Large Number Of Discrete Variables Research Articles

Full operator preconditioning and the accuracy of solving linear systems

Abstract Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error and often slow convergence of an iterative solver. In many cases, such systems arise from the discretization of operator equations with a large number of discrete variables and the ill-conditioning is tackled by means of preconditioning. A key observation in this paper is the sometimes overlooked fact that while traditional preconditioning effectively accelerates convergence of iterative methods, it generally does not improve the accuracy of the solution. Nonetheless, it is sometimes possible to overcome this barrier: accuracy can be improved significantly if the equation is transformed before discretization, a process we refer to as full operator preconditioning (FOP). We highlight that this principle is already used in various areas, including second kind integral equations and Olver–Townsend’s spectral method. We formulate a sufficient condition under which high accuracy can be obtained by FOP. We illustrate this for a fourth order differential equation which is discretized using finite elements.

A Large-Scale Combinatorial Many-Objective Evolutionary Algorithm for Intensity-Modulated Radiotherapy Planning

Intensity-modulated radiotherapy (IMRT) is one of the most popular techniques for cancer treatment. However, existing IMRT planning methods can only generate one solution at a time and, consequently, medical physicists should perform the planning process many times to obtain diverse solutions to meet the requirement of a clinical case. Meanwhile, multiobjective evolutionary algorithms (MOEAs) have not been fully exploited in IMRT planning since they are ineffective in optimizing the large number of discrete variables of IMRT. To bridge the gap, this article formulates IMRT planning into a large-scale combinatorial many-objective optimization problem and proposes a coevolutionary algorithm to solve it. In contrast to the existing MOEAs handling high-dimensional search spaces via variable grouping or dimensionality reduction, the proposed algorithm evolves one population with fine encoding for local exploitation and evolves another population with rough encoding for global exploration. Moreover, the convergence speed is further accelerated by two customized local search strategies. The experimental results verify that the proposed algorithm outperforms state-of-the-art MOEAs and IMRT planning methods on a variety of clinical cases.

General models for topology layout of oil and gas star-tree pipeline network

Oil&gas gathering pipeline network structure is a significant part of oil&gas field construction, and the rational construction of pipeline network is directly related to the efficiency and benefits of oil&gas field production. Therefore, optimizing the gathering and transportation system of oil and gas fields is the key to reducing development costs. The star-tree type pipe network is widely used in the gathering and transportation system. In order to optimize the star-tree pipe networks (STPNs) that has restrictions on the processing capacity and gathering radius of the station as a whole, this paper establishes four models of pipe network layout with specific constraints. They are Mixed-Integer Linear Programming Models with a large number of discrete variables. We take two virtual fields as examples, use CPLEX solver to solve the above four models as a whole, to obtain the optimal scheme, and also figure out the investment of the pipeline network. We further optimize the hierarchical optimization of the pipeline network with special constraints, then compare and analyze results obtained by the overall optimization. Finally, models are applied to an actual oil field and an actual gas field as examples to optimize the layout, which verifies the validity and feasibility of the models.

An efficient approach for solving layout problems

This paper offers an approach that could be useful for diverse types of layout problems or even area allocation problems. By this approach there is no need to large number of discrete variables and only by few continues variables large-scale layout problems could be solved in polynomial time. This is resulted from dividing area into discrete and continuous dimensions. Also defining decision variables as starting and finishing point of departments in area makes it possible to model layout problem so. This paper also provides new technique that models basic constraints of layout problems.

Large neighbourhood search applied to the efficient solution of spatially explicit strategic supply chain management problems

Supply chain management (SCM) has recently gained wider interest in both academia and industry given its potential to improve the benefits of a company through an integrated coordination of all its entities. Optimization problems in SCM are commonly cast as large scale mixed-integer linear programs (MILPs) that are hard to solve in short CPU times. This limitation is critical in spatially explicit SCM models since they require a large number of discrete variables to represent the geographical configuration of the network, which leads to complex MILPs. We present herein a novel solution method for this type of problems that combines the strengths of standard branch and cut techniques with the efficiency of large neighbourhood search (LNS). We illustrate the capabilities of this novel approach through its application to two case studies arising in energy applications: the design of supply chains (SCs) for bioethanol production and the strategic planning of hydrogen infrastructures for vehicle use.

A Memetic Algorithm for Global Optimization in Chemical Process Synthesis Problems

Design optimization problems in chemical engineering and in many other engineering domains are characterized by the presence of a large number of discrete and continuous decision variables, complex nonlinear models that restrict the search space, nonlinear cost functions, and the presence of many local optima. The classical approach to such problems are mixed integer nonlinear program solvers that work on a superstructure formulation which explicitly represents all design alternatives. The structural decisions lead to a large number of discrete variables and an exponential increase in the computational effort. The mathematical programming (MP) methods which are usually employed to solve the continuous subproblems that arise by fixing the discrete variables provide only one local optimum which depends strongly on the initialization. Thus standard methods may not find the global optimum despite long computation times. In this contribution we introduce a memetic algorithm (MA) for the global optimization of a computational demanding real-world design problem from the chemical engineering domain. The MA overcomes the problem of getting stuck in local optima by the use of an evolution strategy (ES) which addresses the global optimization of the design decisions, whereas a robust MP solver is used to handle complex nonlinear constraints as well as to improve the individuals of the ES by performing a local search in continuous sub-spaces in an integrated fashion. The MA is discussed in detail, the novel decomposition of the problem class at hand is analyzed and the MA is tested for the example of the optimal design of a reactive distillation column with several thousand decision variables. The MA is the only algorithm that finds the global solution in reasonable computation times. The introduction of structural decisions and additional constraints and discontinuous penalty terms lead only to a moderate increase in the computational effort which demonstrates the potential of this class of memetic algorithms in real-world design optimization problems.

Imaging Mass Spectrometry Data Reduction: Automated Feature Identification and Extraction

Imaging MS now enables the parallel analysis of hundreds of biomolecules, spanning multiple molecular classes, which allows tissues to be described by their molecular content and distribution. When combined with advanced data analysis routines, tissues can be analyzed and classified based solely on their molecular content. Such molecular histology techniques have been used to distinguish regions with differential molecular signatures that could not be distinguished using established histologic tools. However, its potential to provide an independent, complementary analysis of clinical tissues has been limited by the very large file sizes and large number of discrete variables associated with imaging MS experiments. Here we demonstrate data reduction tools, based on automated feature identification and extraction, for peptide, protein, and lipid imaging MS, using multiple imaging MS technologies, that reduce data loads and the number of variables by >100×, and that highlight highly-localized features that can be missed using standard data analysis strategies. It is then demonstrated how these capabilities enable multivariate analysis on large imaging MS datasets spanning multiple tissues.