Finding the distance between the ellipsoid and the intersection of a linear manifold and ellipsoid

The problem of finding the closest points between an ellipsoid and an intersection of a linear manifold and an ellipsoid is considered. In particular, this problem includes a problem of finding the minimum distance between the ellipsoid and the ellipse. The original constrained optimization problem is reduced to the unconstrained one by means of the theory of exact penalty functions. Constructed exact penalty function is nonsmooth and belongs to the class of hypodifferentiable. Hypodifferential calculus is implied for its study and steepest hypodifferential descent is used to find its stationary points.