Consider symmetric matrix $A \in \mathbb{R}^{n \times n}$ with $n-1$ negative eigenvalues and a positive one such that $\lambda_1 > \lambda_2> \dots > \lambda_n$, where $\lambda_n$ is the most negative eigenvalue of $A$.
Based on how I build $A$, there are cases where $\big|\frac{\lambda_n}{\lambda_1}\big|$ is really small (e.g., $10^{-4}$). If I have an optimization problem whose convexity depends on $A$ being PSD, can I treat it as a convex problem in such cases?
For example, to use convex solvers for a relevant quadratic minimization problem?