LMIs in Control/pages/LMI for Minimizing Condition Number of Positive Definite Matrix

The System:
A related problem is minimizing the condition number of a positive-defnite matrix $$M$$ that depends affinely on the variable $$x$$, subject to the LMI constraint $$F(x)$$ > 0. This problem can be reformulated as the GEVP.

The Optimization Problem:
The GEVP can be formulated as follows:

minimize $$ \gamma $$

subject to $$ F(x)$$ > 0;

$$\mu$$>0;

$$\mu I $$< $$M(x)$$ < $$\gamma \mu I $$.

We can reformulate this GEVP as an EVP as follows. Suppose,

$$M(x)$$= $$M_0$$ +$$\sum_{n=1}^{m}$$$$x_i M_i $$, $$F(x)$$= $$F_0$$+ $$\sum_{n=1}^{m}$$$$x_i F_i $$

The LMI:
Defining the new variables $$\nu$$=$$1/\mu$$, $$ \tilde{x}$$=$$x/\mu$$ we can express the previous formulation as the EVP with variables $$ \tilde{x},\nu$$ and $$\gamma$$:

miminize$$ \gamma $$

subject to $$\nu F_0$$+ $$\sum_{n=1}^{m}$$$$x_i F_i $$ >0; $$I$$ < $$\nu M_0$$+ $$\sum_{n=1}^{m}$$$$x_i M_i $$ < $$ \gamma I $$

Conclusion:
The LMI is feasible.