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I have a quadratic, positive definite function $V(x)$ for which I am attempting to find a level set that is contained within two level sets of another quadratic, positive definite function. The following simple example should clarify my objective.

Given the following contours

$$C_1 = x_1^2 + x_2^2 = x^TH_1x = 1.00$$

$$C_2 = x_1^2 + x_2^2 = x^TH_2x = 0.25$$

and the following function:

$$V(x) = x_1^2 + x_1x_2 + x_2^2 = x^TPx = \gamma$$

determine a value $\gamma$ such that the resulting level set lies "between" the two contours $C_1 , C_2$. The following figure shows the intent.

Contour-Bounded Level Set

For the value $\gamma = 0.22$, the resulting level set is between the two contours.

I believe this is considered a set-containment problem. I initially thought I could solve this using CVX with the following linear matrix inequality:

$$ max. \gamma $$ $$ (x^TH_1x - \beta_1) + (x^TPx - \gamma) > 0 $$ $$ (x^TH_2x - \beta_2) + (x^TPx - \gamma) < 0 $$

For example, for points along the outer contour $$ (x^TH_1x - \beta_1) = 0 $$ we are ensuring that $$ (x^TPx > \gamma) $$ thus the level set equal to $\gamma$ must be inside the outer contour. Similar logic applies for the 2nd matrix equality.

This LMI is not giving me the desired answer, therefore my problem formulation must be off somehow. Any input would be appreciated.

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    So you want $V(x) \geq \gamma \; \forall x : x^TH_2x \leq 0.25$ and $V(x) \geq \gamma \; \forall x : x^TH_1x \geq 1.00$. Verifying this for fixed $\gamma$ is hard, so bisection search is not an option. You can solve an [SDP relaxation based on SOS](http://www.mit.edu/~dbertsim/papers/Robust%20Optimization/A%20hierarchy%20of%20policies%20for%20adaptive%20optimization.pdf) (Section 3.1). – 2017-01-17

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