228,143 research outputs found
Extended Formulations via Decision Diagrams
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 that somehow
represents a given constraint matrix, and then build an equivalent
set of linear constraints over variables. That is, the size of
the resultant extended formulation depends not explicitly on the number 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 -norm
regularized hard margin optimization over the binary instance space
, which can be formulated as a linear programming problem with
constraints with -valued coefficients over variables, where
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 -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
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
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
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
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
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
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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