Centers of convex sets are geometric objects that have received extensive attention in the mathematical and optimization literature, both from a theoretical and practical standpoint. For instance, they serve as initialization points for many algorithms such as interior-point, hit-and-run, or cutting-planes methods. First, we observe that computing a Minkowski center of a convex set can be formulated as … Read more
For a given triangle $\triangle ABC$, Pierre de Fermat posed around 1640 the problem of finding a point $P$ minimizing the sum $s_P$ of the Euclidean distances from $P$ to the vertices $A$, $B$, $C$. Based on geometrical arguments this problem was first solved by Torricelli shortly after, by Simpson in 1750, and by several … Read more
For a conic optimization problem: minimize cx subject to Ax=b, x \in C, we present a geometric relationship between the maximum norms of the level sets of the primal and the inscribed sizes of the level sets of the dual (or the other way around). CitationMIT Operations Research Center Working PaperArticleDownload View PDF