Regression Estimation difficulties

My regression problem is properly formulated, but is encountering serious computational difficulties.

Dependent: $Y$ = multinomial

Independent: $X_1, \dots, X_{90}$ = linearly independent set of variables. (I verified the independence. Afterall, I defined these variables).

Consider design matrix $X$, Hessian $H$, and gradient $G$.

Difficulties:

condition_number($H$) = $10^9$

Variance Inflation Factors (VIFs): all of them $< 5.5$ except for one variable which has $VIF = 15.6$

eigenvalues$(H)$: ranges from $-10^{10}$ more-or-less-smoothly to $-17.3$

This causes parameter estimation to go whacky - any sort of Newton-Raphson approxmation encounters numerical problems when computing $H^{-1} \cdot G$

Any suggestions or ideas?