Conditions for local and global optimality
$begingroup$
Consider the everywhere twice differentiable function $f:mathbb R^nto mathbb R$, the closed and convex set $mathcal S$, and the convex optimization problem
$$
min_{xin mathcal S} ; f(x).
$$
Is there an easy / intuitive way of proving both statements?
$x = x^*$ is a local minimizer if $nabla f(x^*) = 0$ and $nabla^2 f(x^*) succ 0$. Specifically, the condition $nabla^2 f(x^*) succeq 0$ is not sufficient, since as a counterexample we can consider $f(x) = x^3$, where at $x = 0$, $nabla^2 f(0) = 0$ and $nabla f(0) = 0$, but $0$ is not a minimum.
$x = x^*$ is a global minimizer if $nabla f(x^*) = 0$ and $nabla^2 f(x) succeq 0$ for all $xin mathcal S$.
The second statement in particular is quite well-known in convex optimization literature. However, I wonder if there is a nice proof, to reassure ourselves that there are no corner cases (like the one found in case 1).
optimization convex-optimization
asked Dec 23 '18 at 3:10
Y. S.Y. S.
682310
$endgroup$
|
show 7 more comments
$begingroup$
Consider the everywhere twice differentiable function $f:mathbb R^nto mathbb R$, the closed and convex set $mathcal S$, and the convex optimization problem
$$
min_{xin mathcal S} ; f(x).
$$
Is there an easy / intuitive way of proving both statements?
$x = x^*$ is a local minimizer if $nabla f(x^*) = 0$ and $nabla^2 f(x^*) succ 0$. Specifically, the condition $nabla^2 f(x^*) succeq 0$ is not sufficient, since as a counterexample we can consider $f(x) = x^3$, where at $x = 0$, $nabla^2 f(0) = 0$ and $nabla f(0) = 0$, but $0$ is not a minimum.
$x = x^*$ is a global minimizer if $nabla f(x^*) = 0$ and $nabla^2 f(x) succeq 0$ for all $xin mathcal S$.
The second statement in particular is quite well-known in convex optimization literature. However, I wonder if there is a nice proof, to reassure ourselves that there are no corner cases (like the one found in case 1).
optimization convex-optimization
asked Dec 23 '18 at 3:10
Y. S.Y. S.
682310
$endgroup$
1
$begingroup$
I'm a bit confused. Is $f$ convex?
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:14
$begingroup$
In the second case, yes (but not strictly convex). in the first, no. But the only assumptions I'm stating is that $f$ is twice differentiable over $xin mathcal S$; the rest should be from the assumptions.
$endgroup$
– Y. S.
Dec 23 '18 at 3:17
$begingroup$
What is $S$? Is $S$ convex, closed, bounded, etc,etc.
$endgroup$
– copper.hat
Dec 23 '18 at 3:54
$begingroup$
Let's assume $S$ is closed and convex, not necessarily bounded. (question amended)
$endgroup$
– Y. S.
Dec 23 '18 at 3:55
$begingroup$
So I think the answer might be as simple as: all stationary points are either local min, local max, or saddle points. If the function has a PSD hessian everywhere, then in the interior of S the stationary points must be local mins. Clearly at the boundary, saddles can occur, but the "descending" part must happen outside of S. Anyway, the question is probably unnecessarily pedantic; I mostly just wanted to clarify my understanding here to make sure there weren't any strange corner cases in case 2.
$endgroup$
– Y. S.
Dec 23 '18 at 4:01
|
show 7 more comments
$begingroup$
Consider the everywhere twice differentiable function $f:mathbb R^nto mathbb R$, the closed and convex set $mathcal S$, and the convex optimization problem
$$
min_{xin mathcal S} ; f(x).
$$
Is there an easy / intuitive way of proving both statements?
$x = x^*$ is a local minimizer if $nabla f(x^*) = 0$ and $nabla^2 f(x^*) succ 0$. Specifically, the condition $nabla^2 f(x^*) succeq 0$ is not sufficient, since as a counterexample we can consider $f(x) = x^3$, where at $x = 0$, $nabla^2 f(0) = 0$ and $nabla f(0) = 0$, but $0$ is not a minimum.
$x = x^*$ is a global minimizer if $nabla f(x^*) = 0$ and $nabla^2 f(x) succeq 0$ for all $xin mathcal S$.
The second statement in particular is quite well-known in convex optimization literature. However, I wonder if there is a nice proof, to reassure ourselves that there are no corner cases (like the one found in case 1).
optimization convex-optimization
asked Dec 23 '18 at 3:10
Y. S.Y. S.
682310
$endgroup$
Consider the everywhere twice differentiable function $f:mathbb R^nto mathbb R$, the closed and convex set $mathcal S$, and the convex optimization problem
$$
min_{xin mathcal S} ; f(x).
$$
Is there an easy / intuitive way of proving both statements?
$x = x^*$ is a local minimizer if $nabla f(x^*) = 0$ and $nabla^2 f(x^*) succ 0$. Specifically, the condition $nabla^2 f(x^*) succeq 0$ is not sufficient, since as a counterexample we can consider $f(x) = x^3$, where at $x = 0$, $nabla^2 f(0) = 0$ and $nabla f(0) = 0$, but $0$ is not a minimum.
$x = x^*$ is a global minimizer if $nabla f(x^*) = 0$ and $nabla^2 f(x) succeq 0$ for all $xin mathcal S$.
The second statement in particular is quite well-known in convex optimization literature. However, I wonder if there is a nice proof, to reassure ourselves that there are no corner cases (like the one found in case 1).
optimization convex-optimization
optimization convex-optimization
asked Dec 23 '18 at 3:10
Y. S.Y. S.
682310
asked Dec 23 '18 at 3:10
Y. S.Y. S.
682310
edited Dec 23 '18 at 3:55
Y. S.
asked Dec 23 '18 at 3:10
Y. S.Y. S.
682310
asked Dec 23 '18 at 3:10
Y. S.Y. S.
682310
asked Dec 23 '18 at 3:10
Y. S.Y. S.
682310
682310
1
$begingroup$
I'm a bit confused. Is $f$ convex?
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:14
$begingroup$
In the second case, yes (but not strictly convex). in the first, no. But the only assumptions I'm stating is that $f$ is twice differentiable over $xin mathcal S$; the rest should be from the assumptions.
$endgroup$
– Y. S.
Dec 23 '18 at 3:17
$begingroup$
What is $S$? Is $S$ convex, closed, bounded, etc,etc.
$endgroup$
– copper.hat
Dec 23 '18 at 3:54
$begingroup$
Let's assume $S$ is closed and convex, not necessarily bounded. (question amended)
$endgroup$
– Y. S.
Dec 23 '18 at 3:55
$begingroup$
So I think the answer might be as simple as: all stationary points are either local min, local max, or saddle points. If the function has a PSD hessian everywhere, then in the interior of S the stationary points must be local mins. Clearly at the boundary, saddles can occur, but the "descending" part must happen outside of S. Anyway, the question is probably unnecessarily pedantic; I mostly just wanted to clarify my understanding here to make sure there weren't any strange corner cases in case 2.
$endgroup$
– Y. S.
Dec 23 '18 at 4:01
|
show 7 more comments
1
$begingroup$
I'm a bit confused. Is $f$ convex?
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:14
$begingroup$
In the second case, yes (but not strictly convex). in the first, no. But the only assumptions I'm stating is that $f$ is twice differentiable over $xin mathcal S$; the rest should be from the assumptions.
$endgroup$
– Y. S.
Dec 23 '18 at 3:17
$begingroup$
What is $S$? Is $S$ convex, closed, bounded, etc,etc.
$endgroup$
– copper.hat
Dec 23 '18 at 3:54
$begingroup$
Let's assume $S$ is closed and convex, not necessarily bounded. (question amended)
$endgroup$
– Y. S.
Dec 23 '18 at 3:55
$begingroup$
So I think the answer might be as simple as: all stationary points are either local min, local max, or saddle points. If the function has a PSD hessian everywhere, then in the interior of S the stationary points must be local mins. Clearly at the boundary, saddles can occur, but the "descending" part must happen outside of S. Anyway, the question is probably unnecessarily pedantic; I mostly just wanted to clarify my understanding here to make sure there weren't any strange corner cases in case 2.
$endgroup$
– Y. S.
Dec 23 '18 at 4:01
1
1
$begingroup$
I'm a bit confused. Is $f$ convex?
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:14
$begingroup$
I'm a bit confused. Is $f$ convex?
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:14
$begingroup$
In the second case, yes (but not strictly convex). in the first, no. But the only assumptions I'm stating is that $f$ is twice differentiable over $xin mathcal S$; the rest should be from the assumptions.
$endgroup$
– Y. S.
Dec 23 '18 at 3:17
$begingroup$
In the second case, yes (but not strictly convex). in the first, no. But the only assumptions I'm stating is that $f$ is twice differentiable over $xin mathcal S$; the rest should be from the assumptions.
$endgroup$
– Y. S.
Dec 23 '18 at 3:17
$begingroup$
What is $S$? Is $S$ convex, closed, bounded, etc,etc.
$endgroup$
– copper.hat
Dec 23 '18 at 3:54
$begingroup$
What is $S$? Is $S$ convex, closed, bounded, etc,etc.
$endgroup$
– copper.hat
Dec 23 '18 at 3:54
$begingroup$
Let's assume $S$ is closed and convex, not necessarily bounded. (question amended)
$endgroup$
– Y. S.
Dec 23 '18 at 3:55
$begingroup$
Let's assume $S$ is closed and convex, not necessarily bounded. (question amended)
$endgroup$
– Y. S.
Dec 23 '18 at 3:55
$begingroup$
So I think the answer might be as simple as: all stationary points are either local min, local max, or saddle points. If the function has a PSD hessian everywhere, then in the interior of S the stationary points must be local mins. Clearly at the boundary, saddles can occur, but the "descending" part must happen outside of S. Anyway, the question is probably unnecessarily pedantic; I mostly just wanted to clarify my understanding here to make sure there weren't any strange corner cases in case 2.
$endgroup$
– Y. S.
Dec 23 '18 at 4:01
$begingroup$
So I think the answer might be as simple as: all stationary points are either local min, local max, or saddle points. If the function has a PSD hessian everywhere, then in the interior of S the stationary points must be local mins. Clearly at the boundary, saddles can occur, but the "descending" part must happen outside of S. Anyway, the question is probably unnecessarily pedantic; I mostly just wanted to clarify my understanding here to make sure there weren't any strange corner cases in case 2.
$endgroup$
– Y. S.
Dec 23 '18 at 4:01
|
show 7 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Yes to both of the questions (assuming that $nabla^2 f$ is continuous for the first question).
The result follows from the multivariables Taylor theorem:
$$
f(x+v) = f(x) + nabla f(x)cdot v + (nabla^2f(x+theta v): votimes v )
$$
for some $thetain(0,1)$. By letting $x=x^*$ this reduces to
$$
f(x^*+v) - f(x^*) = (nabla^2f(x^*+theta v): votimes v ),
$$
which obviously implies (2.).
For (1.), the assumption that $nabla^2 f(x^*) succ 0$ and continuity of $nabla^2 f(x^*)$ means that $nabla^2 f(x^*+theta v) succ 0$ for sufficiently small $v$ (see this question), thus the above formula shows that $f(x^*+v) - f(x^*) >0$ in a small neighborhood.
answered Dec 23 '18 at 3:31
BigbearZzzBigbearZzz
8,75121652
$endgroup$
$begingroup$
Taylor's theorem only gives inequality for some $thetain (0,1)$, not for all $thetain (0,1)$, which is not sufficient to say that $f(x)$ is a minimizer (local or global). To be more specific, the possible corner case is $nabla^2 f(x^*) succeq 0$ but not $nabla^2 f(x^*) succ 0$, which isn't really resolved by this (since $f(x+v) leq f(x)$ without really requiring global convexity).
$endgroup$
– Y. S.
Dec 23 '18 at 3:36
$begingroup$
What do you mean? For each $yinBbb R^n$ we can let $v=y-x^*$ and for each such $v$ there's a corresponding $theta = theta(v)$.
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:39
$begingroup$
So ok, restricting to case 1, for local optimality we need to say that there exists some $epsilon$ where for ALL $y$ where $|y-x|leq epsilon$, $f(y) leq f(x)$. This only says that there exists SOME $y$ in the $epsilon$ ball around $x$.
$endgroup$
– Y. S.
Dec 23 '18 at 3:41
1
$begingroup$
That's why I said we require continuity of $nabla^2 f$ at $x^*$. The link I gave is about the openess of the set of positive definite matrices. Together they imply that $nabla^2 f(y) > 0$ in a small neighborhood of $x^*$.
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:43
$begingroup$
Sure, ok. I am probably being unnecessarily pedantic about case 1; with a proof for open set PD-ness, I agree it works. But what about case 2? I think you can't use the same argument here.
$endgroup$
– Y. S.
Dec 23 '18 at 3:45
|
show 4 more comments
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1 Answer
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$begingroup$
Yes to both of the questions (assuming that $nabla^2 f$ is continuous for the first question).
The result follows from the multivariables Taylor theorem:
$$
f(x+v) = f(x) + nabla f(x)cdot v + (nabla^2f(x+theta v): votimes v )
$$
for some $thetain(0,1)$. By letting $x=x^*$ this reduces to
$$
f(x^*+v) - f(x^*) = (nabla^2f(x^*+theta v): votimes v ),
$$
which obviously implies (2.).
For (1.), the assumption that $nabla^2 f(x^*) succ 0$ and continuity of $nabla^2 f(x^*)$ means that $nabla^2 f(x^*+theta v) succ 0$ for sufficiently small $v$ (see this question), thus the above formula shows that $f(x^*+v) - f(x^*) >0$ in a small neighborhood.
answered Dec 23 '18 at 3:31
BigbearZzzBigbearZzz
8,75121652
$endgroup$
$begingroup$
Taylor's theorem only gives inequality for some $thetain (0,1)$, not for all $thetain (0,1)$, which is not sufficient to say that $f(x)$ is a minimizer (local or global). To be more specific, the possible corner case is $nabla^2 f(x^*) succeq 0$ but not $nabla^2 f(x^*) succ 0$, which isn't really resolved by this (since $f(x+v) leq f(x)$ without really requiring global convexity).
$endgroup$
– Y. S.
Dec 23 '18 at 3:36
$begingroup$
What do you mean? For each $yinBbb R^n$ we can let $v=y-x^*$ and for each such $v$ there's a corresponding $theta = theta(v)$.
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:39
$begingroup$
So ok, restricting to case 1, for local optimality we need to say that there exists some $epsilon$ where for ALL $y$ where $|y-x|leq epsilon$, $f(y) leq f(x)$. This only says that there exists SOME $y$ in the $epsilon$ ball around $x$.
$endgroup$
– Y. S.
Dec 23 '18 at 3:41
1
$begingroup$
That's why I said we require continuity of $nabla^2 f$ at $x^*$. The link I gave is about the openess of the set of positive definite matrices. Together they imply that $nabla^2 f(y) > 0$ in a small neighborhood of $x^*$.
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:43
$begingroup$
Sure, ok. I am probably being unnecessarily pedantic about case 1; with a proof for open set PD-ness, I agree it works. But what about case 2? I think you can't use the same argument here.
$endgroup$
– Y. S.
Dec 23 '18 at 3:45
|
show 4 more comments
$begingroup$
Yes to both of the questions (assuming that $nabla^2 f$ is continuous for the first question).
The result follows from the multivariables Taylor theorem:
$$
f(x+v) = f(x) + nabla f(x)cdot v + (nabla^2f(x+theta v): votimes v )
$$
for some $thetain(0,1)$. By letting $x=x^*$ this reduces to
$$
f(x^*+v) - f(x^*) = (nabla^2f(x^*+theta v): votimes v ),
$$
which obviously implies (2.).
For (1.), the assumption that $nabla^2 f(x^*) succ 0$ and continuity of $nabla^2 f(x^*)$ means that $nabla^2 f(x^*+theta v) succ 0$ for sufficiently small $v$ (see this question), thus the above formula shows that $f(x^*+v) - f(x^*) >0$ in a small neighborhood.
answered Dec 23 '18 at 3:31
BigbearZzzBigbearZzz
8,75121652
$endgroup$
$begingroup$
Taylor's theorem only gives inequality for some $thetain (0,1)$, not for all $thetain (0,1)$, which is not sufficient to say that $f(x)$ is a minimizer (local or global). To be more specific, the possible corner case is $nabla^2 f(x^*) succeq 0$ but not $nabla^2 f(x^*) succ 0$, which isn't really resolved by this (since $f(x+v) leq f(x)$ without really requiring global convexity).
$endgroup$
– Y. S.
Dec 23 '18 at 3:36
$begingroup$
What do you mean? For each $yinBbb R^n$ we can let $v=y-x^*$ and for each such $v$ there's a corresponding $theta = theta(v)$.
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:39
$begingroup$
So ok, restricting to case 1, for local optimality we need to say that there exists some $epsilon$ where for ALL $y$ where $|y-x|leq epsilon$, $f(y) leq f(x)$. This only says that there exists SOME $y$ in the $epsilon$ ball around $x$.
$endgroup$
– Y. S.
Dec 23 '18 at 3:41
1
$begingroup$
That's why I said we require continuity of $nabla^2 f$ at $x^*$. The link I gave is about the openess of the set of positive definite matrices. Together they imply that $nabla^2 f(y) > 0$ in a small neighborhood of $x^*$.
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:43
$begingroup$
Sure, ok. I am probably being unnecessarily pedantic about case 1; with a proof for open set PD-ness, I agree it works. But what about case 2? I think you can't use the same argument here.
$endgroup$
– Y. S.
Dec 23 '18 at 3:45
|
show 4 more comments
$begingroup$
Yes to both of the questions (assuming that $nabla^2 f$ is continuous for the first question).
The result follows from the multivariables Taylor theorem:
$$
f(x+v) = f(x) + nabla f(x)cdot v + (nabla^2f(x+theta v): votimes v )
$$
for some $thetain(0,1)$. By letting $x=x^*$ this reduces to
$$
f(x^*+v) - f(x^*) = (nabla^2f(x^*+theta v): votimes v ),
$$
which obviously implies (2.).
For (1.), the assumption that $nabla^2 f(x^*) succ 0$ and continuity of $nabla^2 f(x^*)$ means that $nabla^2 f(x^*+theta v) succ 0$ for sufficiently small $v$ (see this question), thus the above formula shows that $f(x^*+v) - f(x^*) >0$ in a small neighborhood.
answered Dec 23 '18 at 3:31
BigbearZzzBigbearZzz
8,75121652
$endgroup$
Yes to both of the questions (assuming that $nabla^2 f$ is continuous for the first question).
The result follows from the multivariables Taylor theorem:
$$
f(x+v) = f(x) + nabla f(x)cdot v + (nabla^2f(x+theta v): votimes v )
$$
for some $thetain(0,1)$. By letting $x=x^*$ this reduces to
$$
f(x^*+v) - f(x^*) = (nabla^2f(x^*+theta v): votimes v ),
$$
which obviously implies (2.).
For (1.), the assumption that $nabla^2 f(x^*) succ 0$ and continuity of $nabla^2 f(x^*)$ means that $nabla^2 f(x^*+theta v) succ 0$ for sufficiently small $v$ (see this question), thus the above formula shows that $f(x^*+v) - f(x^*) >0$ in a small neighborhood.
answered Dec 23 '18 at 3:31
BigbearZzzBigbearZzz
8,75121652
answered Dec 23 '18 at 3:31
BigbearZzzBigbearZzz
8,75121652
answered Dec 23 '18 at 3:31
BigbearZzzBigbearZzz
8,75121652
answered Dec 23 '18 at 3:31
BigbearZzzBigbearZzz
8,75121652
8,75121652
$begingroup$
Taylor's theorem only gives inequality for some $thetain (0,1)$, not for all $thetain (0,1)$, which is not sufficient to say that $f(x)$ is a minimizer (local or global). To be more specific, the possible corner case is $nabla^2 f(x^*) succeq 0$ but not $nabla^2 f(x^*) succ 0$, which isn't really resolved by this (since $f(x+v) leq f(x)$ without really requiring global convexity).
$endgroup$
– Y. S.
Dec 23 '18 at 3:36
$begingroup$
What do you mean? For each $yinBbb R^n$ we can let $v=y-x^*$ and for each such $v$ there's a corresponding $theta = theta(v)$.
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:39
$begingroup$
So ok, restricting to case 1, for local optimality we need to say that there exists some $epsilon$ where for ALL $y$ where $|y-x|leq epsilon$, $f(y) leq f(x)$. This only says that there exists SOME $y$ in the $epsilon$ ball around $x$.
$endgroup$
– Y. S.
Dec 23 '18 at 3:41
1
$begingroup$
That's why I said we require continuity of $nabla^2 f$ at $x^*$. The link I gave is about the openess of the set of positive definite matrices. Together they imply that $nabla^2 f(y) > 0$ in a small neighborhood of $x^*$.
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:43
$begingroup$
Sure, ok. I am probably being unnecessarily pedantic about case 1; with a proof for open set PD-ness, I agree it works. But what about case 2? I think you can't use the same argument here.
$endgroup$
– Y. S.
Dec 23 '18 at 3:45
|
show 4 more comments
$begingroup$
Taylor's theorem only gives inequality for some $thetain (0,1)$, not for all $thetain (0,1)$, which is not sufficient to say that $f(x)$ is a minimizer (local or global). To be more specific, the possible corner case is $nabla^2 f(x^*) succeq 0$ but not $nabla^2 f(x^*) succ 0$, which isn't really resolved by this (since $f(x+v) leq f(x)$ without really requiring global convexity).
$endgroup$
– Y. S.
Dec 23 '18 at 3:36
$begingroup$
What do you mean? For each $yinBbb R^n$ we can let $v=y-x^*$ and for each such $v$ there's a corresponding $theta = theta(v)$.
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:39
$begingroup$
So ok, restricting to case 1, for local optimality we need to say that there exists some $epsilon$ where for ALL $y$ where $|y-x|leq epsilon$, $f(y) leq f(x)$. This only says that there exists SOME $y$ in the $epsilon$ ball around $x$.
$endgroup$
– Y. S.
Dec 23 '18 at 3:41
1
$begingroup$
That's why I said we require continuity of $nabla^2 f$ at $x^*$. The link I gave is about the openess of the set of positive definite matrices. Together they imply that $nabla^2 f(y) > 0$ in a small neighborhood of $x^*$.
$endgroup$
– BigbearZzz
Dec 23 '18 at 3:43
$begingroup$
Sure, ok. I am probably being unnecessarily pedantic about case 1; with a proof for open set PD-ness, I agree it works. But what about case 2? I think you can't use the same argument here.
$endgroup$
– Y. S.
Dec 23 '18 at 3:45
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Taylor's theorem only gives inequality for some $thetain (0,1)$, not for all $thetain (0,1)$, which is not sufficient to say that $f(x)$ is a minimizer (local or global). To be more specific, the possible corner case is $nabla^2 f(x^*) succeq 0$ but not $nabla^2 f(x^*) succ 0$, which isn't really resolved by this (since $f(x+v) leq f(x)$ without really requiring global convexity).
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– Y. S.
Dec 23 '18 at 3:36
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Taylor's theorem only gives inequality for some $thetain (0,1)$, not for all $thetain (0,1)$, which is not sufficient to say that $f(x)$ is a minimizer (local or global). To be more specific, the possible corner case is $nabla^2 f(x^*) succeq 0$ but not $nabla^2 f(x^*) succ 0$, which isn't really resolved by this (since $f(x+v) leq f(x)$ without really requiring global convexity).
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– Y. S.
Dec 23 '18 at 3:36
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What do you mean? For each $yinBbb R^n$ we can let $v=y-x^*$ and for each such $v$ there's a corresponding $theta = theta(v)$.
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– BigbearZzz
Dec 23 '18 at 3:39
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What do you mean? For each $yinBbb R^n$ we can let $v=y-x^*$ and for each such $v$ there's a corresponding $theta = theta(v)$.
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– BigbearZzz
Dec 23 '18 at 3:39
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So ok, restricting to case 1, for local optimality we need to say that there exists some $epsilon$ where for ALL $y$ where $|y-x|leq epsilon$, $f(y) leq f(x)$. This only says that there exists SOME $y$ in the $epsilon$ ball around $x$.
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– Y. S.
Dec 23 '18 at 3:41
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So ok, restricting to case 1, for local optimality we need to say that there exists some $epsilon$ where for ALL $y$ where $|y-x|leq epsilon$, $f(y) leq f(x)$. This only says that there exists SOME $y$ in the $epsilon$ ball around $x$.
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– Y. S.
Dec 23 '18 at 3:41
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That's why I said we require continuity of $nabla^2 f$ at $x^*$. The link I gave is about the openess of the set of positive definite matrices. Together they imply that $nabla^2 f(y) > 0$ in a small neighborhood of $x^*$.
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– BigbearZzz
Dec 23 '18 at 3:43
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That's why I said we require continuity of $nabla^2 f$ at $x^*$. The link I gave is about the openess of the set of positive definite matrices. Together they imply that $nabla^2 f(y) > 0$ in a small neighborhood of $x^*$.
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– BigbearZzz
Dec 23 '18 at 3:43
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Sure, ok. I am probably being unnecessarily pedantic about case 1; with a proof for open set PD-ness, I agree it works. But what about case 2? I think you can't use the same argument here.
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– Y. S.
Dec 23 '18 at 3:45
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Sure, ok. I am probably being unnecessarily pedantic about case 1; with a proof for open set PD-ness, I agree it works. But what about case 2? I think you can't use the same argument here.
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– Y. S.
Dec 23 '18 at 3:45
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I'm a bit confused. Is $f$ convex?
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– BigbearZzz
Dec 23 '18 at 3:14
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In the second case, yes (but not strictly convex). in the first, no. But the only assumptions I'm stating is that $f$ is twice differentiable over $xin mathcal S$; the rest should be from the assumptions.
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– Y. S.
Dec 23 '18 at 3:17
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What is $S$? Is $S$ convex, closed, bounded, etc,etc.
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– copper.hat
Dec 23 '18 at 3:54
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Let's assume $S$ is closed and convex, not necessarily bounded. (question amended)
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– Y. S.
Dec 23 '18 at 3:55
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So I think the answer might be as simple as: all stationary points are either local min, local max, or saddle points. If the function has a PSD hessian everywhere, then in the interior of S the stationary points must be local mins. Clearly at the boundary, saddles can occur, but the "descending" part must happen outside of S. Anyway, the question is probably unnecessarily pedantic; I mostly just wanted to clarify my understanding here to make sure there weren't any strange corner cases in case 2.
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– Y. S.
Dec 23 '18 at 4:01