Is there a finite set $mathcal{D}$ such that $E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2$...
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Let $mathcal{D} subset mathbb{R}^2$ be a finite set. Define a function $E : mathbb{R}^3 rightarrow mathbb{R}$ by
$$large E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$
Does there exist a set $mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.
I proved $E$ is convex, and I know that if $mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $mathcal{D}$ has multiple elements are appreciated.
real-analysis multivariable-calculus convex-analysis convex-optimization
asked Nov 26 at 2:50
Joe Man Analysis
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Let $mathcal{D} subset mathbb{R}^2$ be a finite set. Define a function $E : mathbb{R}^3 rightarrow mathbb{R}$ by
$$large E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$
Does there exist a set $mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.
I proved $E$ is convex, and I know that if $mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $mathcal{D}$ has multiple elements are appreciated.
real-analysis multivariable-calculus convex-analysis convex-optimization
asked Nov 26 at 2:50
Joe Man Analysis
31319
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $mathcal{D} subset mathbb{R}^2$ be a finite set. Define a function $E : mathbb{R}^3 rightarrow mathbb{R}$ by
$$large E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$
Does there exist a set $mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.
I proved $E$ is convex, and I know that if $mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $mathcal{D}$ has multiple elements are appreciated.
real-analysis multivariable-calculus convex-analysis convex-optimization
asked Nov 26 at 2:50
Joe Man Analysis
31319
Let $mathcal{D} subset mathbb{R}^2$ be a finite set. Define a function $E : mathbb{R}^3 rightarrow mathbb{R}$ by
$$large E(a, b, c) = sum_{x in mathcal{D}} (ax_{1}^2+bx_{1}+c-x_{2})^2.$$
Does there exist a set $mathcal{D}$ such that $E$ is strongly convex? Proof or counterexample.
I proved $E$ is convex, and I know that if $mathcal{D}$ has one element then it is not strongly convex, but I am having trouble with the case where $mathcal{D}$ has more than one element. Any hints on how to proceed in the case where $mathcal{D}$ has multiple elements are appreciated.
real-analysis multivariable-calculus convex-analysis convex-optimization
real-analysis multivariable-calculus convex-analysis convex-optimization
asked Nov 26 at 2:50
Joe Man Analysis
31319
asked Nov 26 at 2:50
Joe Man Analysis
31319
edited Nov 26 at 9:54
asked Nov 26 at 2:50
Joe Man Analysis
31319
asked Nov 26 at 2:50
Joe Man Analysis
31319
asked Nov 26 at 2:50
Joe Man Analysis
31319
31319
add a comment |
add a comment |
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Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.
answered Nov 26 at 12:19
gerw
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1 Answer
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1 Answer
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active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.
answered Nov 26 at 12:19
gerw
18.9k11133
add a comment |
up vote
0
down vote
Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.
answered Nov 26 at 12:19
gerw
18.9k11133
add a comment |
up vote
0
down vote
up vote
0
down vote
Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.
answered Nov 26 at 12:19
gerw
18.9k11133
Hint: Try to compute the Hessian. Then, you will see that $x_2$ is irrelevant and you might get a guess what should be done with $x_1$.
answered Nov 26 at 12:19
gerw
18.9k11133
answered Nov 26 at 12:19
gerw
18.9k11133
answered Nov 26 at 12:19
gerw
18.9k11133
answered Nov 26 at 12:19
gerw
18.9k11133
18.9k11133
add a comment |
add a comment |
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