If x* is a local optimum of a function in all directions, could it be optimal in a neighborhood of x*?
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In a Euclidean space whose base is denoted $(e_1,e_2,...e_N)$. suppose that $x*$ is the local minimum of a function $f$ in all direction. Could we say that $x*$ is an optimum of $f$ in a neighborhood of $x*$ ??
optimization nonlinear-optimization
edited Nov 24 '18 at 2:35
Moo
5,53131020
asked Nov 23 '18 at 20:05
yas areyas are
607
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add a comment |
$begingroup$
In a Euclidean space whose base is denoted $(e_1,e_2,...e_N)$. suppose that $x*$ is the local minimum of a function $f$ in all direction. Could we say that $x*$ is an optimum of $f$ in a neighborhood of $x*$ ??
optimization nonlinear-optimization
edited Nov 24 '18 at 2:35
Moo
5,53131020
asked Nov 23 '18 at 20:05
yas areyas are
607
$endgroup$
add a comment |
$begingroup$
In a Euclidean space whose base is denoted $(e_1,e_2,...e_N)$. suppose that $x*$ is the local minimum of a function $f$ in all direction. Could we say that $x*$ is an optimum of $f$ in a neighborhood of $x*$ ??
optimization nonlinear-optimization
edited Nov 24 '18 at 2:35
Moo
5,53131020
asked Nov 23 '18 at 20:05
yas areyas are
607
$endgroup$
In a Euclidean space whose base is denoted $(e_1,e_2,...e_N)$. suppose that $x*$ is the local minimum of a function $f$ in all direction. Could we say that $x*$ is an optimum of $f$ in a neighborhood of $x*$ ??
optimization nonlinear-optimization
optimization nonlinear-optimization
edited Nov 24 '18 at 2:35
Moo
5,53131020
asked Nov 23 '18 at 20:05
yas areyas are
607
edited Nov 24 '18 at 2:35
Moo
5,53131020
asked Nov 23 '18 at 20:05
yas areyas are
607
edited Nov 24 '18 at 2:35
Moo
5,53131020
edited Nov 24 '18 at 2:35
Moo
5,53131020
edited Nov 24 '18 at 2:35
Moo
5,53131020
5,53131020
asked Nov 23 '18 at 20:05
yas areyas are
607
asked Nov 23 '18 at 20:05
yas areyas are
607
asked Nov 23 '18 at 20:05
yas areyas are
607
607
add a comment |
add a comment |
1 Answer
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Thats what a local minimum means. f has a local extremum at x$^*$ if for some $epsilon$-neighborhood of x$^*$, f(x) $ge$ f(x$^*$) for all x in the neighborhood.
answered Nov 23 '18 at 20:14
Joel PereiraJoel Pereira
68819
$endgroup$
$begingroup$
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
$endgroup$
– yas are
Nov 23 '18 at 20:52
add a comment |
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1 Answer
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1 Answer
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oldest
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votes
$begingroup$
Thats what a local minimum means. f has a local extremum at x$^*$ if for some $epsilon$-neighborhood of x$^*$, f(x) $ge$ f(x$^*$) for all x in the neighborhood.
answered Nov 23 '18 at 20:14
Joel PereiraJoel Pereira
68819
$endgroup$
$begingroup$
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
$endgroup$
– yas are
Nov 23 '18 at 20:52
add a comment |
$begingroup$
Thats what a local minimum means. f has a local extremum at x$^*$ if for some $epsilon$-neighborhood of x$^*$, f(x) $ge$ f(x$^*$) for all x in the neighborhood.
answered Nov 23 '18 at 20:14
Joel PereiraJoel Pereira
68819
$endgroup$
$begingroup$
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
$endgroup$
– yas are
Nov 23 '18 at 20:52
add a comment |
$begingroup$
Thats what a local minimum means. f has a local extremum at x$^*$ if for some $epsilon$-neighborhood of x$^*$, f(x) $ge$ f(x$^*$) for all x in the neighborhood.
answered Nov 23 '18 at 20:14
Joel PereiraJoel Pereira
68819
$endgroup$
Thats what a local minimum means. f has a local extremum at x$^*$ if for some $epsilon$-neighborhood of x$^*$, f(x) $ge$ f(x$^*$) for all x in the neighborhood.
answered Nov 23 '18 at 20:14
Joel PereiraJoel Pereira
68819
answered Nov 23 '18 at 20:14
Joel PereiraJoel Pereira
68819
answered Nov 23 '18 at 20:14
Joel PereiraJoel Pereira
68819
answered Nov 23 '18 at 20:14
Joel PereiraJoel Pereira
68819
68819
$begingroup$
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
$endgroup$
– yas are
Nov 23 '18 at 20:52
add a comment |
$begingroup$
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
$endgroup$
– yas are
Nov 23 '18 at 20:52
$begingroup$
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
$endgroup$
– yas are
Nov 23 '18 at 20:52
$begingroup$
The question is: if f has a local minimum in "all ei directions" could it have a local minimum in a neighberhood of x*?
$endgroup$
– yas are
Nov 23 '18 at 20:52
add a comment |
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