Constrained Optimization problem to unconstrained problem
I have a constrained optimization problem, I would like to convert this constrained problem to an unconstrained problem, specifically the constrained problem is constrained on convex set, which is:
$$ A equiv min_{x in mathbb{X}} f(x) mbox{ where } mathbb{X} mbox{ is a convex set.}$$
Furthermore,$ f(x)$ is Lipschitz continuous in which there exists some $L$ that satisfies
$$|f(x) - f(y)| < L|x-y| forall x,y inmathbb{R}^n.$$
I would like to show the above constrained problem is equivalent to the unconstrained problem in the form $$ B equivmin_{x in mathbb{R}^n} f(x) + ccdotoperatorname{dist}(x,mathbb{X}).$$
I would like to show that $A equiv B$ and find the constant $c$.
optimization convex-analysis nonlinear-optimization
asked Nov 26 '18 at 14:25
Ive Xu
83
add a comment |
I have a constrained optimization problem, I would like to convert this constrained problem to an unconstrained problem, specifically the constrained problem is constrained on convex set, which is:
$$ A equiv min_{x in mathbb{X}} f(x) mbox{ where } mathbb{X} mbox{ is a convex set.}$$
Furthermore,$ f(x)$ is Lipschitz continuous in which there exists some $L$ that satisfies
$$|f(x) - f(y)| < L|x-y| forall x,y inmathbb{R}^n.$$
I would like to show the above constrained problem is equivalent to the unconstrained problem in the form $$ B equivmin_{x in mathbb{R}^n} f(x) + ccdotoperatorname{dist}(x,mathbb{X}).$$
I would like to show that $A equiv B$ and find the constant $c$.
optimization convex-analysis nonlinear-optimization
asked Nov 26 '18 at 14:25
Ive Xu
83
You need $f$ to be defined and Lipschitz on $mathbb R^n$, not just $mathbb X$.
– Robert Israel
Nov 26 '18 at 14:33
@RobertIsrael Okay, let $f$ to be defined on $mathbb{R}^n$ then how can I convert to the unconstrained problem.
– Ive Xu
Nov 26 '18 at 14:44
add a comment |
I have a constrained optimization problem, I would like to convert this constrained problem to an unconstrained problem, specifically the constrained problem is constrained on convex set, which is:
$$ A equiv min_{x in mathbb{X}} f(x) mbox{ where } mathbb{X} mbox{ is a convex set.}$$
Furthermore,$ f(x)$ is Lipschitz continuous in which there exists some $L$ that satisfies
$$|f(x) - f(y)| < L|x-y| forall x,y inmathbb{R}^n.$$
I would like to show the above constrained problem is equivalent to the unconstrained problem in the form $$ B equivmin_{x in mathbb{R}^n} f(x) + ccdotoperatorname{dist}(x,mathbb{X}).$$
I would like to show that $A equiv B$ and find the constant $c$.
optimization convex-analysis nonlinear-optimization
asked Nov 26 '18 at 14:25
Ive Xu
83
I have a constrained optimization problem, I would like to convert this constrained problem to an unconstrained problem, specifically the constrained problem is constrained on convex set, which is:
$$ A equiv min_{x in mathbb{X}} f(x) mbox{ where } mathbb{X} mbox{ is a convex set.}$$
Furthermore,$ f(x)$ is Lipschitz continuous in which there exists some $L$ that satisfies
$$|f(x) - f(y)| < L|x-y| forall x,y inmathbb{R}^n.$$
I would like to show the above constrained problem is equivalent to the unconstrained problem in the form $$ B equivmin_{x in mathbb{R}^n} f(x) + ccdotoperatorname{dist}(x,mathbb{X}).$$
I would like to show that $A equiv B$ and find the constant $c$.
optimization convex-analysis nonlinear-optimization
optimization convex-analysis nonlinear-optimization
asked Nov 26 '18 at 14:25
Ive Xu
83
asked Nov 26 '18 at 14:25
Ive Xu
83
edited Nov 26 '18 at 14:46
asked Nov 26 '18 at 14:25
Ive Xu
83
asked Nov 26 '18 at 14:25
Ive Xu
83
asked Nov 26 '18 at 14:25
Ive Xu
83
83
You need $f$ to be defined and Lipschitz on $mathbb R^n$, not just $mathbb X$.
– Robert Israel
Nov 26 '18 at 14:33
@RobertIsrael Okay, let $f$ to be defined on $mathbb{R}^n$ then how can I convert to the unconstrained problem.
– Ive Xu
Nov 26 '18 at 14:44
add a comment |
You need $f$ to be defined and Lipschitz on $mathbb R^n$, not just $mathbb X$.
– Robert Israel
Nov 26 '18 at 14:33
@RobertIsrael Okay, let $f$ to be defined on $mathbb{R}^n$ then how can I convert to the unconstrained problem.
– Ive Xu
Nov 26 '18 at 14:44
You need $f$ to be defined and Lipschitz on $mathbb R^n$, not just $mathbb X$.
– Robert Israel
Nov 26 '18 at 14:33
You need $f$ to be defined and Lipschitz on $mathbb R^n$, not just $mathbb X$.
– Robert Israel
Nov 26 '18 at 14:33
@RobertIsrael Okay, let $f$ to be defined on $mathbb{R}^n$ then how can I convert to the unconstrained problem.
– Ive Xu
Nov 26 '18 at 14:44
@RobertIsrael Okay, let $f$ to be defined on $mathbb{R}^n$ then how can I convert to the unconstrained problem.
– Ive Xu
Nov 26 '18 at 14:44
add a comment |
1 Answer
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Take $c > L$. If $x notin mathbb X$, there is $y in mathbb X$ with $|x - y| le (c/L) text{dist}(x,mathbb X)$, so
$$f(x) + ccdot text{dist}(x,mathbb X) ge f(y) - L |x-y| + c cdot text{dist}(x,mathbb X) < f(y)$$
Thus a global minimum of $f$ can only occur in $mathbb X$.
answered Nov 26 '18 at 17:30
Robert Israel
318k23208457
Thanks for you help and answer.
– Ive Xu
Nov 27 '18 at 6:16
So You assumed $X$ is closed.
– Red shoes
Nov 27 '18 at 23:42
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Take $c > L$. If $x notin mathbb X$, there is $y in mathbb X$ with $|x - y| le (c/L) text{dist}(x,mathbb X)$, so
$$f(x) + ccdot text{dist}(x,mathbb X) ge f(y) - L |x-y| + c cdot text{dist}(x,mathbb X) < f(y)$$
Thus a global minimum of $f$ can only occur in $mathbb X$.
answered Nov 26 '18 at 17:30
Robert Israel
318k23208457
Thanks for you help and answer.
– Ive Xu
Nov 27 '18 at 6:16
So You assumed $X$ is closed.
– Red shoes
Nov 27 '18 at 23:42
add a comment |
Take $c > L$. If $x notin mathbb X$, there is $y in mathbb X$ with $|x - y| le (c/L) text{dist}(x,mathbb X)$, so
$$f(x) + ccdot text{dist}(x,mathbb X) ge f(y) - L |x-y| + c cdot text{dist}(x,mathbb X) < f(y)$$
Thus a global minimum of $f$ can only occur in $mathbb X$.
answered Nov 26 '18 at 17:30
Robert Israel
318k23208457
Thanks for you help and answer.
– Ive Xu
Nov 27 '18 at 6:16
So You assumed $X$ is closed.
– Red shoes
Nov 27 '18 at 23:42
add a comment |
Take $c > L$. If $x notin mathbb X$, there is $y in mathbb X$ with $|x - y| le (c/L) text{dist}(x,mathbb X)$, so
$$f(x) + ccdot text{dist}(x,mathbb X) ge f(y) - L |x-y| + c cdot text{dist}(x,mathbb X) < f(y)$$
Thus a global minimum of $f$ can only occur in $mathbb X$.
answered Nov 26 '18 at 17:30
Robert Israel
318k23208457
Take $c > L$. If $x notin mathbb X$, there is $y in mathbb X$ with $|x - y| le (c/L) text{dist}(x,mathbb X)$, so
$$f(x) + ccdot text{dist}(x,mathbb X) ge f(y) - L |x-y| + c cdot text{dist}(x,mathbb X) < f(y)$$
Thus a global minimum of $f$ can only occur in $mathbb X$.
answered Nov 26 '18 at 17:30
Robert Israel
318k23208457
answered Nov 26 '18 at 17:30
Robert Israel
318k23208457
answered Nov 26 '18 at 17:30
Robert Israel
318k23208457
answered Nov 26 '18 at 17:30
Robert Israel
318k23208457
318k23208457
Thanks for you help and answer.
– Ive Xu
Nov 27 '18 at 6:16
So You assumed $X$ is closed.
– Red shoes
Nov 27 '18 at 23:42
add a comment |
Thanks for you help and answer.
– Ive Xu
Nov 27 '18 at 6:16
So You assumed $X$ is closed.
– Red shoes
Nov 27 '18 at 23:42
Thanks for you help and answer.
– Ive Xu
Nov 27 '18 at 6:16
Thanks for you help and answer.
– Ive Xu
Nov 27 '18 at 6:16
So You assumed $X$ is closed.
– Red shoes
Nov 27 '18 at 23:42
So You assumed $X$ is closed.
– Red shoes
Nov 27 '18 at 23:42
add a comment |
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You need $f$ to be defined and Lipschitz on $mathbb R^n$, not just $mathbb X$.
– Robert Israel
Nov 26 '18 at 14:33
@RobertIsrael Okay, let $f$ to be defined on $mathbb{R}^n$ then how can I convert to the unconstrained problem.
– Ive Xu
Nov 26 '18 at 14:44