2017/01/24 by Amitabh Basu, Basu, Amitabh, Michele Conforti +5
Mathematics · #46E10 #90C10 #90C11 #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC) #math.FA #math.OC #msc:46E10 #msc:90C10 #msc:90C11
paper · pdf · doi:10.48550/arxiv.1701.06700
A subset of these results appeared in Proceedings of IPCO 2016, Lecture Notes in Computer Science, vol. 9682, pp. 190--201
arxiv created 2017/08/27 · arxiv updated 2017/08/29
For the one dimensional infinite group relaxation, we construct a sequence of extreme valid functions that are piecewise linear and such that for every natural number k≥ 2, there is a function in the sequence with k slopes. This settles an open question in this area regarding a universal bound on the number of slopes for extreme functions. The function which is the pointwise limit of this sequence is an extreme valid function that is continuous and has an infinite number of slopes. This provides a new and more refined counterexample to an old conjecture of Gomory and Johnson stating that all extreme functions are piecewise linear. These constructions are extended to obtain functions for the higher dimensional group problems via the sequential-merge operation of Dey and Richard.