Prove that the optimal solution of a fitting term does not effect by the outlier The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)What optimization method should I use here?Minimum of sum of square roots in matrix notationOptimization Problem (Linear Algebra)Showing that mean of vectors minimizes the sum of the squared distances.What is the proximal operator for $f(mathbfx)=-log(x_1x_2dots x_n)$ and its optimal solution?Proving that the optimal solution of an LP is a point on the convex hullHow to show that following is a solution of the optimization problem?Question related to the solution of Problem 3.49 (c) of Convex Optimization by Boyd and VandenbergheWhat is the KKT condition for constraint $M preceq I$?How to prove that the subgradient of a dual function contains the equality constraint for closed and convex function?
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Prove that the optimal solution of a fitting term does not effect by the outlier
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)What optimization method should I use here?Minimum of sum of square roots in matrix notationOptimization Problem (Linear Algebra)Showing that mean of vectors minimizes the sum of the squared distances.What is the proximal operator for $f(mathbfx)=-log(x_1x_2dots x_n)$ and its optimal solution?Proving that the optimal solution of an LP is a point on the convex hullHow to show that following is a solution of the optimization problem?Question related to the solution of Problem 3.49 (c) of Convex Optimization by Boyd and VandenbergheWhat is the KKT condition for constraint $M preceq I$?How to prove that the subgradient of a dual function contains the equality constraint for closed and convex function?
$begingroup$
I'm having difficulty in proving the solution of this problem:
The given vectors $mathbfa_1$, $mathbfa_2 in mathbbR^M$ and the variables $mathbfb_1,mathbfb_2inmathbbR^M$.
Assume that $l<frac12|mathbfa_1-mathbfa_2|_2$, $mathbfc$ is an unit vector orthogonal to $(mathbfa_1-mathbfa_2)$, and $0<p<1$.
How to prove
$$[mathbfa_1,mathbfa_2]=argmin_[mathbfb_1,mathbfb_2]sum_n=1^99Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2^p \+
Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2^p$$
Any help would be highly appreciated.
Update:
I have already tried to use the KKT condition for solving this problem.
The KKT condition of the unconstrained problem is only derivative equal to zeros.
Let the notation of the function in the problem is $J(mathbfb_1,mathbfb_2)$,$$
u_n(mathbfb_1,mathbfb_2) = Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2, n = 1,dots,99,
$$
and
$$
u_100(mathbfb_1,mathbfb_2) = Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2.
$$
The partial derivative are derived as following chain rule
$$
fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1triangleq fracsum_n=1^100partial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial mathbfb_1 =sum_n=1^100fracpartial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial u_n(mathbfb_1,mathbfb_2)fracpartial u_n(mathbfb_1,mathbfb_2)partial mathbfb_1.
$$
Then, the partial derivative of this problem can be obtained as follows:
$$
mathbf0 = fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1 \=sum_n=1^99frac-np100big(u_n(mathbfb_1,mathbfb_2)big)^p-2Bigg(fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg) \+ (-0.5p)big(u_100(mathbfb_1,mathbfb_2)big)^p-2Bigg(0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg).
$$
Can any one help me to check the partial derivative of $mathbfb_1$ in this problem is correct or not?
optimization vector-spaces proof-writing convex-analysis orthogonality
asked Mar 31 at 8:33
Roy HsuRoy Hsu
164
$endgroup$
add a comment |
$begingroup$
I'm having difficulty in proving the solution of this problem:
The given vectors $mathbfa_1$, $mathbfa_2 in mathbbR^M$ and the variables $mathbfb_1,mathbfb_2inmathbbR^M$.
Assume that $l<frac12|mathbfa_1-mathbfa_2|_2$, $mathbfc$ is an unit vector orthogonal to $(mathbfa_1-mathbfa_2)$, and $0<p<1$.
How to prove
$$[mathbfa_1,mathbfa_2]=argmin_[mathbfb_1,mathbfb_2]sum_n=1^99Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2^p \+
Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2^p$$
Any help would be highly appreciated.
Update:
I have already tried to use the KKT condition for solving this problem.
The KKT condition of the unconstrained problem is only derivative equal to zeros.
Let the notation of the function in the problem is $J(mathbfb_1,mathbfb_2)$,$$
u_n(mathbfb_1,mathbfb_2) = Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2, n = 1,dots,99,
$$
and
$$
u_100(mathbfb_1,mathbfb_2) = Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2.
$$
The partial derivative are derived as following chain rule
$$
fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1triangleq fracsum_n=1^100partial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial mathbfb_1 =sum_n=1^100fracpartial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial u_n(mathbfb_1,mathbfb_2)fracpartial u_n(mathbfb_1,mathbfb_2)partial mathbfb_1.
$$
Then, the partial derivative of this problem can be obtained as follows:
$$
mathbf0 = fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1 \=sum_n=1^99frac-np100big(u_n(mathbfb_1,mathbfb_2)big)^p-2Bigg(fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg) \+ (-0.5p)big(u_100(mathbfb_1,mathbfb_2)big)^p-2Bigg(0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg).
$$
Can any one help me to check the partial derivative of $mathbfb_1$ in this problem is correct or not?
optimization vector-spaces proof-writing convex-analysis orthogonality
asked Mar 31 at 8:33
Roy HsuRoy Hsu
164
$endgroup$
1
$begingroup$
Have you tried computing the gradient of that function directly? I'm not sure if it would have a nice, workable form.
$endgroup$
– David M.
Apr 1 at 0:37
$begingroup$
I have derived the KKT condition of this function (the partial derivative of b1 and b2 = 0). It can not solve this problem yet.
$endgroup$
– Roy Hsu
Apr 1 at 3:53
add a comment |
$begingroup$
I'm having difficulty in proving the solution of this problem:
The given vectors $mathbfa_1$, $mathbfa_2 in mathbbR^M$ and the variables $mathbfb_1,mathbfb_2inmathbbR^M$.
Assume that $l<frac12|mathbfa_1-mathbfa_2|_2$, $mathbfc$ is an unit vector orthogonal to $(mathbfa_1-mathbfa_2)$, and $0<p<1$.
How to prove
$$[mathbfa_1,mathbfa_2]=argmin_[mathbfb_1,mathbfb_2]sum_n=1^99Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2^p \+
Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2^p$$
Any help would be highly appreciated.
Update:
I have already tried to use the KKT condition for solving this problem.
The KKT condition of the unconstrained problem is only derivative equal to zeros.
Let the notation of the function in the problem is $J(mathbfb_1,mathbfb_2)$,$$
u_n(mathbfb_1,mathbfb_2) = Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2, n = 1,dots,99,
$$
and
$$
u_100(mathbfb_1,mathbfb_2) = Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2.
$$
The partial derivative are derived as following chain rule
$$
fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1triangleq fracsum_n=1^100partial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial mathbfb_1 =sum_n=1^100fracpartial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial u_n(mathbfb_1,mathbfb_2)fracpartial u_n(mathbfb_1,mathbfb_2)partial mathbfb_1.
$$
Then, the partial derivative of this problem can be obtained as follows:
$$
mathbf0 = fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1 \=sum_n=1^99frac-np100big(u_n(mathbfb_1,mathbfb_2)big)^p-2Bigg(fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg) \+ (-0.5p)big(u_100(mathbfb_1,mathbfb_2)big)^p-2Bigg(0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg).
$$
Can any one help me to check the partial derivative of $mathbfb_1$ in this problem is correct or not?
optimization vector-spaces proof-writing convex-analysis orthogonality
asked Mar 31 at 8:33
Roy HsuRoy Hsu
164
$endgroup$
I'm having difficulty in proving the solution of this problem:
The given vectors $mathbfa_1$, $mathbfa_2 in mathbbR^M$ and the variables $mathbfb_1,mathbfb_2inmathbbR^M$.
Assume that $l<frac12|mathbfa_1-mathbfa_2|_2$, $mathbfc$ is an unit vector orthogonal to $(mathbfa_1-mathbfa_2)$, and $0<p<1$.
How to prove
$$[mathbfa_1,mathbfa_2]=argmin_[mathbfb_1,mathbfb_2]sum_n=1^99Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2^p \+
Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2^p$$
Any help would be highly appreciated.
Update:
I have already tried to use the KKT condition for solving this problem.
The KKT condition of the unconstrained problem is only derivative equal to zeros.
Let the notation of the function in the problem is $J(mathbfb_1,mathbfb_2)$,$$
u_n(mathbfb_1,mathbfb_2) = Bigg|fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg|_2, n = 1,dots,99,
$$
and
$$
u_100(mathbfb_1,mathbfb_2) = Bigg|0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg|_2.
$$
The partial derivative are derived as following chain rule
$$
fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1triangleq fracsum_n=1^100partial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial mathbfb_1 =sum_n=1^100fracpartial big(u_n(mathbfb_1,mathbfb_2)big)^ppartial u_n(mathbfb_1,mathbfb_2)fracpartial u_n(mathbfb_1,mathbfb_2)partial mathbfb_1.
$$
Then, the partial derivative of this problem can be obtained as follows:
$$
mathbf0 = fracpartial J(mathbfb_1,mathbfb_2)partial mathbfb_1 \=sum_n=1^99frac-np100big(u_n(mathbfb_1,mathbfb_2)big)^p-2Bigg(fracn100mathbfa_1 + (1-fracn100)mathbfa_2 - big(fracn100mathbfb_1 + (1-fracn100)mathbfb_2big)Bigg) \+ (-0.5p)big(u_100(mathbfb_1,mathbfb_2)big)^p-2Bigg(0.5mathbfa_1+0.5mathbfa_2-big(0.5mathbfb_1+0.5mathbfb_2big)+lmathbfcBigg).
$$
Can any one help me to check the partial derivative of $mathbfb_1$ in this problem is correct or not?
optimization vector-spaces proof-writing convex-analysis orthogonality
optimization vector-spaces proof-writing convex-analysis orthogonality
asked Mar 31 at 8:33
Roy HsuRoy Hsu
164
asked Mar 31 at 8:33
Roy HsuRoy Hsu
164
edited Apr 2 at 12:41
Roy Hsu
asked Mar 31 at 8:33
Roy HsuRoy Hsu
164
asked Mar 31 at 8:33
Roy HsuRoy Hsu
164
asked Mar 31 at 8:33
Roy HsuRoy Hsu
164
164
1
$begingroup$
Have you tried computing the gradient of that function directly? I'm not sure if it would have a nice, workable form.
$endgroup$
– David M.
Apr 1 at 0:37
$begingroup$
I have derived the KKT condition of this function (the partial derivative of b1 and b2 = 0). It can not solve this problem yet.
$endgroup$
– Roy Hsu
Apr 1 at 3:53
add a comment |
1
$begingroup$
Have you tried computing the gradient of that function directly? I'm not sure if it would have a nice, workable form.
$endgroup$
– David M.
Apr 1 at 0:37
$begingroup$
I have derived the KKT condition of this function (the partial derivative of b1 and b2 = 0). It can not solve this problem yet.
$endgroup$
– Roy Hsu
Apr 1 at 3:53
1
1
$begingroup$
Have you tried computing the gradient of that function directly? I'm not sure if it would have a nice, workable form.
$endgroup$
– David M.
Apr 1 at 0:37
$begingroup$
Have you tried computing the gradient of that function directly? I'm not sure if it would have a nice, workable form.
$endgroup$
– David M.
Apr 1 at 0:37
$begingroup$
I have derived the KKT condition of this function (the partial derivative of b1 and b2 = 0). It can not solve this problem yet.
$endgroup$
– Roy Hsu
Apr 1 at 3:53
$begingroup$
I have derived the KKT condition of this function (the partial derivative of b1 and b2 = 0). It can not solve this problem yet.
$endgroup$
– Roy Hsu
Apr 1 at 3:53
add a comment |
0
active
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1
$begingroup$
Have you tried computing the gradient of that function directly? I'm not sure if it would have a nice, workable form.
$endgroup$
– David M.
Apr 1 at 0:37
$begingroup$
I have derived the KKT condition of this function (the partial derivative of b1 and b2 = 0). It can not solve this problem yet.
$endgroup$
– Roy Hsu
Apr 1 at 3:53