## Approval of the thesis: EFFICIENT SOLUTION OF OPTIMIZATION PROBLEMS WITH CONSTRAINTS AND/OR COST FUNCTIONS EXPENSIVE TO EVALUATE (2009)

### Citations

2542 |
A simplex method for function minimization
- Nelder, Mead
- 1965
(Show Context)
Citation Context ...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... |

812 | Surface reconstruction from unorganized points
- Hoppe, DeRose, et al.
- 1992
(Show Context)
Citation Context ...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 ... |

597 | Convergence properties of the Nelder–Mead simplex method in low dimensions
- Lagarias, Reeds, et al.
- 1998
(Show Context)
Citation Context ...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... |

503 | R.: Reconstruction and representation of 3D objects with radial basis functions - CARR, BEATSON, et al. |

455 |
Direct search solution of numerical and statistical problems
- Hooke, Jeeves
- 1961
(Show Context)
Citation Context ...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... |

434 |
Othogonal least squares learning algorithm for radial basis function networks,"
- Chen, C, et al.
- 1991
(Show Context)
Citation Context ... 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 ... |

413 | KAMVYSSELIS M.: A new Voronoi-based surface reconstruction algorithm
- AMENTA, BERN
- 1998
(Show Context)
Citation Context ...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... |

365 |
Stochastic Approximation Methods for Constrained and Unconstrained Systems
- Kushner, Clark
- 1978
(Show Context)
Citation Context .... 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... |

318 | Multivariate stochastic approximation using a simultaneous perturbation gradient approximation
- Spall
- 1992
(Show Context)
Citation Context ...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 ... |

298 |
Response Surface Methodology
- Myers, Montgomery
- 1995
(Show Context)
Citation Context ...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... |

259 | KOLLURI R.: The power crust
- AMENTA, CHOI
(Show Context)
Citation Context ...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... |

215 | H.: Multi-level partition of unity implicits
- OHTAKE, BELYAEV, et al.
(Show Context)
Citation Context ... 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... |

130 | Direct search methods on parallel machines
- Torczon
- 1991
(Show Context)
Citation Context ...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... |

125 | F.: Modelling with implicit surfaces that interpolate
- TURK, O’BRIEN
(Show Context)
Citation Context ...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... |

120 | Sequential application of simplex designs in optimisation and evolutionary operation - Spendley, Hext, et al. - 1962 |

116 | Interpolating and approximating implicit surfaces from polygon soup
- Shen, OÕBrien, et al.
- 2004
(Show Context)
Citation Context ...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... |

95 | Direct search methods: Then and now - Lewis, Torczon, et al. - 2000 |

92 | Multidimensional stochastic approximation methods - Blum - 1954 |

90 | An overview of the simultaneous perturbation method for efficient optimization,”
- Spall
- 1998
(Show Context)
Citation Context ...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... |

79 |
Introduction to optimization, Optimization
- Polyak
- 1987
(Show Context)
Citation Context ...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 ... |

61 | Variational Implicit Surfaces
- TURK, O’BRIEN
- 1999
(Show Context)
Citation Context ..., 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 ... |

59 | Efficient adaptive simplification of massive meshes
- Shaffer, Garland
- 2001
(Show Context)
Citation Context ... 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 ... |

39 | Global random optimization by simultaneous perturbation stochastic approximation, In:
- Maryak, Chin
- 2001
(Show Context)
Citation Context ...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... |

34 | Penalty function methods for constrained optimization with genetic algorithms - Yeniay - 2005 |

33 |
Stochastic Estimation of a Regression Function.
- Kiefer, Wolfowitz
- 1952
(Show Context)
Citation Context ...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... |

32 | New Advances and Applications. of Combining Simulation and Optimization”,
- Glover
- 1996
(Show Context)
Citation Context ...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 ... |

23 | From evolutionary operation to parallel direct search: Pattern search algorithms for numerical optimization
- Torczon, Trosset
- 1997
(Show Context)
Citation Context ...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... |

21 | Efficient Reconstruction of Large Scattered Geometric Datasets Using the Partition of Unity and Radial Basis Functions,” LaBRI, - Tobor, Reuter, et al. - 2003 |

15 | A revised simplex search procedure for stochastic simulation response-surface optimization - Humphrey, Wilson - 1998 |

15 | A direct search algorithm for optimization with noisy function evaluations
- Anderson, Ferris
(Show Context)
Citation Context ...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... |

12 | I know it when I see it: Toward a definition of direct search methods
- Trosset
- 1997
(Show Context)
Citation Context ...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... |

11 |
Stochastic optimization and the simultaneous perturbation method,”
- Spall
- 1999
(Show Context)
Citation Context ...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... |

10 | Pattern search methods for nonlinear optimization
- Torczon
- 1995
(Show Context)
Citation Context ...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... |

10 |
Inaccuracy in quasi-Newton methods: Local improvement theorems
- Dennis, Walker
- 1984
(Show Context)
Citation Context ...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... |

10 | Simultaneous perturbation stochastic approximation website - Spall - 2008 |

7 |
Response Surface Methodology,
- Carley, Kamneva, et al.
- 2004
(Show Context)
Citation Context ...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... |

4 |
Constrained global optimization with radial basis functions
- Käck
- 2004
(Show Context)
Citation Context ....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... |

3 |
et al, Interpolating implicit surface from scattered surface data using compactly supported radial basis functions
- Morse, Yoo, et al.
- 2001
(Show Context)
Citation Context ... 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... |

3 | Penalty functions,” Section C 5.2 of Handbook of Evolutionary Computation, A Joint Publication - Smith, Coit |

1 |
Direct-search methods and DACE”, Seminar Report
- Isaacs
- 2003
(Show Context)
Citation Context ...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 ............................................................. |

1 |
Penalty and Barrier Methods for Constrained Optimization
- Freud
- 2004
(Show Context)
Citation Context ...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... |

1 | last access time is 17.08.2009 - MathWorks |

1 |
den Hertog, “Constrained Optimization Involving Expensive Function Evaluations: A Sequential Approach”, ISNN
- Brekelmans, Driessen, et al.
- 2001
(Show Context)
Citation Context ...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... |