Approximate symmetric matrix by minimizing condition number
We want to approximate A symmetric semi definite positive by another X that's symmetric and whose condition number $frac{lambda_{max}(X)}{lambda_{min}(X)}$.
The optimization problem can be written as such :
$$underset{y,z}{min} frac{y}{z} ~~~~s.t~~(X,y,z)in C ~~and~ Xge0,~ y,zge0$$
where C is the set formed by the following :
1. $lambda_{max}(X) le y$
2. $lambda_{min}(X) ge z$
3. $||A-X|| le epsilon$ for some epsilon.
Only the problem here is that the objective function is not convex while the conditions are.
The exercice states that the problem can be rewritten as a convex minimization problem whose objective is affine because appropriate modifications, one can force $lambda_{min}(X)=1$.
I can't really see how to do so. Can someone please show how the original problem can be modified properly ?
Thanks.
convex-optimization symmetric-matrices semidefinite-programming condition-number
asked Nov 24 at 18:54
mjab
365
add a comment |
We want to approximate A symmetric semi definite positive by another X that's symmetric and whose condition number $frac{lambda_{max}(X)}{lambda_{min}(X)}$.
The optimization problem can be written as such :
$$underset{y,z}{min} frac{y}{z} ~~~~s.t~~(X,y,z)in C ~~and~ Xge0,~ y,zge0$$
where C is the set formed by the following :
1. $lambda_{max}(X) le y$
2. $lambda_{min}(X) ge z$
3. $||A-X|| le epsilon$ for some epsilon.
Only the problem here is that the objective function is not convex while the conditions are.
The exercice states that the problem can be rewritten as a convex minimization problem whose objective is affine because appropriate modifications, one can force $lambda_{min}(X)=1$.
I can't really see how to do so. Can someone please show how the original problem can be modified properly ?
Thanks.
convex-optimization symmetric-matrices semidefinite-programming condition-number
asked Nov 24 at 18:54
mjab
365
add a comment |
We want to approximate A symmetric semi definite positive by another X that's symmetric and whose condition number $frac{lambda_{max}(X)}{lambda_{min}(X)}$.
The optimization problem can be written as such :
$$underset{y,z}{min} frac{y}{z} ~~~~s.t~~(X,y,z)in C ~~and~ Xge0,~ y,zge0$$
where C is the set formed by the following :
1. $lambda_{max}(X) le y$
2. $lambda_{min}(X) ge z$
3. $||A-X|| le epsilon$ for some epsilon.
Only the problem here is that the objective function is not convex while the conditions are.
The exercice states that the problem can be rewritten as a convex minimization problem whose objective is affine because appropriate modifications, one can force $lambda_{min}(X)=1$.
I can't really see how to do so. Can someone please show how the original problem can be modified properly ?
Thanks.
convex-optimization symmetric-matrices semidefinite-programming condition-number
asked Nov 24 at 18:54
mjab
365
We want to approximate A symmetric semi definite positive by another X that's symmetric and whose condition number $frac{lambda_{max}(X)}{lambda_{min}(X)}$.
The optimization problem can be written as such :
$$underset{y,z}{min} frac{y}{z} ~~~~s.t~~(X,y,z)in C ~~and~ Xge0,~ y,zge0$$
where C is the set formed by the following :
1. $lambda_{max}(X) le y$
2. $lambda_{min}(X) ge z$
3. $||A-X|| le epsilon$ for some epsilon.
Only the problem here is that the objective function is not convex while the conditions are.
The exercice states that the problem can be rewritten as a convex minimization problem whose objective is affine because appropriate modifications, one can force $lambda_{min}(X)=1$.
I can't really see how to do so. Can someone please show how the original problem can be modified properly ?
Thanks.
convex-optimization symmetric-matrices semidefinite-programming condition-number
convex-optimization symmetric-matrices semidefinite-programming condition-number
asked Nov 24 at 18:54
mjab
365
asked Nov 24 at 18:54
mjab
365
asked Nov 24 at 18:54
mjab
365
asked Nov 24 at 18:54
mjab
365
asked Nov 24 at 18:54
mjab
365
365
add a comment |
add a comment |
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