259 research outputs found

    Efficient Maximum A-Posteriori Inference in Markov Logic and Application in Description Logics

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    Maximum a-posteriori (MAP) query in statistical relational models computes the most probable world given evidence and further knowledge about the domain. It is arguably one of the most important types of computational problems, since it is also used as a subroutine in weight learning algorithms. In this thesis, we discuss an improved inference algorithm and an application for MAP queries. We focus on Markov logic (ML) as statistical relational formalism. Markov logic combines Markov networks with first-order logic by attaching weights to first-order formulas. For inference, we improve existing work which translates MAP queries to integer linear programs (ILP). The motivation is that existing ILP solvers are very stable and fast and are able to precisely estimate the quality of an intermediate solution. In our work, we focus on improving the translation process such that we result in ILPs having fewer variables and fewer constraints. Our main contribution is the Cutting Plane Aggregation (CPA) approach which leverages symmetries in ML networks and parallelizes MAP inference. Additionally, we integrate the cutting plane inference (Riedel 2008) algorithm which significantly reduces the number of groundings by solving multiple smaller ILPs instead of one large ILP. We present the new Markov logic engine RockIt which outperforms state-of-the-art engines in standard Markov logic benchmarks. Afterwards, we apply the MAP query to description logics. Description logics (DL) are knowledge representation formalisms whose expressivity is higher than propositional logic but lower than first-order logic. The most popular DLs have been standardized in the ontology language OWL and are an elementary component in the Semantic Web. We combine Markov logic, which essentially follows the semantic of a log-linear model, with description logics to log-linear description logics. In log-linear description logic weights can be attached to any description logic axiom. Furthermore, we introduce a new query type which computes the most-probable 'coherent' world. Possible applications of log-linear description logics are mainly located in the area of ontology learning and data integration. With our novel log-linear description logic reasoner ELog, we experimentally show that more expressivity increases quality and that the solutions of optimal solving strategies have higher quality than the solutions of approximate solving strategies

    The Impact of Symmetry Handling for the Stable Set Problem via Schreier-Sims Cuts

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    Symmetry handling inequalities (SHIs) are an appealing and popular tool for handling symmetries in integer programming. Despite their practical application, little is known about their interaction with optimization problems. This article focuses on Schreier-Sims (SST) cuts, a recently introduced family of SHIs, and investigate their impact on the computational and polyhedral complexity of optimization problems. Given that SST cuts are not unique, a crucial question is to understand how different constructions of SST cuts influence the solving process. First, we observe that SST cuts do not increase the computational complexity of solving a linear optimization problem over any polytope PP. However, separating the integer hull of PP enriched by SST cuts can be NP-hard, even if PP is integral and has a compact formulation. We study this phenomenon more in-depth for the stable set problem, particularly for subclasses of perfect graphs. For bipartite graphs, we give a complete characterization of the integer hull after adding SST cuts based on odd-cycle inequalities. For trivially perfect graphs, we observe that the separation problem is still NP-hard after adding a generic set of SST cuts. Our main contribution is to identify a specific class of SST cuts, called stringent SST cuts, that keeps the separation problem polynomial and a complete set of inequalities, namely SST clique cuts, that yield a complete linear description. We complement these results by giving SST cuts based presolving techniques and provide a computational study to compare the different approaches. In particular, our newly identified stringent SST cuts dominate other approaches

    Almost Symmetries and the Unit Commitment Problem

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    This thesis explores two main topics. The first is almost symmetry detection on graphs. The presence of symmetry in combinatorial optimization problems has long been considered an anathema, but in the past decade considerable progress has been made. Modern integer and constraint programming solvers have automatic symmetry detection built-in to either exploit or avoid symmetric regions of the search space. Automatic symmetry detection generally works by converting the input problem to a graph which is in exact correspondence with the problem formulation. Symmetry can then be detected on this graph using one of the excellent existing algorithms; these are also the symmetries of the problem formulation.The motivation for detecting almost symmetries on graphs is that almost symmetries in an integer program can force the solver to explore nearly symmetric regions of the search space. Because of the known correspondence between integer programming formulations and graphs, this is a first step toward detecting almost symmetries in integer programming formulations. Though we are only able to compute almost symmetries for graphs of modest size, the results indicate that almost symmetry is definitely present in some real-world combinatorial structures, and likely warrants further investigation.The second topic explored in this thesis is integer programming formulations for the unit commitment problem. The unit commitment problem involves scheduling power generators to meet anticipated energy demand while minimizing total system operation cost. Today, practitioners usually formulate and solve unit commitment as a large-scale mixed integer linear program.The original intent of this project was to bring the analysis of almost symmetries to the unit commitment problem. Two power generators are almost symmetric in the unit commitment problem if they have almost identical parameters. Along the way, however, new formulations for power generators were discovered that warranted a thorough investigation of their own. Chapters 4 and 5 are a result of this research.Thus this work makes three contributions to the unit commitment problem: a convex hull description for a power generator accommodating many types of constraints, an improved formulation for time-dependent start-up costs, and an exact symmetry reduction technique via reformulation

    Decomposition and diet problems

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    The purpose of this thesis is to efficiently solve real life problems. We study LPs. We study an NLP and an MINLP based on what is known as the generalised pooling problem (GPP), and we study an MIP that we call the cattle mating problem. These problems are often very large or otherwise difficult to solve by direct methods, and are best solved by decomposition methods. During the thesis we introduce algorithms that exploit the structure of the problems to decompose them. We are able to solve row-linked, column-linked and general LPs efficiently by modifying the tableau simplex method, and suggest how this work could be applied to the revised simplex method. We modify an existing sequential linear programming solver that is currently used by Format International to solve GPPs, and show the modified solver takes less time and is at least as likely to find the global minimum as the old solver. We solve multifactory versions of the GPP by augmented Lagrangian decomposition, and show this is more efficient than solving the problems directly. We introduce a decomposition algorithm to solve a MINLP version of the GPP by decomposing it into NLP and ILP subproblems. This is able to solve large problems that could not be solved directly. We introduce an efficient decomposition algorithm to solve the MIP cattle mating problem, which has been adopted for use by the Irish Cattle Breeding Federation. Most of the solve methods we introduce are designed only to find local minima. However, for the multifactory version of the GPP we introduce two methods that give a good chance of finding the global minimum, both of which succeed in finding the global minimum on test problems

    Precision analysis for hardware acceleration of numerical algorithms

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    The precision used in an algorithm affects the error and performance of individual computations, the memory usage, and the potential parallelism for a fixed hardware budget. However, when migrating an algorithm onto hardware, the potential improvements that can be obtained by tuning the precision throughout an algorithm to meet a range or error specification are often overlooked; the major reason is that it is hard to choose a number system which can guarantee any such specification can be met. Instead, the problem is mitigated by opting to use IEEE standard double precision arithmetic so as to be ‘no worse’ than a software implementation. However, the flexibility in the number representation is one of the key factors that can be exploited on reconfigurable hardware such as FPGAs, and hence ignoring this potential significantly limits the performance achievable. In order to optimise the performance of hardware reliably, we require a method that can tractably calculate tight bounds for the error or range of any variable within an algorithm, but currently only a handful of methods to calculate such bounds exist, and these either sacrifice tightness or tractability, whilst simulation-based methods cannot guarantee the given error estimate. This thesis presents a new method to calculate these bounds, taking into account both input ranges and finite precision effects, which we show to be, in general, tighter in comparison to existing methods; this in turn can be used to tune the hardware to the algorithm specifications. We demonstrate the use of this software to optimise hardware for various algorithms to accelerate the solution of a system of linear equations, which forms the basis of many problems in engineering and science, and show that significant performance gains can be obtained by using this new approach in conjunction with more traditional hardware optimisations

    Combinatorial Optimization

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    This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th
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