title: Statsmodels logistic regression convergence problems
body: <p>I'm trying to run a logistic regression in statsmodels on a large design matrix (~200 columns). The features include a number of interactions, categorical features and semi-sparse (70%) integer features. Although my design matrix is not <em>actually</em> ill-conditioned, it seems to be somewhat close (according to <code>numpy.linalg.matrix_rank</code>, it is full-rank with <code>tol=1e-3</code> but not with <code>tol=1e-2</code>). As a result, I'm struggling to get logistic regression to converge with any of the methods in statsmodels. Here's what I've tried so far:</p>
  
  <ul>
  <li><p><code>method='newton'</code>: Did not converge after 1000 iterations; raised a singular matrix <code>LinAlgError</code> while trying to invert the Hessian.</p></li>
  <li><p><code>method='bfgs'</code>: Warned of possible precision loss. Claimed convergence after 0 iterations, obviously had not actually converged.</p></li>
  <li><p><code>method='nm'</code>: Claimed that it had converged, but model had a negative pseudo-R-squared and many coefficients were still zero (and very different from values they had converged to with better-conditioned submodels). I tried cranking down <code>xtol</code> to <code>1e-8</code> to no avail.</p></li>
  <li><p><code>fit_regularized(method='l1')</code>: reported <code>Inequality constraints incompatible    (Exit mode 4)</code>. Then raised a singular matrix <code>LinAlgError</code> while trying to compute the restricted Hessian inverse.</p></li>
  </ul>