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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 :
$$\underset{y,z}{\min} \frac{y}{z} ~~~~s.t~~(X,y,z)\in C ~~and~ X\ge0,~ y,z\ge0$$ 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.

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