228,143 research outputs found

    Extended Formulations via Decision Diagrams

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    We propose a general algorithm of constructing an extended formulation for any given set of linear constraints with integer coefficients. Our algorithm consists of two phases: first construct a decision diagram (V,E)(V,E) that somehow represents a given m×nm \times n constraint matrix, and then build an equivalent set of ∣E∣|E| linear constraints over n+∣V∣n+|V| variables. That is, the size of the resultant extended formulation depends not explicitly on the number mm of the original constraints, but on its decision diagram representation. Therefore, we may significantly reduce the computation time for optimization problems with integer constraint matrices by solving them under the extended formulations, especially when we obtain concise decision diagram representations for the matrices. We can apply our method to 11-norm regularized hard margin optimization over the binary instance space {0,1}n\{0,1\}^n, which can be formulated as a linear programming problem with mm constraints with {−1,0,1}\{-1,0,1\}-valued coefficients over nn variables, where mm is the size of the given sample. Furthermore, introducing slack variables over the edges of the decision diagram, we establish a variant formulation of soft margin optimization. We demonstrate the effectiveness of our extended formulations for integer programming and the 11-norm regularized soft margin optimization tasks over synthetic and real datasets

    A Micro-Genetic Algorithm Approach for Soft Constraint Satisfaction Problem in University Course Scheduling

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    A university course timetabling problem is a combination of optimization problems. The problems are more challenging when a set of events need to be scheduled in the time slot, to be located to the suitable rooms, which is subjected to several sets of hard and soft constraints. All these constraints that exist as regulations within each resource for the event need to be fulfilled in order to achieve the optimum tasks. In addition, the design of course timetables for universities is a very difficult task because it is a non-deterministic polynomial, (NP) hard problem. This problem can be minimized by using a Micro Genetic Algorithm approach. This approach, encodes a chromosome representation as one of the key elements to ensure the infeasible individual chromosome produced is minimized. Thus, this study proposes an encoding chromosome representation using one-dimensional arrays to improve the Micro Genetic algorithm approach to soft constraint problems in the university course schedule. The research contribution of this study is in developing effective and feasible timetabling software using Micro Genetic Algorithm approach in order to minimize the production of an infeasible individual chromosome compared to the existing optimization algorithm for university course timetabling where UNITAR International University have been used as a data sample. The Micro Genetic Algorithm proposed has been tested in a test comparison with the Standard Genetic algorithm and the Guided Search Genetic algorithm as a benchmark. The results showed that the proposed algorithm is able to generate a minimum number of an infeasible individual chromosome. The result from the experiment also demonstrated that the Micro Genetic Algorithm is capable to produce the best course schedule to the UNITAR International University

    From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz

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    The next few years will be exciting as prototype universal quantum processors emerge, enabling implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation, and which have the potential to significantly expand the breadth of quantum computing applications. A leading candidate is Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the Quantum Alternating Operator Ansatz, is the consideration of general parametrized families of unitaries rather than only those corresponding to the time-evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.Comment: 51 pages, 2 figures. Revised to match journal pape

    Efficient and flexible deformation representation for data-driven surface modeling

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    Effectively characterizing the behavior of deformable objects has wide applicability but remains challenging. We present a new rotation-invariant deformation representation and a novel reconstruction algorithm to accurately reconstruct the positions and local rotations simultaneously. Meshes can be very efficiently reconstructed from our representation by matrix pre-decomposition, while, at the same time, hard or soft constraints can be flexibly specified with only positions of handles needed. Our approach is thus particularly suitable for constrained deformations guided by examples, providing significant benefits over state-of-the-art methods. Based on this, we further propose novel data-driven approaches to mesh deformation and non-rigid registration of deformable objects. Both problems are formulated consistently as finding an optimized model in the shape space that satisfies boundary constraints, either specified by the user, or according to the scan. By effectively exploiting the knowledge in the shape space, our method produces realistic deformation results in real-time and produces high quality registrations from a template model to a single noisy scan captured using a low-quality depth camera, outperforming state-of-the-art methods

    Solving real-world routing problems using evolutionary algorithms and multi-agent-systems

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    This thesis investigates the solving of routing problems using Evolutionary Algorithms (EAs). Routing problems are known to be hard and may possess complex search spaces. Evolutionary algorithms are potentially powerful tools for finding solutions within complex search spaces. The problem investigated is the routing of deliveries to households within an urban environment; the most common instance of this problem is that of daily postal deliveries. A representation known as Street Based Routing (SBR) is presented. This is a problem representation that makes use of the real world groupings of streets and houses. This representation is an indirect problem representation designed specifically for use with EAs. The SBR representation is incorporated within an EA and used to construct delivery routes around a variety of problem instances. The EA based system is compared against a Travelling Salesman Problem (TSP) solver, and the results are presented. The EA based system produces routes that are on average slightly longer than those produced by the TSP solver. Real world problems may often involve the construction of a network of delivery routes that are subject to multiple hard and soft constraints. A Multi Agent System (MAS) based framework for building delivery networks is presented that makes use of the SBR based EA presented earlier. Each agent within the system uses an EA to construct a single route. Agents may exchange work (via auctions or by directly negotiated exchanges) allowing the optimisation of their route. It is demonstrated that this approach has much potential and is capable of constructing delivery networks meeting set constraints, over a range of problem instances and constraint values.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

    A Variational Approach to Particles in Lipid Membranes

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    A variety of models for the membrane-mediated interaction of particles in lipid membranes, mostly well-established in theoretical physics, is reviewed from a mathematical perspective. We provide mathematically consistent formulations in a variational framework, relate apparently different modelling approaches in terms of successive approximation, and investigate existence and uniqueness. Numerical computations illustrate that the new variational formulations are directly accessible to effective numerical methods

    Infrared singularities in QCD amplitudes

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    We review recent progress in determining the infrared singularity structure of on-shell scattering amplitudes in massless gauge theories. We present a simple ansatz where soft singularities of any scattering amplitude of massless partons, to any loop order, are written as a sum over colour dipoles, governed by the cusp anomalous dimension. We explain how this formula was obtained, as the simplest solution to a newly-derived set of equations constraining the singularity structure to all orders. We emphasize the physical ideas underlying this derivation: the factorization of soft and collinear modes, the special properties of soft gluon interactions, and the notion of the cusp anomaly. Finally, we briefly discuss potential multi-loop contributions going beyond the sum-over-dipoles formula, which cannot be excluded at present.Comment: 20 pages, 3 figures. To be published in the proceedings of 23rd Les Rencontres de Physique de la Vallee d'Aoste, La Thuile, Aosta Valley, Italy, 1 - 7 March 200
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