A Sums-of-Squares Extension of Policy Iterations

In order to address the imprecision often introduced by widening operators in static analysis, policy iteration based on min-computations amounts to considering the characterization of reachable value set of a program as an iterative computation of policies, starting from a post-fixpoint. Computing each policy and the associated invariant relies on a sequence of numerical optimizations. While the early research efforts relied on linear programming (LP) to address linear properties of linear programs, the current state of the art is still limited to the analysis of linear programs with at most quadratic invariants, relying on semidefinite programming (SDP) solvers to compute policies, and LP solvers to refine invariants. We propose here to extend the class of programs considered through the use of Sums-of-Squares (SOS) based optimization. Our approach enables the precise analysis of switched systems with polynomial updates and guards. The analysis presented has been implemented in Matlab and applied on existing programs coming from the system control literature, improving both the range of analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure

COOPERATIVE QUERY ANSWERING FOR APPROXIMATE ANSWERS WITH NEARNESS MEASURE IN HIERARCHICAL STRUCTURE INFORMATION SYSTEMS

Cooperative query answering for approximate answers has been utilized in various problem domains. Many challenges in manufacturing information retrieval, such as: classifying parts into families in group technology implementation, choosing the closest alternatives or substitutions for an out-of-stock part, or finding similar existing parts for rapid prototyping, could be alleviated using the concept of cooperative query answering. Most cooperative query answering techniques proposed by researchers so far concentrate on simple queries or single table information retrieval. Query relaxations in searching for approximate answers are mostly limited to attribute value substitutions. Many hierarchical structure information systems, such as manufacturing information systems, store their data in multiple tables that are connected to each other using hierarchical relationships - "aggregation", "generalization/specialization", "classification", and "category". Due to the nature of hierarchical structure information systems, information retrieval in such domains usually involves nested or jointed queries. In addition, searching for approximate answers in hierarchical structure databases not only considers attribute value substitutions, but also must take into account attribute or relation substitutions (i.e., WIDTH to DIAMETER, HOLE to GROOVE). For example, shape transformations of parts or features are possible and commonly practiced. A bar could be transformed to a rod. Such characteristics of hierarchical information systems, simple query or single-relation query relaxation techniques used in most cooperative query answering systems are not adequate. In this research, we proposed techniques for neighbor knowledge constructions, and complex query relaxations. We enhanced the original Pattern-based Knowledge Induction (PKI) and Distribution Sensitive Clustering (DISC) so that they can be used in neighbor hierarchy constructions at both tuple and attribute levels. We developed a cooperative query answering model to facilitate the approximate answer searching for complex queries. Our cooperative query answering model is comprised of algorithms for determining the causes of null answer, expanding qualified tuple set, expanding intersected tuple set, and relaxing multiple condition simultaneously. To calculate the semantic nearness between exact-match answers and approximate answers, we also proposed a nearness measuring function, called "Block Nearness", that is appropriate for the query relaxation methods proposed in this research

Symmetry groups, semidefinite programs, and sums of squares

We investigate the representation of symmetric polynomials as a sum of squares. Since this task is solved using semidefinite programming tools we explore the geometric, algebraic, and computational implications of the presence of discrete symmetries in semidefinite programs. It is shown that symmetry exploitation allows a significant reduction in both matrix size and number of decision variables. This result is applied to semidefinite programs arising from the computation of sum of squares decompositions for multivariate polynomials. The results, reinterpreted from an invariant-theoretic viewpoint, provide a novel representation of a class of nonnegative symmetric polynomials. The main theorem states that an invariant sum of squares polynomial is a sum of inner products of pairs of matrices, whose entries are invariant polynomials. In these pairs, one of the matrices is computed based on the real irreducible representations of the group, and the other is a sum of squares matrix. The reduction techniques enable the numerical solution of large-scale instances, otherwise computationally infeasible to solve.Comment: 38 pages, submitte