Influence Diagrams (ID) are a flexible tool to represent discrete stochastic optimization problems, including Markov Decision Process (MDP) and Partially Observable MDP as standard examples. More precisely, given random variables considered as vertices of an acyclic digraph, a probabilistic graphical model defines a joint distribution via the conditional distributions of vertices given their parents. In ID, the random variables are represented by a probabilistic graphical model whose vertices are partitioned into three types : chance, decision and utility vertices. The user chooses the distribution of the decision vertices conditionally to their parents in order to maximize the expected utility. Leveraging the notion of rooted junction tree, we present a mixed integer linear formulation for solving an ID, as well as valid inequalities, which lead to a computationally efficient algorithm. We also show that the linear relaxation yields an optimal integer solution for instances that can be solved by the "single policy update", the default algorithm for addressing IDs.
Linear Programming Hierarchies Collapse under Symmetry
Department of Industrial Engineering and Operations Research, Columbia University（哥伦比亚大学工业工程与应用运筹学系）；Department of Industrial and Systems Engineering, Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学工业与系统工程系）；Institute for Mathematical and Computatonal Engineering, Pontificia Universidad Católica de Chile（智利天主教 Pontificia 大学数学与计算工程研究所）
Exponential lower bounds on fixed-size psd rank and semidefinite extension complexity
最近有很多兴趣证明表示给定多面体P所需的线性规划的大小的下界。在Fiorini等人的突破性论文[Proceedings of 44 th ACM Symposium on Theory of Computing 2012，第95-106页]中，表明最大割的任何线性规划公式必须具有指数大小。一个自然的问题是，是否可以证明这样强的下界半定规划公式。在本文中，我们朝着这个目标迈出了一步，我们证明了强下界的某一类SDP配方，即SDPs的笛卡儿积的固定大小的半正定锥。在实践中，这对应于具有块对角结构的半定程序，其中块具有恒定大小d。我们表明，任何这样的扩展配方的切割多面体必须有指数的大小（当d是固定的）。Fiorini等人对LP公式的结果是d=1时的特殊情况。对于大小为d=2的块，结果排除了使用二阶锥规划的任何小公式。我们的研究SDP升降机上的笛卡儿积的固定大小的半正定锥的动机主要是从实际的考虑，众所周知，这样的SDP可以更有效地解决比一般的SDP。我们的下界的证明依赖于一些矩阵的稀疏模式与小psd秩的新结果，结合从最近的论文启发Kaibel和Weltge [arXiv：1307.3543]的相关多面体的LP扩展复杂性的归纳参数。
