Projection on convex sets with equality and inequality constraints
$begingroup$
I want to find the projection of a vector called "a" on a closed and convex set with linear constraints. The set is in the following form:
begin{array}{ll}
& Ax = b \
& Bx le d \
&x ge 0.
end{array}
I know I need to solve the following problem to get the projection:
begin{array}{ll}
text{minimize} & || x-a||^2 \
& Ax = b \
& Bx le d \
&x ge 0.
end{array}
However, I am looking for an efficient way (instead of solving with commercial software like CPLEX) using the structure, convexity, and linearity of the set. I was wondering if anyone knows how to find the projection on this type of feasible region and space properties efficiently. Any resource that has addressed this issue will also help a lot.
Note that I found this "Projection of $z$ onto the affine set ${xmid Ax = b}$" here, but the KKT condition requires other conditions in my problem that lead to non-linear constraints.
Thanks in advance for your help.
convex-analysis projection
asked Dec 8 '18 at 2:01
Mehrzad Mehrzad
63
$endgroup$
add a comment |
$begingroup$
I want to find the projection of a vector called "a" on a closed and convex set with linear constraints. The set is in the following form:
begin{array}{ll}
& Ax = b \
& Bx le d \
&x ge 0.
end{array}
I know I need to solve the following problem to get the projection:
begin{array}{ll}
text{minimize} & || x-a||^2 \
& Ax = b \
& Bx le d \
&x ge 0.
end{array}
However, I am looking for an efficient way (instead of solving with commercial software like CPLEX) using the structure, convexity, and linearity of the set. I was wondering if anyone knows how to find the projection on this type of feasible region and space properties efficiently. Any resource that has addressed this issue will also help a lot.
Note that I found this "Projection of $z$ onto the affine set ${xmid Ax = b}$" here, but the KKT condition requires other conditions in my problem that lead to non-linear constraints.
Thanks in advance for your help.
convex-analysis projection
asked Dec 8 '18 at 2:01
Mehrzad Mehrzad
63
$endgroup$
$begingroup$
You could use any quadratic program (QP) solver. For example, SQOPT.
$endgroup$
– copper.hat
Dec 8 '18 at 2:28
$begingroup$
Thanks for your reply. CPLEX can solve quadratic objectives with the set of constraints, and I get the optimal solutions, but I am looking for efficient ways to solve this problem using mathematical properties like getting the solution by multiplying some matrices and doing similar things instead of optimizing using commercial software.
$endgroup$
– Mehrzad
Dec 8 '18 at 23:22
add a comment |
$begingroup$
I want to find the projection of a vector called "a" on a closed and convex set with linear constraints. The set is in the following form:
begin{array}{ll}
& Ax = b \
& Bx le d \
&x ge 0.
end{array}
I know I need to solve the following problem to get the projection:
begin{array}{ll}
text{minimize} & || x-a||^2 \
& Ax = b \
& Bx le d \
&x ge 0.
end{array}
However, I am looking for an efficient way (instead of solving with commercial software like CPLEX) using the structure, convexity, and linearity of the set. I was wondering if anyone knows how to find the projection on this type of feasible region and space properties efficiently. Any resource that has addressed this issue will also help a lot.
Note that I found this "Projection of $z$ onto the affine set ${xmid Ax = b}$" here, but the KKT condition requires other conditions in my problem that lead to non-linear constraints.
Thanks in advance for your help.
convex-analysis projection
asked Dec 8 '18 at 2:01
Mehrzad Mehrzad
63
$endgroup$
I want to find the projection of a vector called "a" on a closed and convex set with linear constraints. The set is in the following form:
begin{array}{ll}
& Ax = b \
& Bx le d \
&x ge 0.
end{array}
I know I need to solve the following problem to get the projection:
begin{array}{ll}
text{minimize} & || x-a||^2 \
& Ax = b \
& Bx le d \
&x ge 0.
end{array}
However, I am looking for an efficient way (instead of solving with commercial software like CPLEX) using the structure, convexity, and linearity of the set. I was wondering if anyone knows how to find the projection on this type of feasible region and space properties efficiently. Any resource that has addressed this issue will also help a lot.
Note that I found this "Projection of $z$ onto the affine set ${xmid Ax = b}$" here, but the KKT condition requires other conditions in my problem that lead to non-linear constraints.
Thanks in advance for your help.
convex-analysis projection
convex-analysis projection
asked Dec 8 '18 at 2:01
Mehrzad Mehrzad
63
asked Dec 8 '18 at 2:01
Mehrzad Mehrzad
63
edited Dec 11 '18 at 1:39
Mehrzad
asked Dec 8 '18 at 2:01
Mehrzad Mehrzad
63
asked Dec 8 '18 at 2:01
Mehrzad Mehrzad
63
asked Dec 8 '18 at 2:01
Mehrzad Mehrzad
63
63
$begingroup$
You could use any quadratic program (QP) solver. For example, SQOPT.
$endgroup$
– copper.hat
Dec 8 '18 at 2:28
$begingroup$
Thanks for your reply. CPLEX can solve quadratic objectives with the set of constraints, and I get the optimal solutions, but I am looking for efficient ways to solve this problem using mathematical properties like getting the solution by multiplying some matrices and doing similar things instead of optimizing using commercial software.
$endgroup$
– Mehrzad
Dec 8 '18 at 23:22
add a comment |
$begingroup$
You could use any quadratic program (QP) solver. For example, SQOPT.
$endgroup$
– copper.hat
Dec 8 '18 at 2:28
$begingroup$
Thanks for your reply. CPLEX can solve quadratic objectives with the set of constraints, and I get the optimal solutions, but I am looking for efficient ways to solve this problem using mathematical properties like getting the solution by multiplying some matrices and doing similar things instead of optimizing using commercial software.
$endgroup$
– Mehrzad
Dec 8 '18 at 23:22
$begingroup$
You could use any quadratic program (QP) solver. For example, SQOPT.
$endgroup$
– copper.hat
Dec 8 '18 at 2:28
$begingroup$
You could use any quadratic program (QP) solver. For example, SQOPT.
$endgroup$
– copper.hat
Dec 8 '18 at 2:28
$begingroup$
Thanks for your reply. CPLEX can solve quadratic objectives with the set of constraints, and I get the optimal solutions, but I am looking for efficient ways to solve this problem using mathematical properties like getting the solution by multiplying some matrices and doing similar things instead of optimizing using commercial software.
$endgroup$
– Mehrzad
Dec 8 '18 at 23:22
$begingroup$
Thanks for your reply. CPLEX can solve quadratic objectives with the set of constraints, and I get the optimal solutions, but I am looking for efficient ways to solve this problem using mathematical properties like getting the solution by multiplying some matrices and doing similar things instead of optimizing using commercial software.
$endgroup$
– Mehrzad
Dec 8 '18 at 23:22
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%2f3030617%2fprojection-on-convex-sets-with-equality-and-inequality-constraints%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%2f3030617%2fprojection-on-convex-sets-with-equality-and-inequality-constraints%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
$begingroup$
You could use any quadratic program (QP) solver. For example, SQOPT.
$endgroup$
– copper.hat
Dec 8 '18 at 2:28
$begingroup$
Thanks for your reply. CPLEX can solve quadratic objectives with the set of constraints, and I get the optimal solutions, but I am looking for efficient ways to solve this problem using mathematical properties like getting the solution by multiplying some matrices and doing similar things instead of optimizing using commercial software.
$endgroup$
– Mehrzad
Dec 8 '18 at 23:22