Suppose I have a quadratic programming problem with objective function of the form:
$x^T \Sigma x\ +\ ...$
The objective function is convex only if $\Sigma$ which is a covariance matrix is semi-definite positive. However under some extreme circumstances $\Sigma$ may become singular / rank deficient and span a set of possible optimization solutions in that case not just one. When $\Sigma$ becomes singular is it still semi-definite positive and the optimization problem convex or not?