appendix a scripts - springer978-3-642-31187-1/1.pdf · appendix a scripts a.1 latin hypercube ......

Appendix AScripts

A.1 Latin Hypercube DOE

M. Cavazzuti, Optimization Methods: From Theory to Design,DOI: 10.1007/978-3-642-31187-1, � Springer-Verlag Berlin Heidelberg 2013

233

A.2 I-Optimal DOE for Full Quadratic or Full CubicPolynomial Response

234 Appendix A: Scripts

Appendix A: Scripts 235

A.3 Ordinary Kriging RSM


A.4 Radial Basis Functions RSM


A.5 Wolfe-Powell Line-Search Algorithm


A.6 Golden Section Line-Search Algorithm


A.7 Nelder and Mead Simplex Algorithm


A.8 BFGS Algorithm


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256 References

Index

AActivation function, 66Active set method, 91Actual reduction, 82Adiabatic analysis, 200Adjusted regression parameter, 48Aim, 42, 75Aliasing, 22Allele, 107Anisotropic kriging, 59Anisotropy, 59Approximating, 44Approximation, 71Architecture, 66Archive, 112Axis orthogonal importance latin hypercube

sampling, 139Axis orthogonal importance sampling

Monte Carlo, 136

BB-spline, 157Bézier curve, 157Backpropagated error, 70Backpropagation algorithm, 61Balanced, 21Barrier function, 91Barrier function method, 96Basis functions, 59Bayesian method, 50Bernstein basis polynomials, 158Best fit, 44Best linear unbiased estimator, 51BFGS formula, 87Bias, 67Binary step function, 66

Bipolar sigmoid function, 66Blend cross-over, 119Blocking, 14Blue, see best linear unbiased

estimator, 51Box-Behnken, 25Bracket, 81Bracketing, 80Branch and bound method, 91, 97Broyden family, 87

CCentral composite, 23

circumscribed, 24faced, 24inscribed, 24scaled, 24

Child, 107Chimney, 176Cholesky decomposition, 35Chromosome, 107Coefficient of variation, 135Cognitive learning factor, 111Cold dead volume ratio, 198Combination factor, 117Compact heat exchanger, 153Compression space, 197Conditional probability, 60Confounding, 22Conjugate direction methods, 87Conjugate gradient method, 87Constraint, 3Constraint satisfaction problem, 78Continuous selection, see steady-state

evolution, 118Contour plot, 44


257

C (cont.)Control factor, see control factor, 15Control points, 157Control variables, 27Cooler, 197Cooperative game, 114Correlation reduction, 34Cost function, 67Courant penalty function, 96Covariance function, 51Covariance matrix, 34, 38Craziness, 111Cross-over, 123Cross-over constant, 117Cross-over operator, 116Crossed array, 27Cumulative distribution, see distribution

curvature, 34Curvature, 78

DData set, 44Degree of freedom, 19Delta rule, 67Derandomized evolution strategy, 116Derandomized evolutionary

strategy, 120Design factor, see primary factor, 15Design of experiments, 6, 13, 149Design point, 134Design resolution, 22Design space, 2Deterministic optimization, 77, 151DFP formula, 87Differential evolution, 116Direct elimination method, 93Direct numerical simulation, 156Direction, 78Direction set method, 87Directional cross-over, 124Displacer, 197Distribution, 34, 121

normal Gaussian, 34Disturbance factors, see

nuisance factor, 15DNA, 107

EEffort, 75Elimination, 90Elitism operator, 124Emissivity, 190

Euler method, 204Evolutionary algorithm, 103Evolutionary algorithms, 116Exact penalty functions, 91Expanded design matrix, 39Expansion space, 197Expected value, see mean value, 30Experiment, 2, 13Experimental design, see design

of experiments, 13External factors, 132

FFactor, 14Failure area, 133Failure probability, 10, 133Faure sequence, 33Feasible point, 90Feasible region, 90Feedforward, 67First order necessary condition, 79First order reliability method, 136Fitness function, 105Follower, 114Fractional factorial, 21

one-half, 21one-quarter, 21

Friction factor, 156Full factorial, 17

adjustable, 19two-levels, 17

Function evaluation, 77Fuzzy recombination, 119

GGame theory, 103Gauss-Newton algorithm, 47Gene, 7, 107General linearly constrained

optimization, 91Generalized elimination method, 94Generation, 107Generational evolution, 124Generational selection, see generational

evolution, 119Generator, 22Genetic algorithm, 103, 121Genotype, 107Global intermediate, 119Golden section method, 81Graeco-latin square, 16Guide, 111

258 Index

HHalton sequence, 33Heater, 197Hessian matrix, 78Hidden layer, 67Hierarchical and distributed metamodel-assis-

ted evolutionary algorithms, 120Hierarchical competitive game, 114Hierarchy, 120Homogeneous covariance function, 62Hot dead volume ratio, 198Hyper-graeco-latin square, 16Hyperbolic tangent sigmoid function, 66

IIdentity function, 66Importance latin hypercube sampling, 138Importance sampling, 137Importance sampling Monte Carlo, 138Individual, 107Inertia factor, 111Initial value problem, 204Inner array, 27Input layer, 67Input parameters, 2Input variable, 2Integrated prediction variance, 38Interaction effect, 19Internal energy, 108Interpolating, 44Interpolation, 71Involute, 182

JJoint probability, 61

KK-nearest, 50Khayyam triangle, see

Tartaglia triangle, 19Kriging, 50

disjunctive, 52indicator, 52IRF-k, 51lognormal, 52multiple-indicator, 52ordinary, 51simple, 51universal, 51

Kriging error, see kriging variance, 52Kriging nearest, 50

Kriging variance, 52Kuhn-Tucker conditions, 92

LLag, 53Lagrange multipliers method, 90Lagrange–Newton method, 97Lagrangian function, 92Lagrangian matrix, 95Lagrangian method, see lagrange

multipliers method, 94Laminar flow, 157Larger-the-better, 29Latin hypercube, 33Latin hypercube sampling, 136, 138Latin square, 16Leader, 11, 114Learning rate, 70Least squares, 44Levels, 14Levenberg–Marquardt methods, 89Levenberg–Marquardt trajectory, 90Limit state function, 133Line, 78Line-search, 79Linear least squares, 45Linear programming, 91Load effect, 133Logistic sigmoid function, 66

MMain interaction, 18Marginal probability, 60Mass flow rate, 178Mathematical programming, 7Mean value, 13, 29, 34Merit function, 98Meta-model, 43Metamodel, 121, 150Micro combined heat and power unit, 195Mixed integer programming, 91, 97Mixing number, 118Model function, 44Mollifier Shepard, 50Moment matrix, 39Monte Carlo simulation, 135Multi-disciplinary optimization, 160Multi-layer, 68Multi-membered

evolution strategy, 116Multi-objective genetic algorithm, 124Multi-objective optimization, 105

Index 259

M (cont.)Multi-objective robust design optimization, 9,

132Mutant individual, 116Mutation constant, 117Mutation operator, 116

NNash equilibrium, 113Neural networks, 66Neuron, 66Newton’s method, 85NLPQLP, 98No free lunch theorem, 130Noise, 71, 131Noise, see noise factors, 13Noise factors, 9Noise variables, 27Nominal-the-best, 30Non uniform rational b-spline, 157Non-smooth optimization, 91Nondimensional analysis, 157Nonlinear least squares, 46Nonlinear programming, 91Nonstationary covariance function, 62Normal regression parameter, 47Normalized average, see integrated prediction

variance, 38Nugget, 51, 55Nuisance factor, 15Number of experiments, 41Number of levels, 41Number of parameters, 41Nusselt number, 156

OObjective, see objective function, 2Objective function, 2Offspring, 107One-point cross-over, 123Operating conditions, 132Operating fluid, 195Optimal design, 36

a-optimal, 40d-optimal, 40e-optimal, 40g-optimal, 40i-optimal, 38

Optimal RSM, 49Optimization problem, 2Optimization, 2, 3

constrained, 7

convex, 8deterministic, 7discrete, 8evolutionary, 7genetic, 7global, 8gradient-based, 7local, 8multi-objective, 3, 8multivariate, 8single objective, 3, 8stochastic, 7unconstrained, 7

Order of convergence, 79Orthogonal, 18Outer array, 27Output layer, 67Output parameters, 2

PParameter, 14, 75Parent, 107Pareto dominance, 105Pareto frontier, 105Pareto optimality, 105Partial sill, 55Particle swarm optimization, 103, 110Pascal triangle, see Tartaglia triangle, 19Penalty function, 91Penalty function method, 96Phenotype, 107Plackett-Burman, 26Player, 113Plenum, 178Population, 107Power piston, 197Practical range, 55Predicted reduction, 82Prediction variance, 39Predictive capability of the model, 48Pressure swing ratio, 199Primal active set method, 95Primary factor, 15Prior probability, see marginal probability, 60Problem, see optimization problem, 2Pseudo-random numbers generator, 32

QQuadratic programming, 91Quality, 131Quasi-Newton condition, 86Quasi-Newton methods, 85

260 Index

RRadial basis function

Gaussian, 62inverse multiquadric, 63multiquadric, 63polyharmonic splines, 63

Radiation heat transfer, 190Random, 32Random search, 109Random seed generator, 77Randomization, 13Randomized complete block design, 15Range, 55Rank one formula, 86Rayleigh number, 177Recirculation, 187Recurrent, 67Reduced dead volume, 198Reduced gradient vector, 94Reduced Hessian matrix, 94Regenerator dead volume ratio, 198Regenerator mean effective temperature, 198Region of interest, 14Regression parameter, 47Regularity, 71Reinforcement learning, 66Reliability, 131Reliability analysis, 9, 132Reliability index, 10, 134Replication, 13Resistance effect, 133Response surface, 20, 43Response surface methodology, see response

surface modelling, 43Response surface modelling, 6, 43, 149Response variable, 14Restricted step, 79Rib, 176Regenerator, 197Robust design analysis, 131, 152, 8Robust engineering design, see robust design

analysis, 8Robust parameter design problem, 27Robustness, 105, 131, 8Rotatability, 25Roulette-wheel selection, 122Runge–Kutta methods, 204

SSafe area, 133Sample, 2Sample size, 15Sample space, 14

Sampling map, 33Scaling factor, 117Schmidt analysis, 197Second order necessary condition, 79Second order reliability method, 137Sectioning, 80Selection, 122Self-adaptive evolution, 116Semivariance, 53Semivariogram, 51Semivariogram cloud, 53Semivariogram model, 53

Bessel, 55circular, 55exponential, 55Gaussian, 55linear, 55pentaspherical, 55spherical, 53

Sequential competitive game, see hierarchicalcompetitive game, 113

Sequential quadratic programming, 91Set of active constraints, 90Shepard, 50Shift vector, 46Signal-to-noise ratio, 29Sill, 55Simple importance latin hypercube

sampling, 139Simplex method for linear optimization, 91Simplex method for nonlinear optimization, 82Simulated annealing, 103, 107Simulated binary cross-over, 119Simulation, 2Simultaneous competitive game, 113Single-layer, 68Sinusoidal wavy channel, 153Slope, 78Smaller-the-better, 29Sobol sequence, 33Social learning factor, 111Solution space, 2Space filling, 30Spatial auto-correlation effect, 53Standard deviation, 13, 29, 34Standard normal space, 134Star points, 23Stationary covariance function, 61Statistical design of experiments, see statistical

experimental design, 14Statistical experimental design, 14Steady-state evolution, 124Steady-state selection, see steady-state

evolution, 118

Index 261

S (cont.)Steepest descent method, 85Stirling cycle, 196Stirling engine, 195Stochastic optimization, 103, 150Strength of the mutation, 119Supervised learning, 66Swarm intelligence, 104

TTaguchi, 27Tartaglia triangle, 19Temperature ratio, 198Tolerance, 132Tournament selection, 122Training algorithm, 66Transformed importance latin hypercube

sampling, 139Transitional flow, 156Travelling salesman problem, 109Treatment factor, see primary factor, 15Trial individual, 116Trust region, 79Turbulence, 111Turbulence model, 156Two-points cross-over, 123

UUncertainty, see noise, 131Uniform cross-over, 123Uniform heat flux condition, 175Uniform wall temperature condition, 175Unimodal normally distributed cross-over, 119Unsupervised learning, 66

VVan der Corput sequence, 32Variable, see input variable, 2Variance, 38Volume ratio, 198

WWavy channel, 153Wear, 132Wetted area, 177Wolfe–Powell conditions, 80Word, see generator, 22Words, 22Working fluid, see operating fluid, 195Working space, 195

262 Index

appendix a scripts - springer978-3-642-31187-1/1.pdf · appendix a scripts a.1 latin hypercube ......

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