  We consider several basic questions pertaining to the geometry of image of a
general quadratic map. In general the image of a quadratic map is non-convex,
although there are several known classes of quadratic maps when the image is
convex. Remarkably, even when the image is not convex it often exhibits hidden
convexity: a surprising efficiency of convex relaxation to address various
geometric questions by reformulating them in terms of convex optimization
problems. In this paper we employ this strategy and put forward several
algorithms that solve the following problems pertaining to the image: verify if
a given point does not belong to the image; find the boundary point of the
image lying in a particular direction; stochastically check if the image is
convex, and if it is not, find a maximal convex subset of the image. Proposed
algorithms are implemented in the form of an open-source MATLAB library CAQM,
which accompanies the paper. Our results can be used for various problems of
discrete optimization, uncertainty analysis, physical applications, and study
of power flow equations.
