Generalized Pattern Search Algorithm Research Articles

Optimization of a residential solar combisystem for minimum life cycle cost, energy use and exergy destroyed

This paper presents the optimization of a model of a solar combisystem in an energy efficient house in Montreal (Qc), Canada. A hybrid particle swarm and Hook–Jeeves generalized pattern search algorithm is used to minimize the life cycle cost, energy use and exergy destroyed of the combisystem. The results presented include four different optimal configurations depending on the objective function used. The optimizations were able to reduce, compared with the base case combisystem, the life cycle cost of the combisystem by 19%, the life cycle energy use by 34%, the life cycle exergy destroyed by 33% and 24% using the technical boundary and physical boundary, respectively. Due to the high cost of the solar collector technologies and the low price of electricity in Quebec, none of the optimal configurations have acceptable financial payback periods. However, they all have energy payback times between 5.8 and 6.6years. The use of technical boundary in the exergy analysis favors the use of electricity over solar energy due to the low exergy efficiency of the solar collectors. The use of the physical boundary, on the other hand, favors the use of solar energy over electricity, and all of the combisystem configurations have exergy payback times between 4.2 and 6.3years.

Optimization method based on Generalized Pattern Search Algorithm to identify bridge parameters indirectly by a passing vehicle

Abstract Generalized Pattern Search Algorithm (GPSA) has rarely been investigated for structural health monitoring, but may have potential application in civil engineering, because it does not require any gradient information of the objective function. Meanwhile, indirect identification is an attractive concept that recognizes the bridge parameters by the vehicle responses. This paper proposes a theoretical indirect identification method based on optimization method, and the implementation is performed by the GPSA. Firstly, the GPSA theory is investigated, and a simple example is employed to describe the process of the algorithm. Secondly, a theoretical indirect identification method is proposed, based on the optimization method rather than the conventional transforms from time domain to frequency domain. The proposed method can identify the parameters of the vehicle–bridge system, including the bridge stiffness and the 1st frequency. Based on the optimization method, the feasibility and accuracy of GPSA are demonstrated with 0.06% of errors. The GPSA shows good robustness in the identifications with various noise levels, and the maximum error is about 3.30% and can be accepted for the engineering application even with a SNR 5 noise level. The computation time relies only on the function evaluation times, and is not positively related to the noise level. Thirdly, the performance of GPSA is compared with that of Genetic Algorithm (GA). The accuracy of GPSA and GA are approximately equivalent with various noise levels. Compared with GA, GPSA needs fewer iterations and much fewer evaluations, therefore is more efficient in the identification with an almost consistent accuracy with various noise levels.

A subclass of generating set search with convergence to second-order stationary points

In generalized pattern search algorithms for linearly constrained optimization, certain limit points have been shown to satisfy a pseudo-second-order necessary condition, in which positive semidefiniteness of the Hessian is achieved only with respect to an orthonormal basis. Satisfaction of the full second-order condition can only be ensured if the finite set of positive spanning directions includes the eigenvectors of the Hessian at the limit point. Of course, this is generally not possible in practice, due to the unavailability of derivatives. The present work takes advantage of a chief feature of the more general parent class of generating set search algorithms – that of allowing the use of an infinite (non-dense) set of directions in the limit. If approximate second derivative information is gathered during the iteration process, then the normalized eigenvectors of the approximate Hessian can be used to construct the set of 2n orthonormal directions used by the algorithm. If the Hessian approximation converges in the limit to the true Hessian (or a matrix in the generalized Hessian), then the full second-order optimality conditions can be achieved. Two approaches for generating such a convergent approximate Hessian are described. Numerical examples show the effectiveness of the approach.

New horizons in sphere-packing theory, part II: lattice-based derivative-free optimization via global surrogates

Derivative-free algorithms are frequently required for the optimization of nonsmooth scalar functions in n dimensions resulting, for example, from physical experiments or from the statistical averaging of numerical simulations of chaotic systems such as turbulent flows. The core idea of all efficient algorithms for problems of this type is to keep function evaluations far apart until convergence is approached. Generalized pattern search (GPS) algorithms, a modern class of methods particularly well suited to such problems, accomplish this by coordinating the search with an underlying grid which is refined, and coarsened, as appropriate. One of the most efficient subclasses of GPS algorithms, known as the surrogate management framework (SMF; see Booker et al. in Struct Multidiscip Optim 17:1---13, 1999), alternates between an exploratory search over an interpolating function which summarizes the trends exhibited by existing function evaluations, and an exhaustive poll which checks the function on neighboring points to confirm or confute the local optimality of any given candidate minimum point (CMP) on the underlying grid. The original SMF algorithm implemented a GPS step on an underlying Cartesian grid, augmented with a Kriging-based surrogate search. Rather than using the n-dimensional Cartesian grid (the typical choice), the present work introduces for this purpose the use of lattices derived from n-dimensional sphere packings. As reviewed and analyzed extensively in Part I of this series (see Belitz, PhD dissertation, University of California, San Diego, 2011, Chap. 2), such lattices are significantly more uniform and have many more nearest neighbors than their Cartesian counterparts. Both of these facts make them far better suited for coordinating GPS algorithms, as demonstrated here in a variety of numerical tests.

Amplitude-only pattern synthesis of nonuniform linear arrays using a generalized pattern search optimization

In this work, synthesis of linear array geometry and excitation amplitudes is first formulated as a linearly constrained multiobjective optimization problem with the goals of minimum sidelobe level, null control and high directivity and then solved by a generalized pattern search (GPS) algorithm for the optimum element locations and excitation amplitudes. The constraints are imposed on the interelement spacing and dynamic range ratio of the amplitude tapering to reduce mutual coupling effects between the elements. GPS methods are newly discovered, derivative-free methods where the current iterate is updated by sampling the fitness function at a finite number of points along a suitable set of search directions to find a decrease in the function value. Thus, GPS methods can be exploited efficiently in solving optimization problems without requiring any information about the gradient of the fitness function which may be even discontinuous, nondifferentiable, stochastic or highly nonlinear. Finally, four worked examples are presented that illustrate the use of the whole GPS synthesis method, and the optimization goal in each example is easily achieved. Furthermore the full-wave simulations of the synthesized arrays are also completed to examine the mutual coupling effects. Finally the results of the GPS algorithm are validated by comparing with results obtained using the genetic algorithm, and the results of the uniform and Dolph - Chebyshev arrays, having the same number of element and the same aperture length. © 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2011.

Interference suppression by optimising the positions of selected elements using generalised pattern search algorithm

We propose the generalised pattern search (GPS) optimisation algorithm for steering nulls in the directions of interference. This algorithm is a deterministic, fast and efficient derivative-free optimisation process searching for function minima. A new technique based on step amplitudes for excitations of array elements is used as the template. In order to reduce the complexity and cost, only the positions of selected elements of the array are optimised for suppression of narrow or wide interferences by the GPS algorithm. The results show that the GPS algorithm is an efficient and robust technique in imposing single null, multiple nulls or wide suppressed sectors by perturbating only the selected few array elements while the other elements freeze. Furthermore, the full-wave simulations of the synthesised arrays are also completed to examine the effects of mutual coupling on the radiation patterns.

Strategy for optimising cutting parameters in the dry turning of 6061-T6 aluminium alloy based on design of experiments and the generalised pattern search algorithm

This paper proposes a strategy for optimising cutting parameters in the dry turning process to improve surface roughness and machining time and minimise machining cost. Based on this strategy, average surface roughness is expressed using design of experiments and regression analysis in terms of the investigated process parameters: cutting speed, feed rate and depth of cut. Machining time is expressed analytically as a function of cutting speed and feed rate. Machining cost per unit is then derived and minimised subject to the relevant set of constraints using the generalised pattern search (GPS) algorithm. The proposed strategy is presented through a case study using the widely used aluminium alloy 6061-T6. An optimal solution was achieved at values for cutting speed, feed rate and depth of cut of 142.8 m/min, 0.05 mm/rev and 0.05 mm, respectively, corresponding to a machining cost of 0.211 $/unit. Results obtained showed that the optimal solution was efficiently reached by the GPS algorithm.

Optimal Design of a Parallel Micromanipulator for Precision Machining

A parallel micromanipulator (PmM) which can be applied into the precision machining is optimized in the paper. This paper adopts a methodology to determine a set of optimal design parameters of PmM whose workspace is as close as possible of being equal to a prescribed cuboid dexterous workspace (PCDW). The kinematic problem is analyzed in brief to determine the design parameters and their relation. Then, an optimal design procedure which adopts the generalized pattern search algorithm in the genetic algorithm and direct search toolbox of Matlab is proposed to solve these problems. As an applying example, the results of four cases PCDW to PmM are presented. And the design result is compared with a concept of the distance between the best state of the PmM and the requirement of the operation task. The method and result of this paper are very useful for the design of micromanipulator for the machining field which requires the high precision and accuracy.

Insights into performance of pattern search algorithms for high-frequency surface wave analysis

Inversion of high-frequency surface wave dispersion curves is challenging for most local-search methods due to its high nonlinearity and to its multimodality. In this paper, we implemented an investigation to fully exploit and utilize the potentiality of pattern search algorithms and to further enhance their performance for surface wave analysis. We first investigate effects of different inversion strategies, initial mesh size and final mesh size, expansion factor, and contraction factor, as well as inclusion of noise in surface wave data on the performance of the approaches, by three synthetic earth models. Then, a comparative analysis with genetic algorithms is made to further highlight the advantages of the proposed inverse procedure. Finally, the insights issued from this analysis are verified by a real-world example from Henan, China. Results from both synthetic and field data demonstrate: (a) generalized pattern search (GPS) algorithm with the maximal positive basis set 2 N vectors works better than GPS algorithm with the minimal positive basis set N+1 vectors; (b) if one gets a suitable initial mesh size by taking some experimentation, then setting expansion factor Λ=1 (i.e., not allow expansions) and contraction factor θ=1/2 can greatly enhance the performance of pattern search algorithms. This is particularly true as the algorithm converges and final mesh size should go to zero; and (c) pattern search algorithms possess stronger immunity with respect to noise and should be considered good not only in terms of accuracy but also in terms of computation effort, especially when compared to the application of genetic algorithms to Rayleigh wave inversion.

Hyperspectral source prediction based on an optimal selection of multispectral data

Two optimization techniques from the class of direct search solvers, the genetic and generalized pattern search algorithms, are applied to hyperspectral source prediction. The number of filter bands to be sampled and their placements are the variables subject to optimization. The algorithms provide the optimal placements by sampling the stable source radiances at multispectral resolution, estimating the original source spectrum upon interpolation to hyperspectral resolution, and retaining the most accurate prediction. The integrated absolute error between the source prediction and the hyperspectral source truth is minimized. The interpolation estimate of the source approaches zero error asymptotically, as the number of allowed band samples is increased. The source spectrum of 2.15 μm bandwidth is reconstructed with 13 bands and relative absolute error of less than 0.5%. The method can be applied to any radiance spectrum or subspectrum partition, and is useful in transfer radiometer design or minimizing the number of spectrometer measurements required for hyperspectral source prediction.

Buckling optimization of laminated composite plates using genetic algorithm and generalized pattern search algorithm

In this paper, genetic algorithm and generalized pattern search algorithm are used for optimal stacking sequence of a composite panel, which is simply supported on four sides and is subject to biaxial in-plane compressive loads. The problem has several global optimum configurations in the vicinity of local optima. The composite plate under consideration is 64-ply laminate made of graphite/epoxy. The laminate is taken to be symmetric and balanced, comprised of two-ply stacks with discrete fiber angles of 02, ± 45, 902 in the laminate sequence. The critical buckling loads are maximized for several combinations of load case and plate aspect ratio, and are compared with published results. Performance of both algorithms is compared in terms of capability of identifying global optima. It is found that genetic algorithm is efficient for problems with global optima.

Generalized Pattern Search Algorithm for Peptide Structure Prediction

Finding the near-native structure of a protein is one of the most important open problems in structural biology and biological physics. The problem becomes dramatically more difficult when a given protein has no regular secondary structure or it does not show a fold similar to structures already known. This situation occurs frequently when we need to predict the tertiary structure of small molecules, called peptides. In this research work, we propose a new ab initio algorithm, the generalized pattern search algorithm, based on the well-known class of Search-and-Poll algorithms. We performed an extensive set of simulations over a well-known set of 44 peptides to investigate the robustness and reliability of the proposed algorithm, and we compared the peptide conformation with a state-of-the-art algorithm for peptide structure prediction known as PEPstr. In particular, we tested the algorithm on the instances proposed by the originators of PEPstr, to validate the proposed algorithm; the experimental results confirm that the generalized pattern search algorithm outperforms PEPstr by 21.17% in terms of average root mean-square deviation, RMSD C α .

Pattern search in the presence of degenerate linear constraints

This paper deals with generalized pattern search (GPS) algorithms for linearly constrained optimization. At each iteration, the GPS algorithm generates a set of directions that conforms to the geometry of any nearby linear constraints. This set is then used to construct trial points to be evaluated during the iteration. In a previous work, Lewis and Torczon developed a scheme for computing the conforming directions, however, the issue of degeneracy merits further investigation. The contribution of this paper is to provide a detailed algorithm for constructing the set of directions whether or not the constraints are degenerate. One difficulty in the degenerate case is the classification of constraints as redundant or nonredundant. We give a short survey of the main definitions and methods for treating redundancy and propose an approach to identify nonredundant ϵ-active constraints. We also introduce a new approach for handling nonredundant linearly dependent constraints, which maintains GPS convergence properties without significantly increasing computational cost. Some simple numerical tests illustrate the effectiveness of the algorithm.

Collision-free trajectory planning for manipulators using generalized pattern search

A Generalized Pattern Search algorithm (GPS) is introduced to design a collision-free trajectory for planar redundant manipulators. Cubic-spline interpolation trajectory for the end-effector is designed a priori by selecting three intermediate points around the workspace obstacles. The GPS algorithm ensures that none of the selected points lies inside any of the obstacles. The end-effector will follow the prescribed trajectory in excellent manner without colliding with the obstacles. Simulation studies are carried out for three different obstacles to prove the validity of the proposed algorithm.

Generalized pattern search methods for linearly equality constrained optimization problems

Torczon and several persons have designed pattern search methods for unconstrained optimization problem, bound constrained, and linearly constrained optimization problems. In this paper, we shall consider linear equality constrained problems as unconstrained ones using generalized pattern search algorithm, except for a feasible starting point.

Precision Control for Generalized Pattern Search Algorithms with Adaptive Precision Function Evaluations

In the literature on generalized pattern search algorithms, convergence to a stationary point of a once continuously differentiable cost function is established under the assumption that the cost function can be evaluated exactly. However, there is a large class of engineering problems where the numerical evaluation of the cost function involves the solution of systems of differential algebraic equations. Since the termination criteria of the numerical solvers often depend on the design parameters, computer code for solving these systems usually defines a numerical approximation to the cost function that is discontinuous with respect to the design parameters. Standard generalized pattern search algorithms have been applied heuristically to such problems, but no convergence properties have been stated. In this paper we extend a class of generalized pattern search algorithms to include a subprocedure that adaptively controls the precision of the approximating cost functions. The numerical approximations to the cost function need not define a continuous function. Our algorithms can be used for solving linearly constrained problems with cost functions that are at least locally Lipschitz continuous. Assuming that the cost function is smooth, we prove that our algorithms converge to a stationary point. Under the weaker assumption that the cost function is only locally Lipschitz continuous, we show that our algorithms converge to points at which the Clarke generalized directional derivatives are nonnegative in predefined directions. An important feature of our adaptive precision scheme is the use of coarse approximations in the early iterations, with the approximation precision controlled by a test. We show by numerical experiments that such an approach leads to substantial time savings in minimizing computationally expensive functions.

Convergence of Mesh Adaptive Direct Search to Second‐Order Stationary Points

A previous analysis of second-order behavior of generalized pattern search algorithms for unconstrained and linearly constrained minimization is extended to the more general class of mesh adaptive direct search (MADS) algorithms for general constrained optimization. Because of the ability of MADS to generate an asymptotically dense set of search directions, we are able to establish reasonable conditions under which a subsequence of MADS iterates converges to a limit point satisfying second-order necessary or sufficient optimality conditions for general set-constrained optimization problems.

Objective Function Effect Based Pattern Search—An Implementation for 3D Component Layout

Generalized pattern search (GPS) algorithms have been used successfully to solve three-dimensional (3D) component layout problems. These algorithms use a set of patterns and successively decreasing step sizes of these patterns to explore the search space before converging to good local minima. A shortcoming of conventional GPS algorithms is the lack of recognition of the fact that patterns affect the objective function by different amounts and hence it might be efficient to introduce them into the search in a certain order rather than introduce all of them at the beginning of the search. To address this shortcoming, it has been shown by the authors in previous work that it is more efficient to schedule patterns in decreasing order of their effect on the objective function. The effect of the patterns on the objective function was estimated by the a priori expectation of the objective function change due to the patterns. However, computing the a priori expectation is expensive, and to practically implement the scheduling of patterns, an inexpensive estimate of the effect on the objective function is necessary. This paper introduces a metric for geometric layout called the sensitivity metric that is computationally inexpensive, to estimate the effect of pattern moves on the objective function. A new pattern search algorithm that uses the sensitivity metric to schedule patterns is shown to perform as well as the pattern search algorithm that used the a priori expectation of the objective function change. Though the sensitivity metric applies to the class of geometric layout or placement problems, the foundation and approach is useful for developing metrics for other optimization problems.

Convergence Results for Generalized Pattern Search Algorithms are Tight

The convergence theory of generalized pattern search algorithms for unconstrained optimization guarantees under mild conditions that the method produces a limit point satisfying first order optimality conditions related to the local differentiability of the objective function. By exploiting the flexibility allowed by the algorithm, we derive six small dimensional examples showing that the convergence results are tight in the sense that they cannot be strengthened without additional assumptions, i.e., that certain requirement imposed on pattern search algorithms are not merely artifacts of the proofs. In particular, we first show the necessity of the requirement that some algorithmic parameters are rational. We then show that, even for continuously differentiable functions, the method may generate infinitely many limit points, some of which may have non-zero gradients. Finally, we consider functions that are not strictly differentiable. We show that even when a single limit point is generated, the gradient may be non-zero, and zero may be excluded from the generalized gradient, therefore, the method does not necessarily produce a Clarke stationary point.

Catalytic Combustion Kinetics: Using a Direct Search Algorithm to Evaluate Kinetic Parameters from Light‐Off Curves

AbstractThis paper describes a method for evaluating the kinetic constants in a rate expression for catalytic combustion applications using experimental light‐off curves. The method uses a transient one‐dimensional single channel monolith finite element reactor model to simulate reactor performance. The heat and mass transfer models used account for developing flow in the entrance region. A parameter global optimization routine based on a generalized pattern Search algorithm is used to determine the best fit parameters in the rate expression for the oxidation of CO and mixtures of CO and CH4. The algorithm is compared to a more classical gradient method, the Fletcher‐Reeves method.