Rosenbrock Function Research Articles

Immune Algorithm with Rearrangement

The mammalian immune system is a subject of great research interest because of its powerful information processing capabilities, namely, adaptivity. The adaptivity of the immune system is characterized by mainly two aspects, responsibility and diversity. The responsibility is a result of the response network of the immune system and the diversity is arise from gene rearrengement of the immune system. Recentry many artificial immune algorithms were devised by inspiring the adaptivity of the immune system. In terms of the two aspects of the immune system, however, those artificial immune algorithms only utilize the responsibility using models of response and regulation networks in the immune system. This paper proposes a new scheme of artificial immune algorithm, called Rearrangement Immune Algorithm (RIA), in which the rearrengement of the immune system is utilized combining evolution of the gelmline with an optimization of genetic algorithms. We empirically shows the effectiveness of our new scheme, RIA, with applying the Rosenbrock function and an HP folding problem.

Identification of strategy parameters for particle swarm optimizer through Taguchi method

Particle swarm optimization (PSO), like other evolutionary algorithms is a population-based stochastic algorithm inspired from the metaphor of social interaction in birds, insects, wasps, etc. It has been used for finding promising solutions in complex search space through the interaction of particles in a swarm. It is a well recognized fact that the performance of evolutionary algorithms to a great extent depends on the choice of appropriate strategy/operating parameters like population size, crossover rate, mutation rate, crossover operator, etc. Generally, these parameters are selected through hit and trial process, which is very unsystematic and requires rigorous experimentation. This paper proposes a systematic based on Taguchi method reasoning scheme for rapidly identifying the strategy parameters for the PSO algorithm. The Taguchi method is a robust design approach using fractional factorial design to study a large number of parameters with small number of experiments. Computer simulations have been performed on two benchmark functions—Rosenbrock function and Griewank function—to validate the approach.

A Note on the Extended Rosenbrock Function

The Rosenbrock function is a well-known benchmark for numerical optimization problems, which is frequently used to assess the performance of Evolutionary Algorithms. The classical Rosenbrock function, which is a two-dimensional unimodal function, has been extended to higher dimensions in recent years. Many researchers take the high-dimensional Rosenbrock function as a unimodal function by instinct. In 2001 and 2002, Hansen and Deb found that the Rosenbrock function is not a unimodal function for higher dimensions although no theoretical analysis was provided. This paper shows that the n-dimensional (n = 4 approximately 30) Rosenbrock function has 2 minima, and analysis is proposed to verify this. The local minima in some cases are presented. In addition, this paper demonstrates that one of the "local minima" for the 20-variable Rosenbrock function found by Deb might not in fact be a local minimum.

A Note on the Extended Rosenbrock Function

The Rosenbrock function is a well-known benchmark for numerical optimization problems, which is frequently used to assess the performance of Evolutionary Algorithms. The classical Rosenbrock function, which is a two-dimensional unimodal function, has been extended to higher dimensions in recent years. Many researchers take the high-dimensional Rosenbrock function as a unimodal function by instinct. In 2001 and 2002, Hansen and Deb found that the Rosenbrock function is not a unimodal function for higher dimensions although no theoretical analysis was provided. This paper shows that the n-dimensional (n = 4 approximately 30) Rosenbrock function has 2 minima, and analysis is proposed to verify this. The local minima in some cases are presented. In addition, this paper demonstrates that one of the "local minima" for the 20-variable Rosenbrock function found by Deb might not in fact be a local minimum.

Saving MGG: Reducing Fitness Evaluations for Real-coded GA/MGG.

In this paper, we propose an extension of the Minimal Generation Gap (MGG) to reduce the number of fitness evaluation for the real-coded GAs (RCGA). When MGG is applied to actual engineering problems, for example applied to optimization of design parameters, the fitness calculating time is usually huge because MGG generates many children from one pair of parents and the fitness is calculated by repetitive simulation or analysis. The proposed method called Saving MGG reduces the number of fitness evaluation by estimating the promising degrees of children using individual distribution and fitness information of population, and selecting children based on the promising degree before evaluating the fitness. Experimental results show that RCGA with Saving MGG can provide large reducing effects on 20 or 30 dimensional Sphere functions, Rosenbrock functions, ill-scaled Rosenbrock functions, and Rastrigin function.

EM-based optimization exploiting partial space mapping and exact sensitivities

We present a family of robust techniques for exploiting sensitivities in electromagnetic (EM)-based circuit optimization through space mapping (SM) technology. We utilize derivative information for parameter extractions and mapping updates. We exploit a partial SM (PSM) concept, where a reduced set of parameters is sufficient for parameter extraction optimization. It reflects the idea of tuning and execution time is reduced. Upfront gradients of both EM (fine) model and coarse surrogates can initialize possible mapping approximations. We introduce several effective approaches for updating the mapping during the optimization iterations. Examples include the classical Rosenbrock function, modified to illustrate the approach, a two-section transmission-line 10:1 impedance transformer and a microstrip bandstop filter with open stubs.

Probabilistic Modeling of Errors from Structural Optimization Based on Multiple Starting Points

With optimization increasingly used in engineering applications, a series of optimization runs may be required, and it may be too expensive to converge them to very high accuracy. A procedure for estimating average optimization convergence errors from a set of poorly converged optimization runs is developed. A probabilistic model is fitted to the errors in optimal objective function values of poorly converged runs. The Weibull distribution was identified as a reasonable error model both for the Rosenbrock function problem and the structural optimization of a high speed civil transport. Once a statistical model for the error is identified, it can be used to estimate average errors from a set of pairs of runs. In particular, by performing pairs of optimization runs from two starting points, accurate estimates of the mean and standard deviation of the convergence errors can be obtained.

Detection and Repair of Poorly Converged Optimization Runs

The use of iteratively reweighted least squares (IRLS) for detecting design points where optimizations converge poorly is demonstrated. Because optimization error tends to he one-sided with poor results producing overweight designs, a nonsymmetrical version of IRLS (NIRLS) that accounts for the asymmetry in optimization errors is also developed. A parameterized Rosenbrock function problem, where a programming error caused poor optimizations, is used to demonstrate the techniques. The identified outliers were repaired by more accurate optimizations to improve response surface approximations. Structural optimization of a high-speed civil transport (HSCT) produced inaccurate wing structural weight. Optimization studies with various sets of convergence criteria showed that the optimization errors were due to incomplete convergence. The IRLS technique could identify many of the points with very large optimization errors, but the NIRLS techniques were much more reliable in this task. However, results of HSCT configuration optimizations using wing weight response surface models show that it may be preferable to identify and repair only a subset of the poorly converged optima.

Searching minima of an N-dimensional surface: A robust valley following method

A procedure is proposed to follow the “minimum path” of a hypersurface starting anywhere in the catchment region of the corresponding minimum. The method uses a modification of the so-called “following the reduced gradient” [1]. The original method connects points where the gradient has a constant direction. In the present letter, this is replaced by the successive directions of the tangent of the searched curve. The resulting pathway is that valley floor gradient extremal which belongs to the smallest (absolute) eigenvalue of the Hessian. The new method avoids third derivatives of the objective function. The effectiveness of the algorithm is demonstrated by using a polynomial test, the notorious Rosenbrock function in two, 20, and in 100 dimensions.

Distributed evolutionary optimization, in Manifold: Rosenbrock's function case study

A competitive coevolutionary approach using loosely coupled genetic algorithms is proposed for a distributed optimization of Rosenbrock's function. The computational scheme is a coevolutionary system of agents with only local interaction among them, without any central synchronization. We use a recently developed coordination language, called Manifold, to implement our distributed optimization algorithm. We show that this implementation outperforms a sequential optimization algorithm based on standard genetic algorithms.

Non-linear multiparameter inversion using a hybrid global search algorithm: applications in reflection seismology

SUMMARY Many interesting inverse problems in geophysics are non-linear and multimodal. Parametrization of these problems leads to an objective function, or measure of agreement between data and model predictions, that has a complex topography with many local minima. Optimization algorithms that rely on local gradients in the objective function or that search the model space locally may become trapped in these local minima. By combining simulated annealing with the downhill simplex method, a hybrid global search algorithm is presented in this paper for non-linear, multimodal, inverse problems. The hybrid algorithm shares the advantages of both local search methods that perform well if the local model is suitable, and global methods that are able to explore efficiently the full model space. The hybrid algorithm also utilizes a larger and more complex memory to store information on the objective function than simulated annealing algorithms. The effectiveness of this new scheme is evaluated in three problems: minimization of the multidimensional Rosenbrock function, non-linear, 1 -D, acoustic waveform inversion, and residual statics. The performance of the hybrid algorithm is compared with simulated annealing and genetic algorithms and is shown to converge more rapidly and to have a higher success rate of locating the global minimum for the cases investigated.

Modified quadratic hill-climbing with SAS/IML

This article describes EZClimb, a set of SAS/IML steps useful in solving numerical optimization problems. The program uses the method of modified quadratic hill-climbing with either analytical or numerical derivatives to maximize a user-defined criterion function. Modified quadratic hill-climbing is one of the more powerful algorithms known for function optimization but is not widely available outside of the software package GQOPT. The efficacy of the SAS steps is illustrated using Rosenbrock's function, Savin and White's Box-Cox extended autoregressive model, and Klein's Model.

Nonlinear programming on a microcomputer

AbstractThe Davidon‐Fletcher‐Powell (DFP) algorithm for multivariable unconstrained nonlinear function minimization is implemented on a desk microcomputer—the Hindustan Computers MICRO‐2200—with a floppy disk memory extension peripheral. The program is developed using the programming keys of the system. The program is closely based on FMFP, a well‐known FORTRAN subroutine of the algorithm. Among the exercises performed are two standard nonlinear programming (NLP) test problems, namely the Rosenbrock and Wood function minimization and two SUMT (Sequential Unconstrained Minimization Technique) illustration problems in structural optimization, i.e. two‐bar and three‐bar truss weight minimization examples. Experience bears out the potential of desk calculating systems as tools in nonlinear programming algorithm research. Also, the small‐scale structural optimization capability afforded by such systems is expected to be of significance in teaching and preliminary design contexts.