Constrained Optimization problem to unconstrained problem












1














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







share|cite|improve this question
























  • 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


















1














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







share|cite|improve this question
























  • 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
















1












1








1







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







share|cite|improve this question















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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 26 '18 at 14:46

























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




















  • 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












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 $mathbb X$.






share|cite|improve this answer





















  • 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











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1 Answer
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active

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1 Answer
1






active

oldest

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active

oldest

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active

oldest

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1














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 $mathbb X$.






share|cite|improve this answer





















  • 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
















1














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 $mathbb X$.






share|cite|improve this answer





















  • 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














1












1








1






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 $mathbb X$.






share|cite|improve this answer












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 $mathbb X$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










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


















  • 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


















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