Intuitive explanation for self-concordance in convex optimization
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Can someone give some intuition on what self-concordance means in optimization? The best I understand it is that it provides some type of bound on the growth of barrier functions. Specifically, what's an intuitive interpretation for the definition:
$|F'''(x)| le 2|F''(x)|^{3/2}$
optimization convex-optimization
asked Jan 4 at 21:12
BackstromBackstrom
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Can someone give some intuition on what self-concordance means in optimization? The best I understand it is that it provides some type of bound on the growth of barrier functions. Specifically, what's an intuitive interpretation for the definition:
$|F'''(x)| le 2|F''(x)|^{3/2}$
optimization convex-optimization
asked Jan 4 at 21:12
BackstromBackstrom
1
$endgroup$
1
$begingroup$
That's basically what it is, yes. It's a particular bound on the growth of barrier function that happens to allow for proofs of convergence of Newton's method. And it happens to be valid for a variety of genuinely useful barrier functions, too. I'm not inside Nesterov or Nemirovskii's head, so I can't tell you how they came up with it. But if I had to guess, they were trying to prove convergence, and they found that their proof would work if only they could bound the Hessian in this particular way...
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– Michael Grant
Jan 5 at 1:51
add a comment |
0
0
0
$begingroup$
Can someone give some intuition on what self-concordance means in optimization? The best I understand it is that it provides some type of bound on the growth of barrier functions. Specifically, what's an intuitive interpretation for the definition:
$|F'''(x)| le 2|F''(x)|^{3/2}$
optimization convex-optimization
asked Jan 4 at 21:12
BackstromBackstrom
1
$endgroup$
Can someone give some intuition on what self-concordance means in optimization? The best I understand it is that it provides some type of bound on the growth of barrier functions. Specifically, what's an intuitive interpretation for the definition:
$|F'''(x)| le 2|F''(x)|^{3/2}$
optimization convex-optimization
optimization convex-optimization
asked Jan 4 at 21:12
BackstromBackstrom
1
asked Jan 4 at 21:12
BackstromBackstrom
1
asked Jan 4 at 21:12
BackstromBackstrom
1
asked Jan 4 at 21:12
BackstromBackstrom
1
asked Jan 4 at 21:12
BackstromBackstrom
1
1
1
$begingroup$
That's basically what it is, yes. It's a particular bound on the growth of barrier function that happens to allow for proofs of convergence of Newton's method. And it happens to be valid for a variety of genuinely useful barrier functions, too. I'm not inside Nesterov or Nemirovskii's head, so I can't tell you how they came up with it. But if I had to guess, they were trying to prove convergence, and they found that their proof would work if only they could bound the Hessian in this particular way...
$endgroup$
– Michael Grant
Jan 5 at 1:51
add a comment |
1
$begingroup$
That's basically what it is, yes. It's a particular bound on the growth of barrier function that happens to allow for proofs of convergence of Newton's method. And it happens to be valid for a variety of genuinely useful barrier functions, too. I'm not inside Nesterov or Nemirovskii's head, so I can't tell you how they came up with it. But if I had to guess, they were trying to prove convergence, and they found that their proof would work if only they could bound the Hessian in this particular way...
$endgroup$
– Michael Grant
Jan 5 at 1:51
1
1
$begingroup$
That's basically what it is, yes. It's a particular bound on the growth of barrier function that happens to allow for proofs of convergence of Newton's method. And it happens to be valid for a variety of genuinely useful barrier functions, too. I'm not inside Nesterov or Nemirovskii's head, so I can't tell you how they came up with it. But if I had to guess, they were trying to prove convergence, and they found that their proof would work if only they could bound the Hessian in this particular way...
$endgroup$
– Michael Grant
Jan 5 at 1:51
$begingroup$
That's basically what it is, yes. It's a particular bound on the growth of barrier function that happens to allow for proofs of convergence of Newton's method. And it happens to be valid for a variety of genuinely useful barrier functions, too. I'm not inside Nesterov or Nemirovskii's head, so I can't tell you how they came up with it. But if I had to guess, they were trying to prove convergence, and they found that their proof would work if only they could bound the Hessian in this particular way...
$endgroup$
– Michael Grant
Jan 5 at 1:51
add a comment |
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1
$begingroup$
That's basically what it is, yes. It's a particular bound on the growth of barrier function that happens to allow for proofs of convergence of Newton's method. And it happens to be valid for a variety of genuinely useful barrier functions, too. I'm not inside Nesterov or Nemirovskii's head, so I can't tell you how they came up with it. But if I had to guess, they were trying to prove convergence, and they found that their proof would work if only they could bound the Hessian in this particular way...
$endgroup$
– Michael Grant
Jan 5 at 1:51