Structural optimization using the cross-entropy method

Multidisciplinary design optimization (MDO) was performed on a helicopter rotor blade. The blade was modeled as a rigid flapping blade for dynamics; Blade Element Theory (BET) was the analysis approach to model the aerodynamic loading, and a simple linearly elastic hollowed rectangular section was the main structural component. MATLAB was used to solve the flapping differential equations and its Sequential Quadratic Programming (SQP) and Genetic Algorithm (GA) were used for the optimization. A Particle Swarm Optimization (PSO) routine was also tested. The optimization process consisted of three cases. The first case was a simple cantilever beam under centrifugal and an assumed bending loads. The optimization was performed using the SQP, GA, and PSO algorithms. The SQP resulted in the superior design with 75.45 compared to the GA's 87.1 and the PSO's 79.2, but a local minimum was present. The second case was an expansion of the first case by turning it into multidisciplinary problem. Aerodynamics was included in the design variables and objective function. Only the SQP algorithm was used and there was a reduction in hub vertical shear by 33.6%. The blade mass increased by 36.84%. The last case was an improvement to the second by creating a multiobjective problem by including the hub radial shear and the results were improved significantly by reducing the hub vertical shear by 34.06% and radial shear by 17.87% with a reduction of blade mass by 23.86%.

In order to increase productivity and reduce energy consumption of steelmaking-continuous casting (SCC) production process, especially with complicated technological routes, the cross entropy (CE) method was adopted to optimize the SCC production scheduling (SCCPS) problem. Based on the CE method, a matrix encoding scheme was proposed and a backward decoding method was used to generate a reasonable schedule. To describe the distribution of the solution space, a probability distribution model was built and used to generate individuals. In addition, the probability updating mechanism of the probability distribution model was proposed which helps to find the optimal individual gradually. Because of the poor stability and premature convergence of the standard cross entropy (SCE) algorithm, the improved cross entropy (ICE) algorithm was proposed with the following improvements: individual generation mechanism combined with heuristic rules, retention mechanism of the optimal individual, local search mechanism and dynamic parameters of the algorithm. Simulation experiments validate that the CE method is effective in solving the SCCPS problem with complicated technological routes and the ICE algorithm proposed has superior performance to the SCE algorithm and the genetic algorithm (GA).

New optimisation algorithm for planar antenna array synthesis

With the increase of the number of IP cores integrated in SOC, the functions of SOC ar e becoming more complexed. Test time and test cost grow rapidly, therefore it becomes bo ttleneck of SOC test. Test scheduling is one of the efficient approaches to solve the forenamed ques tion. Cross-entropy method, which is ba sed on probability density function, has been used to solve the SOC test scheduling problem. Experimental results on ITC’02 benchmarks show that the proposed method provides better test time results compared to the Linear-Programming. Keywords: System on chip, test scheduling, cross-entropy method 1. INTRODUCTION In order to shorten the time to market, core based system-on-chip (SOC) design has becomed a widely used technique of integrated circuit design. Compared with system-on-board, work frequency of SOC has improved observably because signal transfers delay is avoid. However, the testing of SOC and the reusable cores becomes a bottleneck of the SOC development. The SOC test includes test information exchange mechanism, test wrapper generation, test access mechanism (TAM) design, test controller design, test schedulin g and test integration, etc. Among the SOC test issues, the test scheduling is one of the most important challenges. Many works have forced on core test scheduling to minimize the overall testing time, in [7], the test scheduling is mapped to the linear programming (LP) problem, but the computation time is too long. In this paper, we use a cross-entropy (CE) method, which is based on probability density function, to solve the test scheduling problem. The paper is organized as follows. In section II, the test scheduling problem is briefly described. We formulate the problem of test scheduling using CE method in section III. Experimental results are given in section IV, and section V concludes the paper.

Project scheduling is essential to be planned before activities. Standard methods of project scheduling based on precedence constraints scheduling events and the assumption that resources are not unlimited capacity. But in fact the project activity has resource limits. The main objective is to minimize the total duration of the project with precedence constraints and resource constraints for all project activities. Many optimization methods are used to improve the quality of scheduling and speed up the calculation time. This paper proposed the use of Cross Entropy (CE) method to solve resource constraints project scheduling problem, as well as comparing the advantages and disadvantages between the CE methods with Differential Evolution (DE) method. The purpose CE method consist of four critical steps including generating a sample of random solution, calculating the performance according to the specific fitness function, selecting elite sample and finally updating the previous parameters to get a better sample in the next iteration. To speed up the computation time, in this study decrease the number of samples for each iteration in Cross Entropy algorithm. Numerical experiments with several data sets from the Project Scheduling library (PSPLIB) showed that CE can provide the optimal total duration of the project same as the DE with calculation time is faster than DE.

This paper discusses an alternative approach to parameter optimization of prototype-based learning algorithms that aim to minimize an objective function based on gradient search. The proposed approach is a stochastic optimization method called the Cross Entropy (CE) method. The CE method is used to tackle the initialization sensitiveness problem associated with the original generalized Learning Vector Quantization (GLVQ) algorithm and its variants and to locate the globally optimal solutions. We will focus our study on a variant which deals with a weighted norm instead of the Euclidean norm in order to select the most relevant features. The results in this paper indicate that the CE method can successfully be applied to this kind of problems and efficiently generate high quality solutions. Also, highly competitive numerical results on real world data sets are reported.KeywordsDirichlet DistributionHigh Quality SolutionLearn Vector QuantizationCross EntropyNatural Exponential Function FamilyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this dissertation we present several papers about two optimization methods that are derived from rare-event simulation techniques, i.e., the splitting method and the cross-entropy (CE) method. The splitting method is a well-known method for rare-event simulation, where sample paths of a Markov process are split into multiple copies during the simulation, so as to make the occurrence of a rare event more frequent. Motivated by the splitting algorithm we introduce a novel global optimization method for continuous optimization (SCO) that is both very fast and accurate. We also introduce a new multi-objective optimization (MOO) methodology based the SCO algorithm. The method, called MOS, generalizes the elite set selection of the traditional splitting framework, and uses both local and global sampling to sample in the decision space. Both SCO and MOS were compared with state-of-the art algorithms using a prevailing set of benchmark problems. Numerical experiments demonstrate that the new splitting methods are competitive with famous methods in the fields. The thesis also contains two new applications of the CE method on Laguerre fitting problems. The Laguerre fitting problems are about recovering the weighted generator points from a given Laguerre tessellation. When the description of the Laguerre tessellation is exact, the solutions are not unique and different weighted generator points can create the same tessellation. To recover pertinent generator points we formulate the problem as an optimization problem and apply the modified CE method to solve it. However, the representation of a Laguerre tessellation is typically not exact when working with real data, such as tomographic image data. In this case, we formulate an optimization problem where we minimize the discrepancy between the image data and our approximated tessellations. We then solve the optimization problem using the CE method. Last, we introduce a new R package that implements the general CE methodology for optimization and well as some useful modifications. The usage and efficacy of CEoptim is demonstrated through a variety of optimization examples, including model fitting, combinatorial optimization, and maximum likelihood estimation.

With large-scale wind farms integrated into power systems, the computational requirements for reliability evaluation of composite power systems are increasing. This paper introduces a new concept of dynamic orthogonal list, which is a form of multiple failure retrievals to avoid repeatedly calling the optimal power flow (OPF) program in the stage of state evaluation. An existing algorithm of cross-entropy (CE) method was developed to deal with the rare events in sampling. The major contribution of CE is to achieve convergence faster and reduce the number of samples, thus shortening the consuming time in the stage of sampling. This paper proposes a fast reliability evaluation process which combines the CE method with dynamic orthogonal list (CE-DOL), comprehensively improving the computational efficiency of Monte Carlo simulation (MCS). The proposed method is tested on a modified IEEE-RTS 79 system. In addition, for comparison purposes, the efficiencies of MCS, CE, and CE-DOL are estimated in the cases of different system adequacies. It is shown that, in range of certain reliability level, the proposed method is much more efficient compared with the CE method. Finally, the applicability of different methods is analyzed under different system reliability levels.

The sequential quadratic programming and genetic algorithms are finely tuned on the optimal design of a brushless DC wheel motor. Combining both methods, a hybrid approach is tested. Multiobjective optimizations are performed using Nondominated Sorting Genetic Algorithm, Strength Pareto Evolutionary Algorithm, and a variable weighted sum of objectives with the sequential quadratic programming. The Pareto fronts are compared in term of accuracy, uniformity, and coverage criteria using some quality indicators.

The logistics of moving grain from silos to ports is constrained by the number of trains available and the capacity of loading and unloading facilities. We aim to schedule the available trains to move grain from silos to the port as quickly as possible. The sequence of trips that minimises the time required to complete all trips is a permutation of a basic sequence that has the required number of trips to each silo. We use the Cross Entropy Optimisation method to search for a permutation that minimises span. For small problems, where the optimal solution can be found by enumeration, the Cross Entropy Optimisation method achieves solutions within 5% of the optimum. It can also find good solutions for large problems. References Unveren A. and A. Acan. Multi-objective optimisation with cross entropy method: Stochastic learning with clustered Pareto fronts. IEEE Congress on Evolutionary Computations, pages 3065--3071, 2007. Boer P. T., Kroese D. P., Mannor S., and Rubinstein R. Y. A tutorial on the cross-entropy method. http://www.springerlink.com/index/KPW596202975755N.pdf. Rubinstein R. Y. and D. P. Kroese. The cross entropy method: A unified approach to combinatorial optimization, monte-carlo simulation, and machine learning. Springer--Verlag, New York, 2004.

The current paper considers the joint precoding and transmit antenna selection to reduce hardware cost, such as radio-frequency chains, associated with antennas in the downlink of multiuser multiple-input multiple-output systems with limited feedback. The joint precoding and transmit antenna selection algorithm requires an exhaustive search of all possible combinations and permutations to find the optimum solution at the transmitter, thus resulting in extremely high computational complexity. To reduce the computational load while still maximizing channel capacity, the cross-entropy (CE) method is adopted to determine the suboptimum solution. Compared with the conventional genetic algorithm and random search method, the CE method provides better performance under the same computational complexity, as shown by the simulation results.

An optimization procedure is developed for the design of structural components under crashworthiness requirements. The optimization procedure consists of first developing a system of parallel neural networks able to reproduce the structural behavior during the crash phenomena. Training and test sets are generated by finite element analyses. Then, to find the optimal configuration, both sequential quadratic programming and genetic algorithms are implemented to evaluate the objective and constraint functions using the response surfaces generated by the neural networks. The procedure is applied for the optimization of helicopter subfloor components. In this application, the objective function is taken equal to the sum of two terms: the crush force efficiency and the specific absorbed energy. The values of maximum and mean forces are constrained to meet crashworthiness requirements, whereas other constraints are applied to define the feasibility of the structural solution. The optimal configuration, obtained measuring orders of magnitude savings in the CPU time, allows an increase of the crush force efficiency equal to 18% and a decrease of the mass equal to 8%.

The idea of multidisciplinary design optimization (MDO) is applied to synthesize optimum robots. The main idea and frame of collaborative optimization are introduced. Three disciplines of working stroke L driving performance and hydraulic component are involved in the design optimization of a 3-RRS robot. The optimization model of system-level and optimization model of subsystem are established. Sequential quadratic programming (SQP) and genetic algorithm (GA) are respectively used to execute the optimization of system-level and the optimization of subsystems using popular commercialized MATLAB. The numerical results indicate that the collaborative optimization can be successfully applied to dealing with the complex robot system, and lay a foundation to solve more complex mechanical system. (5 pages)

In this article, the artificial intelligence techniques have been used for the solution of Falkner–Skan (FS) equations based on neural networks optimized with three methods including active set technique, sequential quadratic programming and genetic algorithms (GA) hybridization. Log-sigmoid activation function is used in artificial neural network architecture. The proposed techniques are applied to a number of cases for Falkner–Skan problems, and results were compared with GA hybrid results in all cases and were found accurate. The level of accuracy is examined through statistical analyses based on a sufficiently large number of independent runs.

The authors present a general framework of the construction of biorthogonal wavelets based on Bernstein bases along with theory analysis and application. The presented framework possesses the largest possible regularity, the required vanishing moments and the passband flatness of frequency response of filters. Based on this concept, the authors establish explicit formulas for filters of biorthogonal wavelets with arbitrary odd lengths. Meanwhile, a new family of parametric filters with symmetry is constructed. The choice of filter bank in wavelet compression is a critical issue that affects image quality. In this study, an optimal model of FIR aiming at image compression is brought forward, and the optimal finite impulse response (FIR) filters can be obtained correspondingly through sequential quadratic programming (SQP) and genetic algorithm (GA). The authors demonstrate the performance of the new family of filters given in this study for image compression with very encouraging results.