We prove that the classic policy-iteration method (Howard 1960) and the original simplex method with the most-negative-reduced-cost pivoting rule (Dantzig 1947) are strongly polynomial-time algorithms for solving the Markov decision problem (MDP) with a fixed discount rate. Furthermore, the computational complexity of the policy-iteration and simplex methods is superior to that of the only know...

The methods for computing economic equilibria, based on variational inequalities are currently much in vogue (see e.g. [1, 2, 3, 4, 5, 6, 7, 8]). These are iterative and a natural interpretation is that they describe how individual players sequentially adjust their actions. At any stage, players solve a decentralized optimization problem depending on what the rivals are actually doing. Thus, me...

An infinite sequence of 0’s and 1’s evolves by flipping each 1 to a 0 exponentially at rate one. When a 1 flips, all bits to its right also flip. Starting from any configuration with finitely many 1’s to the left of the origin, we show that the leftmost 1 moves right with bounded speed. Upper and lower bounds are given on the speed. A consequence is that a lower bound for the run time of the ra...

The d-dimensional Goldfarb cube is a polytope with the property that all its 2 vertices appear on some shadow of it (projection onto a 2-dimensional plane). The Goldfarb cube is the solution set of a system of 2d linear inequalities with at most 3 variables per inequality. We show in this paper that the d-dimensional Klee-Minty cube — constructed from inequalities with at most 2 variables per i...