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...ficients ofsreflection ( ) , expansion ( )X , contraction ( ) , and shrinkage ( ) . The material ofsthis part is based on the following reference, [16].sAccording to the original Nelder-Mead paper =-=[9]-=-, parameters of the method shouldssatisfy the conditions below.s0 , 1 , , 0 1X X s(2.7)sUsually the values of the parameters in standard Nelder-Mead simplex algorithm aresselected as:s1 1...

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...riance matrix of a point sample set obtained from a certainspart of the surface can be used to estimate the curvature of that part of the surface, assit has been demonstrated in studies referenced by =-=[18]-=-, [19].sLet sbe the centroid of the point sample setD , i.e.,s1 1 m i im s(2.26)sThen, the nxncovariance matrix, Csfor this set is given bys1 1 T m m C ...

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...eters has to be defined; these parameters are coefficients ofsreflection ( ) , expansion ( )X , contraction ( ) , and shrinkage ( ) . The material ofsthis part is based on the following reference, =-=[16]-=-.sAccording to the original Nelder-Mead paper [9], parameters of the method shouldssatisfy the conditions below.s0 , 1 , , 0 1X X s(2.7)sUsually the values of the parameters in standard N...

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...961, Robert Hooke and T.A. Jeeves developed a method for optimization andsintroduced “direct search” term. They presented the following description of directssearch in the introduction of their paper =-=[10]-=-:s“We use the phrase “direct search” to describe sequential examination of trialssolutions involving comparison of each trial solution with the “best” obtained up tosthat time together with a strategy...

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... center points.sTheoretical investigation and practical results shows that the choice of thesnonlinearity of the function (.)sis not very important for the performance of thesconstructed RBF network =-=[27]-=-. For example, the thin-plate-spline functions2( ) log( )v v v s(3.3)sand the gaussian functions2 2( ) exp( / )v v s(3.4)swhere sis a real constant, are two typical choices for radial basis ...

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...earchers in this field offer useful strategies and methods to solve the problem ofssurface reconstruction from an unorganized point cloud. Hoppe’s [18] signedsdistance function based method, Amenta’s =-=[28]-=-, [29], Voronoi and crust basedsmethod, moving least square (MLS) of Shen [22], etc.sIn general, subset of suitable centers is selected arbitrarily or randomly from theswhole point set. It is clear th...

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.... It is observed that these methods are relativelysinsensitive to inaccuracies in the cost and/or the constraint functions [1], [2], [3], [4].sBesides derivative free methods, Keifer–Wolfowitz method =-=[5]-=-, [6] is an approachsfrom the area of stochastic approximation. The gradient estimation with this methodsrequires the evaluation of the cost function at pointssk ix e , where iesrepresentssthe i ths...

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...the gradients, where they can be estimatedseither numerically or analytically. The Kiefer-Wolfowitz [12] algorithm uses thesfinite difference gradient approximation to estimate gradients. Spall [13], =-=[14]-=-sproposed the Simultaneous Perturbation Stochastic Approximation (SPSA) methodsfor gradient estimation, which requires only two function evaluations per estimation,sregardless of the dimension of the ...

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...nters among thesinput sample set.s42s3.1. RBF BASED SURFACE GENERATIONsResponse surface methodology is a collection of statistical and mathematicalstechniques that can be used to approximate a system =-=[20]-=-. In this section,sdevelopment procedure of a response surface model based on radial basis functions,smatching the measurements obtained from the original model by experimentation orsnumerical simulat...

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...rs in this field offer useful strategies and methods to solve the problem ofssurface reconstruction from an unorganized point cloud. Hoppe’s [18] signedsdistance function based method, Amenta’s [28], =-=[29]-=-, Voronoi and crust basedsmethod, moving least square (MLS) of Shen [22], etc.sIn general, subset of suitable centers is selected arbitrarily or randomly from theswhole point set. It is clear that thi...

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... robustlysreconstruction of a continuous surface from the unorganized points is desirable, forsthat reason, some useful methods are offered by researchers to overcome thissimportant problem [18],[22],=-=[23]-=-.sRBF attracts more attentions recently in multidimensional data interpolation [22],s[24],[25],[26]. It is identified as one of the most accurate and stable methods to solvesscattered data interpolati...

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...vely used to solve design optimization problems,scalled direct search methods. It is observed that these methods are relativelysinsensitive to inaccuracies in the cost and/or the constraint functions =-=[1]-=-, [2], [3], [4].sBesides derivative free methods, Keifer–Wolfowitz method [5], [6] is an approachsfrom the area of stochastic approximation. The gradient estimation with this methodsrequires the evalu...

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...sthat reason, some useful methods are offered by researchers to overcome thissimportant problem [18],[22],[23].sRBF attracts more attentions recently in multidimensional data interpolation [22],s[24],=-=[25]-=-,[26]. It is identified as one of the most accurate and stable methods to solvesscattered data interpolation problems.s44s3.1.2.1. RADIAL BASIS FUNCTIONSsA radial basis function is a real-valued funct...

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...y and robustlysreconstruction of a continuous surface from the unorganized points is desirable, forsthat reason, some useful methods are offered by researchers to overcome thissimportant problem [18],=-=[22]-=-,[23].sRBF attracts more attentions recently in multidimensional data interpolation [22],s[24],[25],[26]. It is identified as one of the most accurate and stable methods to solvesscattered data interp...

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...2: The steps of the algorithm ......................................................................... 7sFigure 2.1: Relative search patterns for two dimensional problem for SPSA andsFDSA algorithms =-=[13]-=- ........................................................................................ 15sFigure 2.2: After a reflection and expansion step Nelder-Mead Simplex is shown.sThe dashed lines refer to o...

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...is observed that these methods are relativelysinsensitive to inaccuracies in the cost and/or the constraint functions [1], [2], [3], [4].sBesides derivative free methods, Keifer–Wolfowitz method [5], =-=[6]-=- is an approachsfrom the area of stochastic approximation. The gradient estimation with this methodsrequires the evaluation of the cost function at pointssk ix e , where iesrepresentssthe i thsunit ...

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..., forsthat reason, some useful methods are offered by researchers to overcome thissimportant problem [18],[22],[23].sRBF attracts more attentions recently in multidimensional data interpolation [22],s=-=[24]-=-,[25],[26]. It is identified as one of the most accurate and stable methods to solvesscattered data interpolation problems.s44s3.1.2.1. RADIAL BASIS FUNCTIONSsA radial basis function is a real-valued ...

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... matrix of a point sample set obtained from a certainspart of the surface can be used to estimate the curvature of that part of the surface, assit has been demonstrated in studies referenced by [18], =-=[19]-=-.sLet sbe the centroid of the point sample setD , i.e.,s1 1 m i im s(2.26)sThen, the nxncovariance matrix, Csfor this set is given bys1 1 T m m C ...

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...astic approximation is a popular stochastic optimization technique. In particular,sit may be called as "gradient-free" SA sometimes, because it only requiressapproximation of the gradient information =-=[45]-=-. Consider the problem given below.s( ) ( ) f g s(2.1)sfor a differentiable function : nf R R . When direct evaluations of gsare notscostly, there are, of course, many approaches which can...

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...oximation (SA) area is the Keifer–Wolfowitz method [5], [6].sThe SA methods essentially approximate the gradients, where they can be estimatedseither numerically or analytically. The Kiefer-Wolfowitz =-=[12]-=- algorithm uses thesfinite difference gradient approximation to estimate gradients. Spall [13], [14]sproposed the Simultaneous Perturbation Stochastic Approximation (SPSA) methodsfor gradient estimati...

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...y function inaccuracies [8].sLocal search, tabu search, simulated annealing are some other approaches to thesdesign optimization problem generally considered in the field of discrete-eventssimulation =-=[7]-=-. These methods assume that one simulation run or one evaluation ofsthe function can be performed quickly. Moreover, they only deal with (simple)sconstraints on the design parameters.sDerivative-free ...

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...to solve design optimization problems,scalled direct search methods. It is observed that these methods are relativelysinsensitive to inaccuracies in the cost and/or the constraint functions [1], [2], =-=[3]-=-, [4].sBesides derivative free methods, Keifer–Wolfowitz method [5], [6] is an approachsfrom the area of stochastic approximation. The gradient estimation with this methodsrequires the evaluation of t...

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...ionsproblems is utilizing derivative free methods which do not try to estimate or calculatesthe gradients. These are heuristic and simple methods called direct search methodss[15].sAs it is stated in =-=[44]-=-, direct search methods for unconstrained optimization problemsshave been more popular for the past few years. Since these methods do not makesgradient estimates and involve relatively few function ev...

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...lve design optimization problems,scalled direct search methods. It is observed that these methods are relativelysinsensitive to inaccuracies in the cost and/or the constraint functions [1], [2], [3], =-=[4]-=-.sBesides derivative free methods, Keifer–Wolfowitz method [5], [6] is an approachsfrom the area of stochastic approximation. The gradient estimation with this methodsrequires the evaluation of the co...

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...time and correspondingsmeasurements (.)ysare obtained. Next, each element of the gradient estimate issformed by differencing the corresponding output values and then dividing by asdifference interval =-=[46]-=-. This type is the standard Kiefer-Wolfowitz approach tosapproximating gradient components and is originated from the definition of asgradient as a vector of p partial derivatives [46].sIf gradient ap...

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...used to solve design optimization problems,scalled direct search methods. It is observed that these methods are relativelysinsensitive to inaccuracies in the cost and/or the constraint functions [1], =-=[2]-=-, [3], [4].sBesides derivative free methods, Keifer–Wolfowitz method [5], [6] is an approachsfrom the area of stochastic approximation. The gradient estimation with this methodsrequires the evaluation...

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...first because derivative information is unavailable or hard to obtain. Even if it wassavailable quasi-Newton methods might be poor choices because they are adverselysaffected by function inaccuracies =-=[8]-=-.sLocal search, tabu search, simulated annealing are some other approaches to thesdesign optimization problem generally considered in the field of discrete-eventssimulation [7]. These methods assume t...

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...thematical aspect, a response surface model is ansinterpolation technique that fits complex multi dimensional function to its functionsvalues on specified function domain at some finite sample points =-=[21]-=-.sThe developed response surface model accuracy largely depends on the number ofsmeasurements or evaluations employed and their distribution on the designsparameter space, as well as the approximation...

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....sThe user in these cases almost usually forced to construct a penalized model, wheresthe violation of the constraints is added to the cost function; thereby enforcing thessatisfaction of constraints =-=[38]-=-. In this way, an optimization problem with constraintsscan be transformed into an unconstrained form.sBarrier and penalty methods are designed to solve these types of problems. In aspenalty method, t...

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... reason, some useful methods are offered by researchers to overcome thissimportant problem [18],[22],[23].sRBF attracts more attentions recently in multidimensional data interpolation [22],s[24],[25],=-=[26]-=-. It is identified as one of the most accurate and stable methods to solvesscattered data interpolation problems.s44s3.1.2.1. RADIAL BASIS FUNCTIONSsA radial basis function is a real-valued function w...

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...nd shrink a Nelder-Mead simplex issshown. The dashed lines refer to original simplexs[16] .................................... 22sFigure 2.4: One at a time search method for function of two variables =-=[15]-=- ............. 24sFigure 2.5: Hooke and Jeeves Pattern Search for two dimensions [15] ..................... 25sFigure 2.6: Sampling procedure .............................................................

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...en in Equation 1.1sis expanded from feasible region Fsto whole space dR , but a large cost or “penalty”sis added to the cost function for the points that lie outside of the original feasiblesregion F =-=[33]-=-.sIn a barrier method, we start with an assumption that a feasible point is given and aslarge cost is added to the cost function on feasible points that ever get closer to thesboundary of F . That mea...

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...the developed approximate model enables the designer to simulate thesperformance of the original system, the problem of finding a setting for possibly aslarge number of design variables still exists. =-=[42]-=- Then a crucial question arises asshow to find the best possible setting requiring minimum number of evaluations. [42]s2sSpecification of the appropriate values for the variables of these kinds of eng...