Stability and convergence in discrete convex monotone dynamical systems

We study the stable behaviour of discrete dynamical systems where the map is convex and monotone with respect to the standard positive cone. The notion of tangential stability for fixed points and periodic points is introduced, which is weaker than Lyapunov stability. Among others we show that the set of tangentially stable fixed points is isomorphic to a convex inf-semilattice, and a criterion is given for the existence of a unique tangentially stable fixed point. We also show that periods of tangentially stable periodic points are orders of permutations on $n$ letters, where $n$ is the dimension of the underlying space, and a sufficient condition for global convergence to periodic orbits is presented.Comment: 36 pages, 1 fugur

Block diagonalization

summary:We study block diagonalization of matrices induced by resolutions of the unit matrix into the sum of idempotent matrices. We show that the block diagonal matrices have disjoint spectra if and only if each idempotent matrix in the inducing resolution double commutes with the given matrix. Applications include a new characterization of an eigenprojection and of the Drazin inverse of a given matrix

Consecutive optimizers for a partitioning problem with applications to optimal inventory groupings for joint replenishment

We consider several subclasses of the problem of grouping n items (indexed 1, 2,.., n) into m subsets so as to minimize the function g(S 1,.., S,). In general, these problems are very difficult to solve to optimality, even for the case m = 2. We provide several sufficient conditions on g(') that guarantee that there is an optimum partition in which each subset consists of consecutive integers (or else the partition S,,-, S,, satisfies a more general condition called semiconsecutiveness"). Moreover, by restricting attention to 'consecutive" (or serniconsecutive " ) partitions, we can solve the partition problem in polynomial time for small values of m. If, in addition, g is symmetric, then the partition problem is solvable in purely polynomial time. We apply these results to generalizations of a problem in inventory groupings considered by the authors in a previous paper. We also relate the results to the Neyman-Pearson lemma in statistical hypothesis testing and to a graph partitioning problem of Barnes and Hoffman. C · lg · · ·II CIL ·I D···1C- ·------- · 111-ET a, , a and b,-, b be real numbers ordered so that for some integer 0 r n, b, *.., b, are negative, b,+,.., b are nonnegative and al ar-- c.-- and tbi I b ar+l an br+i- bn For b, = 0, we consider adb, to be +cc or- according to a> 0 or a, < 0. If ai = bi = 0, al/b1 is defined arbitrarily so that inequality (1) holds. As usual, we let a and b denote the vectors whose coordinates are a, and bi, respectively. Subject clasification: 334 partitioning items into subgroups, 625 optimal inventory groupings

A partitioning problem with additive objective with an application to optimal inventory grouping for joint replenishment

We consider a problem of optimal grouping and provide conditions under which an optimal partition of an ordered set S = {r1,-*t, r,) consists of subsets of consecutive elements. We transform the problem into the problem of finding a shortest path on a directed acyclic graph with n + 1 vertices (for which efficient algorithms exist). These results may be used to solve the problem of grouping n items in stock into subgroups with a common order cycle per group so as to minimize the resulting economic order quantity costs