Аннотация:
Quadratic problems are common in optimization, uncertainty analysis, physical applications. In general they are nonconvex, nevertheless sometimes they have hidden convexity structure. There are several results where this structure can be discovered; examples are Brickman theorem on convexity of 2D image of a sphere or the theorem on convexity of nonlinear image of a small ball. We address slightly different problem formulation: given a quadratic transformation, recognize convexity or nonconvexity of the image of the unit ball under this transformation. Some convexity/nonconvexity specifications are provided; algorithms for sampling boundary points of the image are developed.