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
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
add a comment |
up vote
2
down vote
favorite
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
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
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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
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
optimization
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
add a comment |
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
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%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
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
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