Publications [#319755] of Alexandre Belloni

Papers Published

1. Belloni, A, Norm-induced densities and testing the boundedness of a convex set, Mathematics of Operations Research, vol. 33 no. 1 (February, 2008), pp. 235-256, Institute for Operations Research and the Management Sciences (INFORMS) [doi]
   (last updated on 2023/06/01)

   Abstract:
   In this paper, we explore properties of a family of probability density functions, called norm-induced densities, defined as f t(x)={e -t||x||p/∫ Ke -t||y||pdy, x ∈ K 0, x ∉ K, where K is a n-dimensional convex set that contains the origin, parameters t > 0 and p > 0, and ||·|| is any norm. We also develop connections between these densities and geometric properties of K such, as diameter, width of the recession cone, and others. Since f t is log-concave only if p ≥ 1, this framework also covers nonlog-concave densities. Moreover, we establish a new set inclusion characterization for convex sets. This leads to a new concentration of measure phenomena for unbounded convex sets. Finally, these properties are used to develop an efficient probabilistic algorithm to test whether a convex set, represented only by membership oracles (a membership oracle for K and a membership oracle for its recession cone), is bounded or not, where the algorithm reports an associated certificate of boundedness or unboundedness. © 2008 INFORMS.