Optimization Algorithms for Faster Computational Geometry

We study two fundamental problems in computational geometry: finding the maximum inscribed ball (MaxIB) inside a bounded polyhedron defined bym hyperplanes in a d-dimensional space, and finding the minimum enclosing ball (MinEB) of a set of n points in a d-dimensional space. We translate both these geometric problems into optimization problems and apply firstorder methods for smooth and saddle-point optimization to obtain simpler, faster nearly-lineartime algorithms. For MaxIB, the best known running time is Õ(mdα/ε) [XSX06], where α ≥ 1 is the aspect ratio of the polyhedron. We obtain two new algorithms for MaxIB: one runs in Õ(mdα/ε) time using smooth optimization, and the other runs in Õ(md + m √ dα/ε) time using saddle-point optimization. For MinEB, the best known running time is Õ(nd/ √ ε) [SVZ11]. We obtain two new algorithms for MinEB: one runs in Õ(nd/ √ ε) time using smooth optimization, and the other runs in Õ(nd+ n √ d/ √ ε) time using saddle-point optimization.