Large Optimization Problems Research Articles

A discrete mechanics approach to dislocation dynamics in BCC crystals

A discrete mechanics approach to modeling the dynamics of dislocations in BCC single crystals is presented. Ideas are borrowed from discrete differential calculus and algebraic topology and suitably adapted to crystal lattices. In particular, the extension of a crystal lattice to a CW complex allows for convenient manipulation of forms and fields defined over the crystal. Dislocations are treated within the theory as energy-minimizing structures that lead to locally lattice-invariant but globally incompatible eigendeformations. The discrete nature of the theory eliminates the need for regularization of the core singularity and inherently allows for dislocation reactions and complicated topological transitions. The quantization of slip to integer multiples of the Burgers’ vector leads to a large integer optimization problem. A novel approach to solving this NP-hard problem based on considerations of metastability is proposed. A numerical example that applies the method to study the emanation of dislocation loops from a point source of dilatation in a large BCC crystal is presented. The structure and energetics of BCC screw dislocation cores, as obtained via the present formulation, are also considered and shown to be in good agreement with available atomistic studies. The method thus provides a realistic avenue for mesoscale simulations of dislocation based crystal plasticity with fully atomistic resolution.

Diversity-adaptive parallel memetic algorithm for solving large scale combinatorial optimization problems

Parallel memetic algorithms (PMAs) are a class of modern parallel meta-heuristics that combine evolutionary algorithms, local search, parallel and distributed computing technologies for global optimization. Recent studies on PMAs for large-scale complex combinatorial optimization problems have shown that they converge to high quality solutions significantly faster than canonical GAs and MAs. However, the use of local learning for every individual throughout the PMA search can be a very computationally intensive and inefficient process. This paper presents a study on two diversity-adaptive strategies, i.e., (1) diversity-based static adaptive strategy (PMA-SLS) and (2) diversity-based dynamic adaptive strategy (PMA-DLS) for controlling the local search frequency in the PMA search. Empirical study on a class of NP-hard combinatorial optimization problem, particularly large-scale quadratic assignment problems (QAPs) shows that the diversity-adaptive PMA converges to competitive solutions at significantly lower computational cost when compared to the canonical MA and PMA. Furthermore, it is found that the diversity-based dynamic adaptation strategy displays better robustness in terms of solution quality across the class of QAP problems considered. Static adaptation strategy on the other hand requires extra effort in selecting suitable parameters to suit the problems in hand.

On the convergence of partitioning group correction algorithms

This paper studies a successive partitioning group correction algorithm and its some modified algorithms for solving large scale sparse unconstrained optimization problems. The methods depend on a symmetric consistent partition of the columns of the Hessian matrix. A q-superlinear convergence result and an r-convergence rate estimate show that the methods have good local convergence properties. The numerical results show that the methods, especially the modified algorithms, may be competitive with some current used algorithms.

Selecting a minimal number of relevant genes from microarray data to design accurate tissue classifiers

It is essential to select a minimal number of relevant genes from microarray data while maximizing classification accuracy for the development of inexpensive diagnostic tests. However, it is intractable to simultaneously optimize gene selection and classification accuracy that is a large parameter optimization problem. We propose an efficient evolutionary approach to gene selection from microarray data which can be combined with the optimal design of various multiclass classifiers. The proposed method (named GeneSelect) consists of three parts which are fully cooperated: an efficient encoding scheme of candidate solutions, a generalized fitness function, and an intelligent genetic algorithm (IGA). An existing hybrid approach based on genetic algorithm and maximum likelihood classification (GA/MLHD) is proposed to select a small number of relevant genes for accurate classification of samples. To evaluate the performance of GeneSelect, the gene selection is combined with the same maximum likelihood classification (named IGA/MLHD) for convenient comparisons. The performance of IGA/MLHD is applied to 11 cancer-related human gene expression datasets. The simulation results show that IGA/MLHD is superior to GA/MLHD in terms of the number of selected genes, classification accuracy, and robustness of selected genes and accuracy.

Locally adapted hierarchical basis preconditioning

This paper develops locally adapted hierarchical basis functions for effectively preconditioning large optimization problems that arise in computer graphics applications such as tone mapping, gradient-domain blending, colorization, and scattered data interpolation. By looking at the local structure of the coefficient matrix and performing a recursive set of variable eliminations, combined with a simplification of the resulting coarse level problems, we obtain bases better suited for problems with inhomogeneous (spatially varying) data, smoothness, and boundary constraints. Our approach removes the need to heuristically adjust the optimal number of preconditioning levels, significantly outperforms previously proposed approaches, and also maps cleanly onto data-parallel architectures such as modern GPUs.

Partitioning group correction Cholesky techniques for large scale sparse unconstrained optimization

This paper studies a successive partitioning group correction Cholesky algorithm and its modified algorithm for solving large scale sparse unconstrained optimization problems. These methods employ an initial Cholesky factorization of their approximation Hessians and then correct the partitioning group(s) of their diagonal factor and lower triangular factor directly and successively at each step. Iterations are generated using forward and backward substitution employing the update factorizations. A self-correcting property, a q-superlinear convergence result and an r-convergence rate estimate show that the two methods both have good local convergence properties. The numerical results show that the two methods, especially the modified algorithm may be competitive with some current used algorithms.

Grid computing for parallel bioinspired algorithms

This paper focuses on solving large size combinatorial optimization problems using a Grid-enabled framework called ParadisEO–CMW ( Para ̲ llel and Dis ̲ tributed EO ̲ on top on C ̲ ondor and the M ̲ aster W ̲ orker Framework). The latter is an extension of ParadisEO, an open source framework originally intended to the design and deployment of parallel hybrid meta-heuristics on dedicated clusters and networks of workstations. Relying on the Condor–MW framework, it enables the execution of these applications on volatile heterogeneous computational pools of resources. The motivations, architecture and main features will be discussed. The framework has been experimented on a real-world problem: feature selection in near-infrared spectroscopic data mining. It has been solved by deploying a multi-level parallel model of evolutionary algorithms. Experimentations have been carried out on more than 100 PCs originally intended for education. The obtained results are convincing, both in terms of flexibility and easiness at implementation, and in terms of efficiency, quality and robustness of the provided solutions at run time.

Application of cultural algorithm to generation scheduling of hydrothermal systems

The daily generation scheduling of hydrothermal power systems plays an important role in the operation of electric power systems for economics and security, which is a large scale dynamic non-linear constrained optimization problem. It is difficult to solve using traditional optimization methods. This paper proposes a new cultural algorithm to solve the optimal daily generation scheduling of hydrothermal power systems. The approach takes the water transport delay time between connected reservoirs into consideration and can conveniently deal with the complicated hydraulic coupling simultaneously. An example is used to verify the correctness and effectiveness of the proposed cultural algorithm, comparing with both the Lagrange method and the genetic algorithm method. The simulation results demonstrate that the proposed algorithm has rapid convergence speed and higher solution precision. Thus, an effective method is provided to solve the optimal daily generation scheduling of hydrothermal systems.

Routing for the Pennsylvania Maglev system by a genetic algorithm

The scheduling of vehicle movement activities is an important component in highspeed railway transit operations planning process. This paper presents a mathematical programming approach to vehicle routing for the Pennsylvania Transrapid Maglev Train System. A mathematical model of the routing procedure is first derived. Based on this model, the vehicle routing problem is formulated into a constrained optimization problem, in which the train miles travelled are maximized subject to various operational and safety requirements. This large scale non-linear optimization problem is then solved by a genetic algorithm.

Hybrid-GAを用いた二段式スペースプレーンの最適設計

This paper proposes a Hybrid Genetic Algorithm (Hybrid-GA) which is suitable for a large scale multidisciplinary optimization problem such as Two-Stage-To-Orbit (TSTO) spaceplane optimal design problem. The Hybrid-GA is implemented by combining Sequential Quadratic Programming (SQP) method with GA. When constructing the Hybrid-GA, there are three problems that should be solved; 1) decision of optimized variables and discrete method, 2) how to use results of local search, and 3) selection for survival method. These problems are discussed and solved in order to effectively combine SQP method with GA. In order to demonstrate the effectiveness of the proposed Hybrid-GA, the TSTO spaceplane optimal design problem, which consists of weight, aerodynamics, propulsion, and flight trajectory analyses, is investigated using the proposed Hybrid-GA, and the optimal results of Hybrid-GA are compared with that of Simple-GA, and SQP methods. Finally, strategy to achieve the TSTO spaceplane is proposed by comparing the optimal results of changing payload weight and maximum wing load factor of booster.

Ridge regression based hybrid genetic algorithms for multi-locus quantitative trait mapping

Genetic algorithms (GAs) are increasingly used in large and complex optimisation problems. Here we use GAs to optimise fitness functions related to ridge regression, which is a classical statistica...

Lower bound limit analysis of cohesive‐frictional materials using second‐order cone programming

AbstractThe formulation of limit analysis by means of the finite element method leads to an optimization problem with a large number of variables and constraints. Here we present a method for obtaining strict lower bound solutions using second‐order cone programming (SOCP), for which efficient primal‐dual interior‐point algorithms have recently been developed. Following a review of previous work, we provide a brief introduction to SOCP and describe how lower bound limit analysis can be formulated in this way. Some methods for exploiting the data structure of the problem are also described, including an efficient strategy for detecting and removing linearly dependent constraints at the assembly stage. The benefits of employing SOCP are then illustrated with numerical examples. Through the use of an effective algorithm/software, very large optimization problems with up to 700 000 variables are solved in minutes on a desktop machine. The numerical examples concern plane strain conditions and the Mohr–Coulomb criterion, however we show that SOCP can also be applied to any other problem of lower bound limit analysis involving a yield function with a conic quadratic form (notable examples being the Drucker–Prager criterion in 2D or 3D, and Nielsen's criterion for plates). Copyright © 2005 John Wiley & Sons, Ltd.

Dynamic scaling based preconditioning for truncated Newton methods in large scale unconstrained optimization

This paper deals with the preconditioning of truncated Newton methods for the solution of large scale nonlinear unconstrained optimization problems. We focus on preconditioners which can be naturally embedded in the framework of truncated Newton methods, i.e. which can be built without storing the Hessian matrix of the function to be minimized, but only based upon information on the Hessian obtained by the product of the Hessian matrix times a vector. In particular we propose a diagonal preconditioning which enjoys this feature and which enables us to examine the effect of diagonal scaling on truncated Newton methods. In fact, this new preconditioner carries out a scaling strategy and it is based on the concept of equilibration of the data in linear systems of equations. An extensive numerical testing has been performed showing that the diagonal preconditioning strategy proposed is very effective. In fact, on most problems considered, the resulting diagonal preconditioned truncated Newton method performs better than both the unpreconditioned method and the one using an automatic preconditioner based on limited memory quasi-Newton updating (PREQN) recently proposed by Morales and Nocedal [Morales, J.L. and Nocedal, J., 2000, Automatic preconditioning by limited memory quasi-Newton updating. SIAM Journal on Optimization, 10, 1079–1096].

The Substitution Secant/Finite Difference Method for Large Scale Sparse Unconstrained Optimization

This paper studies a substitution secant/finite difference (SSFD) method for solving large scale sparse unconstrained optimization problems. This method is a combination of a secant method and a finite difference method, which depends on a consistent partition of the columns of the lower triangular part of the Hessian matrix. A q-superlinear convergence result and an r-convergence rate estimate show that this method has good local convergence properties. The numerical results show that this method may be competitive with some currently used algorithms.

Solving Large-Scale Fuzzy and Possibilistic Optimization Problems

The semantic and algorithmic differences between fuzzy and possibilistic optimization methods are presented in the context of three methods for solving large fuzzy and possibilistic optimization problems. In particular, an optimization problem in radiation therapy with various orders of complexity, 1,000-55,000 constraints, possessing (i) soft constraints, (ii) fuzzy right-hand side values and (iii) possibilistic right-hand side values, are used to illustrate the semantics and to test the performance of the three fuzzy and possibilistic optimization methods. We focus on the uncertainty in the right side which arises, in the context of the radiation therapy problem, from the fact that minimal/maximal radiation tolerances are target values rather than fixed real numbers. The results indicate that fuzzy/possibilistic optimization is a natural way to model various types of optimization under uncertainty problems and large optimization problems can be solved efficiently.

Optimisation enhancement using self‐organising fuzzy control

Purpose – To show the successful use of self‐organising fuzzy control in enhancing dynamic optimisation, a controller is used to direct the type of optimisation appropriate in each new dynamic problem. The system uses its experiences to determine which approach is most suitable under varying circumstances.Design/methodology/approach – A knowledge extraction tool is used to gain basic information about the solution space with a simple computation. This information is compared with the fuzzy rules stored in the system. These rules hold a collection of facts on previous successes and failures, which were acquired through the performance monitor. Using this system the controller directs the algorithms, deciphering the most appropriate strategy for the current problem.Research limitations/implications – This procedure is designed for large scale dynamic optimisation problems, where a portion of the computational time is sacrificed to allow the controller to direct the best possible solution strategy. The resul...

Signal reconstruction in sensor arrays using sparse representations

We propose a technique of multisensor signal reconstruction based on the assumption, that source signals are spatially sparse, as well as have sparse representation in a chosen dictionary in time domain. This leads to a large scale convex optimization problem, which involves combined l1-l2 norm minimization. The optimization is carried by the truncated Newton method, using preconditioned conjugate gradients in inner iterations. The byproduct of reconstruction is the estimation of source locations.

Novel orthogonal simulated annealing with fractional factorial analysis to solve global optimization problems

A classical simulated annealing (SA) method is a generic probabilistic and heuristic approach to solving global optimization problems. It uses a stochastic process based on probability, rather than a deterministic procedure, to seek the minima or maxima in the solution space. Although the classical SA method can find the optimal solution to most linear and nonlinear optimization problems, the algorithm always requires numerous numerical iterations to yield a good solution. The method also usually fails to achieve optimal solutions to large parameter optimization problems. This study incorporates well-known fractional factorial analysis, which involves several factorial experiments based on orthogonal tables to extract intelligently the best combination of factors, with the classical SA to enhance the numerical convergence and optimal solution. The novel combination of the classical SA and fractional factorial analysis is termed the orthogonal SA herein. This study also introduces a dynamic penalty function to handle constrained optimization problems. The performance of the proposed orthogonal SA method is evaluated by computing several representative global optimization problems such as multi-modal functions, noise-corrupted data fitting, nonlinear dynamic control, and large parameter optimization problems. The numerical results show that the proposed orthogonal SA method markedly outperforms the classical SA in solving global optimization problems with linear or nonlinear objective functions. Additionally, this study addressed two widely used nonlinear functions, proposed by Keane and Himmelblau to examine the effectiveness of the orthogonal SA method and the presented penalty function when applied to the constrained problems. Moreover, the orthogonal SA method is applied to two engineering optimization design problems, including the designs of a welded beam and a coil compression spring, to evaluate the capacity of the method for practical engineering design. The computational results show that the proposed orthogonal SA method is effective in determining the optimal design variables and the value of objective function.

Multistage Trajectory Optimization with Radar Range Constraints

The problem of finding an optimal aircraft trajectory for long-distance flights in three dimensions subject to radar detection constraints is considered. A general point-mass model previously developed is not suitable because the time discretization needs to be very fine to resolve the rigid-body dynamics resulting in very large optimization problems. Different reduced mathematical models are derived and compared to the more general performance model. Finally, a long-range mission involving two subsonic jet trainers approaching a radar target is numerically simulated. The results indicate that the time a hostile radar has an approaching aircraft under surveillance can be greatly reduced by using the proposed methodology.

The Million-Variable “March” for Stochastic Combinatorial Optimization

Combinatorial optimization problems have applications in a variety of sciences and engineering. In the presence of data uncertainty, these problems lead to stochastic combinatorial optimization problems which result in very large scale combinatorial optimization problems. In this paper, we report on the solution of some of the largest stochastic combinatorial optimization problems consisting of over a million binary variables. While the methodology is quite general, the specific application with which we conduct our experiments arises in stochastic server location problems. The main observation is that stochastic combinatorial optimization problems are comprised of loosely coupled subsystems. By taking advantage of the loosely coupled structure, we show that decomposition-coordination methods provide highly effective algorithms, and surpass the scalability of even the most efficiently implemented backtracking search algorithms.