What is $maxlimits_{sum_{i=k}^{n}x_ileqsum_{i=k}^{n}y_i \forall k=1,2,cdots,n} prod_{i=1}^{n} x_i$











up vote
2
down vote

favorite
2












begin{array}{ll} text{maximize} & prod_{i=1}^{n}x_i\ text{subject to} & mathrm sum_{i=k}^{n}x_ileqsum_{i=k}^{n}y_i \forall k=1,2,cdots,nend{array}



if $x_1geqcdotsgeq x_n$ and $y_1geqcdotsgeq y_nquad$ ($x_i,y_iin mathbb{R}^+$ and all $y_i$ are given).



My attempt: By induction



For $n=2$, we need to maximize $x_1x_2$ when $x_1+x_2leq y_1+y_2$ and $x_2leq y_2$. Let $x_2=y_2-tquad$ ($tgeq0$), then $x_1leq y_1+t$, thus to maximize $x_1x_2$, we let $x_1=y_1+t$. Then $x_1x_2=(y_2-t)(y_1+t)=y_1y_2-(y_1-y_2)t-t^2leq y_1y_2$. Thus $max x_1x_2=y_1y_2$ when $t=0$.



Suppose $maxlimits_{{sum_{i=k}^{n}x_ileqsum_{i=k}^{n}y_i \forall k=1,2,cdots,n}} prod_{i=1}^{n}x_i=prod_{i=1}^{n}y_i$, then begin{array}{ll}maxlimits_{{sum_{i=k}^{n+1}x_ileqsum_{i=k}^{n+1}y_i \forall k=1,2,cdots,n+1}} prod_{i=1}^{n+1}x_i=(prod_{i=1}^{n}y_i)x_{n+1}leq(prod_{i=1}^{n}y_i)y_{n+1}.end{array}



I think I cannot use induction like that because of constraints, any other method to try to prove my hypothesis that maximum is achieved when $x_i=y_i$?










share|cite|improve this question






















  • I cannot just let $k=1$, because inequality in the constraint should hold for all $k$, i.e. $x_1+cdots+x_nleq y_1+cdots+y_n$ when $k=1$ till $x_nleq y_n$ when $k=n$
    – Lee
    5 hours ago















up vote
2
down vote

favorite
2












begin{array}{ll} text{maximize} & prod_{i=1}^{n}x_i\ text{subject to} & mathrm sum_{i=k}^{n}x_ileqsum_{i=k}^{n}y_i \forall k=1,2,cdots,nend{array}



if $x_1geqcdotsgeq x_n$ and $y_1geqcdotsgeq y_nquad$ ($x_i,y_iin mathbb{R}^+$ and all $y_i$ are given).



My attempt: By induction



For $n=2$, we need to maximize $x_1x_2$ when $x_1+x_2leq y_1+y_2$ and $x_2leq y_2$. Let $x_2=y_2-tquad$ ($tgeq0$), then $x_1leq y_1+t$, thus to maximize $x_1x_2$, we let $x_1=y_1+t$. Then $x_1x_2=(y_2-t)(y_1+t)=y_1y_2-(y_1-y_2)t-t^2leq y_1y_2$. Thus $max x_1x_2=y_1y_2$ when $t=0$.



Suppose $maxlimits_{{sum_{i=k}^{n}x_ileqsum_{i=k}^{n}y_i \forall k=1,2,cdots,n}} prod_{i=1}^{n}x_i=prod_{i=1}^{n}y_i$, then begin{array}{ll}maxlimits_{{sum_{i=k}^{n+1}x_ileqsum_{i=k}^{n+1}y_i \forall k=1,2,cdots,n+1}} prod_{i=1}^{n+1}x_i=(prod_{i=1}^{n}y_i)x_{n+1}leq(prod_{i=1}^{n}y_i)y_{n+1}.end{array}



I think I cannot use induction like that because of constraints, any other method to try to prove my hypothesis that maximum is achieved when $x_i=y_i$?










share|cite|improve this question






















  • I cannot just let $k=1$, because inequality in the constraint should hold for all $k$, i.e. $x_1+cdots+x_nleq y_1+cdots+y_n$ when $k=1$ till $x_nleq y_n$ when $k=n$
    – Lee
    5 hours ago













up vote
2
down vote

favorite
2









up vote
2
down vote

favorite
2






2





begin{array}{ll} text{maximize} & prod_{i=1}^{n}x_i\ text{subject to} & mathrm sum_{i=k}^{n}x_ileqsum_{i=k}^{n}y_i \forall k=1,2,cdots,nend{array}



if $x_1geqcdotsgeq x_n$ and $y_1geqcdotsgeq y_nquad$ ($x_i,y_iin mathbb{R}^+$ and all $y_i$ are given).



My attempt: By induction



For $n=2$, we need to maximize $x_1x_2$ when $x_1+x_2leq y_1+y_2$ and $x_2leq y_2$. Let $x_2=y_2-tquad$ ($tgeq0$), then $x_1leq y_1+t$, thus to maximize $x_1x_2$, we let $x_1=y_1+t$. Then $x_1x_2=(y_2-t)(y_1+t)=y_1y_2-(y_1-y_2)t-t^2leq y_1y_2$. Thus $max x_1x_2=y_1y_2$ when $t=0$.



Suppose $maxlimits_{{sum_{i=k}^{n}x_ileqsum_{i=k}^{n}y_i \forall k=1,2,cdots,n}} prod_{i=1}^{n}x_i=prod_{i=1}^{n}y_i$, then begin{array}{ll}maxlimits_{{sum_{i=k}^{n+1}x_ileqsum_{i=k}^{n+1}y_i \forall k=1,2,cdots,n+1}} prod_{i=1}^{n+1}x_i=(prod_{i=1}^{n}y_i)x_{n+1}leq(prod_{i=1}^{n}y_i)y_{n+1}.end{array}



I think I cannot use induction like that because of constraints, any other method to try to prove my hypothesis that maximum is achieved when $x_i=y_i$?










share|cite|improve this question













begin{array}{ll} text{maximize} & prod_{i=1}^{n}x_i\ text{subject to} & mathrm sum_{i=k}^{n}x_ileqsum_{i=k}^{n}y_i \forall k=1,2,cdots,nend{array}



if $x_1geqcdotsgeq x_n$ and $y_1geqcdotsgeq y_nquad$ ($x_i,y_iin mathbb{R}^+$ and all $y_i$ are given).



My attempt: By induction



For $n=2$, we need to maximize $x_1x_2$ when $x_1+x_2leq y_1+y_2$ and $x_2leq y_2$. Let $x_2=y_2-tquad$ ($tgeq0$), then $x_1leq y_1+t$, thus to maximize $x_1x_2$, we let $x_1=y_1+t$. Then $x_1x_2=(y_2-t)(y_1+t)=y_1y_2-(y_1-y_2)t-t^2leq y_1y_2$. Thus $max x_1x_2=y_1y_2$ when $t=0$.



Suppose $maxlimits_{{sum_{i=k}^{n}x_ileqsum_{i=k}^{n}y_i \forall k=1,2,cdots,n}} prod_{i=1}^{n}x_i=prod_{i=1}^{n}y_i$, then begin{array}{ll}maxlimits_{{sum_{i=k}^{n+1}x_ileqsum_{i=k}^{n+1}y_i \forall k=1,2,cdots,n+1}} prod_{i=1}^{n+1}x_i=(prod_{i=1}^{n}y_i)x_{n+1}leq(prod_{i=1}^{n}y_i)y_{n+1}.end{array}



I think I cannot use induction like that because of constraints, any other method to try to prove my hypothesis that maximum is achieved when $x_i=y_i$?







optimization






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 5 hours ago









Lee

706




706












  • I cannot just let $k=1$, because inequality in the constraint should hold for all $k$, i.e. $x_1+cdots+x_nleq y_1+cdots+y_n$ when $k=1$ till $x_nleq y_n$ when $k=n$
    – Lee
    5 hours ago


















  • I cannot just let $k=1$, because inequality in the constraint should hold for all $k$, i.e. $x_1+cdots+x_nleq y_1+cdots+y_n$ when $k=1$ till $x_nleq y_n$ when $k=n$
    – Lee
    5 hours ago
















I cannot just let $k=1$, because inequality in the constraint should hold for all $k$, i.e. $x_1+cdots+x_nleq y_1+cdots+y_n$ when $k=1$ till $x_nleq y_n$ when $k=n$
– Lee
5 hours ago




I cannot just let $k=1$, because inequality in the constraint should hold for all $k$, i.e. $x_1+cdots+x_nleq y_1+cdots+y_n$ when $k=1$ till $x_nleq y_n$ when $k=n$
– Lee
5 hours ago















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',
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
});


}
});














 

draft saved


draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3004633%2fwhat-is-max-limits-sum-i-knx-i-leq-sum-i-kny-i-forall-k-1-2-cd%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown






























active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes
















 

draft saved


draft discarded



















































 


draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3004633%2fwhat-is-max-limits-sum-i-knx-i-leq-sum-i-kny-i-forall-k-1-2-cd%23new-answer', 'question_page');
}
);

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







Popular posts from this blog

android studio warns about leanback feature tag usage required on manifest while using Unity exported app?

SQL update select statement

WPF add header to Image with URL pettitions [duplicate]