A special class of nonconvex problems of optimization is considered. The class involves some kinds of quadratic disturbances of the convex problems. It is substantiated the absence of local optima (in the sense of their coincidence with global optima) for this class of convex problems. It is shown also that for the problems the second order extremum necessary conditions are sufficient for optimality and that the conditions may be simplified substantially in comparison with a general case of the problem with limitations.