Solving variational inequalities with a quadratic cut method: a primal-dual, Jacobian-free approach
Computers and Operations Research
Journal of Global Optimization
Note: A note on approximation of a ball by polytopes
Discrete Optimization
| Hi-index | 0.00 |
Let [math not displayed] be a convex set. We assume that ||x||infinity ≤ 1 for all x ε C, and that C contains a ball of radius 1/R. For x ε Rn, r ε R, and B an n × n symmetric positive definite matrix, let E(x, B, r) = {y|(y - x)T B(y - x)≤ r2}. A β-rounding of C is an ellipsoid E(x, B, r) such that [math not displayed]. In the case that C is characterized by a separation oracle, it is well known that an O(n3/2)-rounding of C can be obtained using the shallow cut ellipsoid method in O(n3 ln(nR)) oracle calls. We show that a modification of the volumetric cutting plane method obtains an O(n3/2)-rounding of C in O(n2 ln(nR)) oracle calls. We also consider the problem of obtaining an O(n)-rounding of C when C has an explicit polyhedral description. Our analysis uses a new characterization of circumscribing ellipsoids centered at, or near, the volumetric center of a polyhedral set.