For a convex optimization problem, the first-order necessary condition says that at an optimum, the gradient is equal to zero.
$\triangledown L(w*) = 0 $
The second-order sufficient condition ensures that the optimum is a minimum (not a maximum or saddle-point) using the Hessian matrix, which is the matrix of second derivatives:
$H(w*) := \dfrac{\partial^2L(w*)}{\partial w \partial w^T}$
is positive semi-definite. The Hessian is also related to the convexity of a function: a twice differentiable function is convex if and only if the Hessian is positive semi-definite at all points.
What is the difference between a necessary and a sufficient condition ? The definition of necessary in this context is clear to me since we need the gradient at the optimum to be equal to zero. What then, is the sufficient condition about ? Is the problem still convex if the sufficient condition is not satisfied ?