(x;y^{(k)})$
- $y^{(k+1)} = argmin_{y} f_2(y;x^{(k+1)})$
My question: How can we prove (or disprove) the convergence of such an optimization algorithm, where convergence should yield $(x',y')$ s.t.
$x' = argmin_{x} f_1(x;y')$ and- $y' = argmin_{y} f_2(y;x')$
i.e. both functions $f_1$ and $f_2$ are at the minimum with respect to their adjustable variable, given the fixed variable.
(I am not interested in the global minima of $f_1$ or $f_2$.)
convergence optimization convex-optimization
asked Jan 3 at 8:17
simaschsimasch
13
$endgroup$
I have two functions $f_1(x;y)$ and $f_2(y;x)$, which are both optimized in terms of the same variables $x$ and $y$. The argument notation with semicolon (e.g. $x;y$ in $f_1(x;y)$) should demonstrate that when optimizing $f_1$, only $x$ can be adapted and $y$ is considered as fixed (symmetric for $f_2$).
The two functions $f_1$ and $f_2$ are:
- $f_1(x;y) = xcdot(x+y)^2 + (1-x)cdot(2-x-y)$
- $f_2(y;x) = ycdot(x+y)^2 + (1-y)cdot(2-x-y)$
An optimization round works as follows (a kind of coordinate descent with multiple objective functions):
- $x^{(k+1)} = argmin_{x} f_1(x;y^{(k)})$
- $y^{(k+1)} = argmin_{y} f_2(y;x^{(k+1)})$
My question: How can we prove (or disprove) the convergence of such an optimization algorithm, where convergence should yield $(x',y')$ s.t.
$x' = argmin_{x} f_1(x;y')$ and- $y' = argmin_{y} f_2(y;x')$
i.e. both functions $f_1$ and $f_2$ are at the minimum with respect to their adjustable variable, given the fixed variable.
(I am not interested in the global minima of $f_1$ or $f_2$.)
convergence optimization convex-optimization
convergence optimization convex-optimization
asked Jan 3 at 8:17
simaschsimasch
13
asked Jan 3 at 8:17
simaschsimasch
13
edited Jan 4 at 7:59
simasch
asked Jan 3 at 8:17
simaschsimasch
13
asked Jan 3 at 8:17
simaschsimasch
13
asked Jan 3 at 8:17
simaschsimasch
13
13
add a comment |
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%2f3060356%2fconvergence-in-multi-objective-coordinate-descent%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%2f3060356%2fconvergence-in-multi-objective-coordinate-descent%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
