Mini-Max Type Robust Optimal Design Combined with Function Regularization

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1 6 th World Congresses of Structural and Multidisciplinary Optimization Rio de Janeiro, 30 May - 03 June 2005, Brazil Mini-Max Type Robust Optimal Design Combined with Function Regularization Noriyasu Hirokawa 1, Kikuo Fujita 2 and Fam Chiou Tzi 3 1 Department of Mechanical Engineering and Biomimetics, Kinki University, Wakayama, Japan, hirokawa@waka.kindai.ac.jp 2 Department of Mechanical Engineering, Osaka University, Osaka, Japan, fujita@mech.eng.osaka-u.ac.jp 3 Department of Mechanical Engineering, Osaka University, Osaka, Japan 1. Abstract This paper proposes a mini-max type robust optimal design and its enhancement by using function regularization. A mini-max type robust optimal design seeks the optimal solution based on the bounding points of the objective function and constraints within a distribution region. In order to obtain the bounding point strictly and efficiently, a distribution region which generally forms an oblique ellipsoid in the design space is diagonalized and isoparameterized into a hyper sphere and functions are approximated with quadratic response surfaces. Further, the accuracy of the bounding points are enhanced with a function regularization which transforms function values at sample points to construct more precise approximations. The effectiveness is ascertained through applications to two example problems. 2. Keywords: Robust Optimal Design, Intermediate Model, Mini-Max Type Formulation, Quadratic Response Surface, Function Regularization 3. Introduction Robust optimal design seeks the optimal solution considering the robustness of a design against manufacturing error, environmental change, wear-out and so forth. It optimizes a solution by using an intermediate model which approximates the objective function and constraints within a distribution region of the design variables and design parameters, and evaluates a solution with the approximation. Various types of intermediate models have been proposed for robust optimal design, e.g. worst-case analysis (WCA) (e.g. [1]), statistic analysis (SA) (e.g. [1]), design variable hyper sphere (DVHS) [2], most probable point (MPP) [3], a mini-max type robust optimal design [4] and so forth. These models assume a distribution region with a hyper rectangular parallelepiped, a hyper ellipsoid etc., approximate a function with a first-order or second-order Taylor series or a quadratic polynomial etc. within it and evaluate the feasibility, optimality and sensitivity of a solution with a worst-case, statistic features etc. They are developed from relatively simple ones toward complicated ones to obtain an accurate solution. A mini-max type robust optimal design [4] evaluates a solution with the bounding points of the functions within a distribution region directly to strictly guarantee the robustness of a design. For obtaining the bounding point strictly and efficiently, it estimates the bounding points by transforming the distribution region into a hyper sphere and approximating functions with quadratic polynomials. Further, in order to enhance the quality of the optimal design, this paper proposes a function regularization for constructing more precise approximations. This paper describes its concept and algorithm, and shows the applications to two numerical example problems to ascertain its effectiveness. 4. Robust optimal design and its development 4.1. Definition of optimal design problem Before discussing a robust optimal design, this paper assumes the formulation of the nominal optimal design problem as follows: find x = [ ] T x 1, x 2,, x nx that minimizes f (x, p) subject to g k (x, p) 0 (k = 1, 2,,n g ) The nominal optimal design seeks the best values of the design variables x that minimizes the objective function f (x, p) within the feasible region assigned by the constraints g k (x, p) under the given design parameters p = [ ] T [ p1, p 2,, p np. For a convenience, a notation of v = x T, p T ] [ ] T T = v 1, v 2,, v j,, v nv is introduced, where n v = n x + n p, v j = x j for 1 j n x and v j = p j nx for n x + 1 j n x + n p. (1) 1

2 Approximation Linear Quadratic Evaluation Worst-case Weighted-sum Evaluation Worst-case Weighted-sum WCA SI SA Mini-Max Type DVHS, MPP, etc. Hyper rectanglar parallelepiped Hyper ellipsoid Oblique hyper ellipsoid Variance Variance+Covariance Distribution region Figure 1: Development of intermediate model for robust optimal design 4.2. Intermediate models in robust optimal design A robust optimal design optimizes a solution evaluating its optimality, feasibility and sensitivity based on the variation of the functions within a distribution region. While its native meaning includes integral and probability calculation over design space, they are substituted with an intermediate model. In early days intermediate models assume an independent variation of variables and approximate the functions with linear expression. Worst-case analysis (WCA) and statistic analysis (SA) [1] are typical ones. The former assumes a distribution region with a hyper rectangular parallelepiped which has vertices at ( v o1 ± v 1, v o2 ± v 2, ) T, v ± v on v n v, where vo j is the average of v j and v j is its deviation, and approximates a constraint with a n v g linear expression. And it evaluates the feasibility with g i (v o ) + i v j=1 v j 0. The latter assumes a distribution region with a hyper ellipsoid which has its center point at v o = v o1, v o2,, v on j ( ) T v and radius κσ j for each axis, where κ is a parameter for the size of the variation and σ j is the standard deviation of v j. It approximates a constraints with a linear expression and evaluates the feasibility with g i (v o ) + κσ gi 0, where σ gi is the standard deviation of g i (v) under the linear approximation. Then, more precise intermediate models were proposed by introducing correlative relationship among variables or by approximating the functions using quadratic expressions. Figure 1 shows the development of intermediate models for a robust optimal design under the viewpoints of assumed distribution region (horizontal axis), evaluation of robustness (right part of the vertical axis) and approximation of functions (left part of the vertical axis). In the figure, covariance among variables are considered as well as their variances, which corresponds to the movement from Variance to Variance + Covariance in the horizontal axis. Worst-case tends to be used rather than weighted-sum for evaluating the robustness of a solution strictly, which corresponds to Weighted-sum and Worst-case in the vertical axis. And a quadratic polynomial is used rather than a linear expression for approximating a function with high fidelity, which corresponds to the movement from Linear to Quadratic in the vertical axis Mini-max type robust optimal design In order to evaluate the robustness of a solution strictly, the intermediate model proposed in this paper combines the developments in the three directions: assumption of a distribution region, approximation of the functions and evaluation of a solution. Namely, it evaluates the robustness of a solution with the bounding points of the objective function and constraints obtained by using their quadratic approximations considering the covariance among variables. This means that the proposed intermediate model locates in the upper-right part in Fig. 1. The proposed method formulates the robust optimal design with the following mini-max type optimization 2

3 (a) Nominal optimal design: v h (b) (c) Robust optimal design: Robust optimal design with function regularization: v v f(x) h f(x) h h ζ () f(x) h h f(x) ζ h ζ h (v) ζ Figure 2: Framework for evaluating a solution in optimal design problem: find that minimizes subject to x max v R(v f (v) o,σ,α ) max v R(v g k (v) 0 (k = 1, 2,,n g ) o,σ,α ) where R(v o, Σ, α ) is a distribution region of v. In the formulation, v vary with the mean vector v o = [ Exp [ ] [ ] ] T ] v 1, Exp v2,, Exp[vnv ], where Exp [v j is the expectation of v j, and the variance-covariance matrix Σ = Exp (v v o )(v v o ) T σ 21 σ 11 σ 12 σ [ ] 1nv σ 22 σ 2nv =......, where σ i j is a covariance between v i and v j. σ nv 1 σ nv 2 σ nv n v The distribution region generally forms an n v -dimensional ellipsoid which is formulated as follows: ( v v o ) T Σ 1 ( v v o ) χ 2 (n v, α) (3) where χ 2 (n v, α) is the value of chi-square distribution with n v -degrees of freedom that leaves probability α in the upper tail. The mini-max type robust optimal design problem includes a bounding point search problem of each function. It is formulated as a maximization problem as follows: find v that maximizes (4) subject to v R(v o, Σ, α ) where h( ) is a general expression of the objective function f ( ) or a constraint g k ( ). It is indispensable to obtain the bounding point efficiently and strictly to obtain the the robust optimal solution. Therefore, the proposed method transforms the distribution region into a hyper sphere and approximates a function with a quadratic response surface Enhancement of mini-max type robust optimal design with function regularization Since the fidelity of the obtained solution depends on the fidelity of the approximated functions in robust optimal design, this paper introduces a function regularization to obtain the bounding points strictly for a highly nonlinear function. Figure 2 shows the framework of a robust optimal design with function regularization in comparison with other optimization schema. A nominal optimal design evaluates a solution with a function value as shown in Fig. 2 (a). And an ordinary robust optimal design evaluates a solution with an approximated function which is constructed based on a set of sample points as shown in Fig. 2 (b). A function regularization transforms function values at sample points through a regularization filter h ζ so that the transformed values h ζ () can build a precise approximation h ζ (v) as shown in Fig. 2 (c). Figure 3 illustrates the effect of function regularization for a quadratic response surface approximation. In Fig. 3 (a) and (b), the horizontal axis is a variable v and vertical one is a function value. Figure 3 (a) shows the original function and its quadratic approximation constructed with a set of sample points. In the (2) 3

4 Sample point Boundary of feasible region hζ() hζ(v) Sample point Regularized sample point Original function Boundary of feasible region Quadratic approximation of original function Original function O Approximated bounding point True bounding point v O Quadratic approximation of regularized function hζ(v) Regularized function hζ() Approximated bounding point v True bounding point (a) Intermediate model of original function (b) Intermediate model of regularized function Figure 3: Effect of function regularization case that the original function is highly nonlinear, an approximation is different from the original function. This situation causes an inaccurate robust optimal solution, especially if the approximated bounding point of a constraint is far from that of the true one. On the other hand, Fig. 3 (b) shows the original function, its regularized one h ζ () and its quadratic approximation of the regularized one h ζ (v). h ζ (v) has high fidelity and the approximated bounding point obtained by using h ζ (v) is expected to be closer to the true one. 5. Algorithm and implementation of mini-max type robust optimal design with function regularization 5.1. Outline of the algorithm The concept of the proposed robust optimal design is itemized as follows: (i) In order to evaluate the feasibility and optimality of a solution precisely, a distribution region which generally forms an oblique ellipsoid is transformed to a hyper sphere and each function is approximated with a quadratic response surface. (ii) A regularization filter transforms function values at sample points so as that the approximated function fits the sample points. (iii) A regularization filter must be monotonically increasing against the function value within an assumed region, which is called regularization region, so as that one-to-one correspondence relationship is guaranteed. (iv) A regularization region should be narrow enough in the later stages of the optimization in order to precisely obtain the optimal solution, and it should be wide enough so as to construct an effective filter which is valid in searching wide region in the earlier stages of the optimization. Figure 4 shows the whole algorithm of the proposed method. First, the algorithm initializes a regularization region which center point is at the initial solution and constructs a regularization filter for each function based on the function values at sample points (from (a) to (d) in Fig. 4). Then, it optimize a solution by iteratively updating a tentative solution based on the function values at their bounding points which are obtained with quadratic polynomials built by the regularized function values at sample points (from (e) to (l) in Fig. 4). The procedure are repeated predefined times reducing the size of a regularization region Representation of distribution region with hyper sphere For solving a bounding point search problem strictly and efficiently, the distribution region is transformed into a hyper sphere by the technique of design variation hyper sphere (DVHS) [2, 4]. First, Eq. (3) is reformed as follows by a diagonalization operation: v T Σ 1 v χ 2 (n v, α) (5) 4

5 Optimize the critical optimality over the design space Start (a) Build suitabilization filters (e) Generate a tentative design (f) Test the tentative design (g) Regularize a region (h) Build quadratic polynomial approximations Build regularization filters over regularization region (b) Initialize or shrink suitabilization region (c) Execute sampling over regularization region (d) Determine filter parameters Search the extremes of functions within a distribution region (j) Discriminate the type of an extreme in the distribution region (i) Calculate its optimality Converged? (k) Calculate the inner extreme analytically (l) Search the boundary extreme numerically Satisfy terminal condition? End The feasibility of a distribution region and the inferior extreme of the nominal objective within it Figure 4: Algorithm of mini-max type robust optimal design with function regularization where v = T 1 ( v v o ) and Σ = diag { } λ 1,,λ nv by using T1 = [ ] T e 1, e 2,, e nv, in which e j is a unit eigenvector of matrix Σ, and λ j is an eigenvalue of matrix Σ. Then, Eq. (5) is reformed by an isoparameterization matrix operation: v T v χ 2 (n v, α) (6) ( where v is determined by v = T 2 v 1, by using T 2 = diag,, 1 λ1 λn v ). These equations transform the distribution region from an oblique hyper ellipsoid (Fig. 5 (a)) to an orthogonal hyper ellipsoid which has half axes λ j χ 2 (n v, α) (Fig. 5 (b)) and then transform it to a hyper sphere which has radius χ 2 (n v, α) (Fig. 5 (c)) Quadratic function approximation The quadratic polynomial approximation of each function is generated by the least square regression method around v o. The method approximates each function so as that the approximation is exactly matched with the original function at v o, which corresponds to v o. Since engineering optimal design problems tend to have the optimal solution on the boundary of the feasible region, a set of sample points are arranged on the boundary of the hyper sphere isoparametrically to approximate with high fidelity on the boundary. Finally, the bounding point search problem is formulated as follows: find v that maximizes h(v ) = h(v o) + where β i and β i j are unknown coefficients. n v i=1 subject to v T v χ 2 (n v,α ) β i v n v n v i + i=1 j=i β i j v i v j (7) 5

6 (a) v 2 v ο2 + κ σ 2 v ο2 v ο2 κ σ 2 Ο v 2 oblique hyper ellipsoid v 1 v ο1 κ σ 1 v ο1 v ο1 + κ σ 1 v 1 (b) v 2 λ2 χ2(nv, α) λ1 χ2(nv, α) ( v - vo ) T -1 Σ ( v - vo ) v v χ2(nv, α) χ2(nv, α) v T -1 Σ v χ2(nv, α) diagonalize : v = T 1 ( v - vo ) v 1 orthogonal hyper ellipsoid (c) hyper sphere isoparameterize : v" = T 2 v Figure 5: Coordinate transformation on the design space T v 2 χ2(nv, α) χ2(nv, α) v Search of the bounding point of a function The bounding point search problem Eq. (7) is a nonlinear constrained optimization problem and has either inner or boundary optimum. The proposed method firstly discriminates the type of bounding point, and then obtains the bounding point analytically or numerically. (1) Inner optimum When the objective function is upward convex, that is, 2 h(v ) < 0 and v = [ 2 h(v o) ] 1 h(v o), the pole of h(v ), is inside of the distribution region, Eq. (7) has the inner optimum v. (2) Boundary optimum Otherwise, the bounding point is on the boundary of the hyper sphere. Since several local optima may exist on the boundary, the objective function is converted to the following form so as that the function ĥ(v ) is upward convex and ĥ(v ) has the unique optimum. ĥ(v ) = h(v ) 1 2 s v T I v = h(v o) + h(v o)v v T ( 2 h(v o) s I ) v (8) where I is an n v -dimensional identify matrix. By determinating an appropriate s, h(v ) can be converted to an upward convex type and has the maximum point within the hyper sphere on its boundary. This is because, since 1 2 s v T I v takes the same value on the boundary of a hyper sphere, ĥ(v ) takes the maximum where h(v ) takes the maximum on boundary of the hyper sphere. The condition is summarized by the following equations: ( 2 h(v o) s I ) v = h(v o) (9) v = χ 2 (n v,α ) (10) s > 0 (11) h(v o) v s I 0 (12) where, is a Euclid norm. Eq. (9) through Eq. (11) mean that the stationary point is on the boundary of the hyper sphere and Eq. (12) means that the matrix h(v o) v s I is semi-negative definite so as that ĥ(v ) is upward convex Adjustment of regularization region For defining a regularization filter, a regularization region is set as a hyper rectangular parallelepiped which has its center point at a tentative solution v. The size of the region is reduced over K iterations. At k-th ( k = 1,2,,K ) iteration, its upper and lower bounds are set to v i ±C (k) κσ i (i = 1, 2,, n v ), where σ i is the standard deviation of v i, κ is the factor for the size of the distribution region and C (k) is the factor for the size of the regularization region. C (k) is reduced by multiplying the factor C (k+1) /C (k) = K C (0) at every iteration. At the final iteration, K-th iteration, C (K) = 1 and the regularization region is circumscribed with the distribution region. 6

7 5.6. Construction of regularization filter Fourier series is used for building a regularization filer with a small number of parameters. In order to guarantee that a regularized filter is monotonically increasing, regularization filter is configured as a superposition of linear transformation and Fourier series which includes only sine curves. A regularization filter forms as follows: ) h ζ (α S ζ,h S () = h S () + N n=1 b n sinnπ h S () (13) where h S () is a linear function which transforms the minimum h min and the maximum h max of in the regularization region to 0 and 1, respectively. N is the order of Fourier series, and α ζ = [ b 1, b 2,,b N ] T are its ) ( )) coefficients. Under this expression, a regularization filter is defined as h ζ (α ζ, = h 1 S (h ζ α S ζ,h S (). In order to construct a quadratic polynomial h ζ (α ζ,v) which has high fidelity against the regularized function, is determined by solving the following mathematical programming problem: α ζ find that minimizes E ζ = 1 n ζ subject to α ζ = [ b 1, b 2,,b N ] T n ζ k=1 [ h 1 S ( )) ] 2 (h ζ α S ζ,h S () h ζ (α ζ,v k ) h ζ (α S ζ,h S ()) 0 over 0 h h S () S () 1 where n ζ is the number of sample points located randomly within a regularization region. The constraint means that a regularization filter must be monotonically increasing within a range of 0 h S () 1, which is replaced with an algebraic form in computation. Since it is difficult to obtain accurate h max and h min values in the regularization region, they are substituted with h max = hmax s + h (hmax s hmin s ), h min = h min s h (h max s hmin s ), where hmax s and hmin s is the maximum and minimum function values at all sample points, respectively, where h ( 0 < h < 0.5 ) is the margins against them. Further, in the case that is a constraint, the regularization filter is compensated so as that the boundary of the constraint, namely the points at which function value is 0, must not move before and after regularization. A regularization filter for a constraint is redefined as follows: ) ( )) h ζ (α ζ,h 0, = h 1 S (h ζ α S ζ,h S () h 0 (15) where h 0 is the parameter for compensation. It is determined as follows: (i): if hmin s 0 h max s, h 0 is determined so as that the points at which the function takes 0 matches before and after regularization, (ii): if hmin s > 0, h 0 is determined so as that the points at which the function takes hmin s matches before and after regularization, (iii): otherwise, namely hmax s < 0, h 0 is determined so as that the points at which the function takes hmax s matches before and after regularization. 6. Numerical examples 6.1. Two-dimensional optimization problem The proposed mini-max type formulation for a robust optimal design is applied to a two-dimensional optimization problem in comparison with other conventional robust optimal design to show the its effectiveness visually. The optimization problem is formulated as follows: find x = [ ] T x 1, x 2 that minimizes f (x) = x 3 1 sin(x 1 + 4) + 10x x 1 + 5x 1 x 2 + 2x x subject to g 1 (x) = x x 1 x 1 sinx 1 + x g 2 (x) = log(0.1x ) + x 2 e x 1 +3x x [ Standard deviations ] of the design variables are σ = [ 0.13, 0.13 ] T and their variance-covariance matrix is Σ = Figure 7 shows the contour plots of the objective function and the constraints (14) (16) 7

8 x2 0.0 Nominal optimal design Inferior extreme of the nominal -1.0 f ( x ) objective Distribution region g1 ( x ) Feasible region g2 ( x ) x1 Bounding point of g2 0 Bounding point of g1 0 Robust optimal solution by the proposed formulation Figure 6: Contour plot of two-dimensional example problem Table 1: Comparison of robust optimality formulations Case 1 Case 2 Case 3 Case 4 Case 5 Nominal Method of feasibility analysis SQ SL SQ SL SQ optimi- Method of optimality & sensitivity analysis SQ SQ WS WS SQ zation Method for building quadratic polynomials R R R R D Optimized design x variables x Extremes of Values in g 1 (x ) functions formulated g 2 (x ) within the model f (x ) distributed Values g 1 (x ) region in real g 2 (x ) behavior f (x ) Abbr.: SL : Strict analysis with linear approximation, R: Least square regression as response surface, SQ : Strict analysis with quadratic polynomial approximation, D: Direct differentiation at the point. WS: Weighted-sum of mean value and standard deviation of the nominal objective Table 1 shows the optimization results. In the table, Case 1 is the proposed method, Case 2 evaluates the feasibility with the bounding point under linear approximations of constraints (SL), Case 3 evaluates the optimality and sensitivity by the weighted-sum of the mean and standard deviation of the objective function (WS), Case 4 is a combination of Case 2 and Case 3. Case 5 is similar with Case 1 but it approximates the functions with second-order Taylor series using the direct differentiation at the solution instead of response surfaces. The difference on the values of g 1 (x ) and g 2 (x ) in real behavior between Case 1 and Case 2, and between Case 3 and Case 4 indicates that strict analysis with quadratic polynomial approximation (SQ) is superior to one with linear approximation (SL) for feasibility analysis. While this difference is caused by nonlinearity of g 2 (x) around g 2 (x) = 0, which is observed in Fig. 7, this result highlights the advantage of the proposed formulation in the cases in which constraints are nonlinear. The difference on the values of f (x ) in real behavior between Case 1 and Case 3, and between Case 2 and Case 4 indicates that strict analysis with quadratic polynomial approximation (SQ) is superior to one with the conventional method with the weighted-sum of mean value and standard deviation for optimality and sensitivity analysis (WS). While this difference is caused by nonlinearity of f (x), this result highlights the advantage of the proposed formulation in the cases in which the nominal objective is nonlinear. The comparison between Case 1 and Case 5 indicates that local approximation of functions performs better that exact values of functions at the point for robust design optimization. This is obviously because the critical 8

9 x Distribution region Robust optimal solution Feasible region x1 Bounding point of g2(x) 0 Bounding point of g1(x) 0 f ( x ) g1 ( x ) g2 ( x ) Figure 7: Contour plots of two-dimensional highly nonlinear example problem points on strict robust optimality are on the boundary of a distributed region rather than its center point. In summary, the numerical result on a two-dimensional example validates the effectiveness of the proposed mini-max type formulation in seeking the strictly robust optimal design Two-dimensional optimization problem with highly nonlinear functions The proposed mini-max type robust optimal design with function regularization is applied to a two-dimensional highly nonlinear optimization problem formulated below: find that minimizes x = [ x 1, x 2 ] T f (x) = 2 subject to g 1 (x) = 40 (x 1 3) 2 + (3 x 2 ) 2 50 (x ) 2 (x 2 5) 4 0 g 2 (x) = log(0.1x ) + x 2 e (x x 2 4) x [ Standard deviations of ] the design variables are σ = [ 0.13, 0.13 ] T and their variance-covariance matrix is Σ = Figure 7 shows the contour plots of the objective function and the constraints. g (x) is highly nonlinear as shown in the figure. In the execution, the number of iteration K = 5, the initial factor for regularization region C (0) = 3.0, the order of Fourier series in regularization filter N = 2, the number of sample points n ζ = 400, the margins for compensating h max and h min by hmax s and hmin s, h = 0.33 are used. Table 2 shows the comparison of optimization results to ascertain the effectiveness of function regularization. In the table, the second and the third columns are the results obtained without function regularization while the forth and fifth columns are ones with function regularization. And the second and the forth columns are the function values at the optimum that are calculated with quadratic approximations while the third and the fifth columns are the function values at the optimum that are calculated with exact functions. The result shows that the highly nonlinear constraint g 2 (v) at the optimal solution is less violated with function regularization than that without function regularization. Figure 8 (a) and (b) show the quadratic approximation g 2 (v) and g 2ζ (α ζ,g 20,g 2 (v)) in comparison with their original function g 2 (v), respectively. The differences of contours shown in Fig. 8 (b) is smaller than that of Fig. 8 (a). It is ascertained that function regularization enhances the approximation. 7. Concluding remarks This paper proposed a mini-max type robust optimal design and its enhancement with a function regularization. The proposed method transforms a distribution region into a hyper sphere and approximates the functions with quadratic polynomials. This procedure enables to obtain the bounding points strictly and efficiently. Further, function regularization makes it possible to build a more precise approximation even if a function is highly nonlinear (17) 9

10 Table 2: Comparison of robust optimal design with/without function regularization Robust optimum without function regularization Vaules in Vaules in formulated real model behavior Robust optimum with function regularization Vaules in Vaules in formulated real model behavior x x g 1 (x ) g 2 (x ) f (x ) x g2=0 g2= x1 x g2=0 g2 =0 ζ (a) Original function (b) Regularized function x1 Original or regularized function Quadratic approximation Regularization region Distribution region Optimal design Figure 8: Approximation of g 2 (v) at the final iteration and the form of an approximation is different from that of the original one. This paper introduced the concept of function regulation and its implementation to a mini-max type robust optimal design. Finally, its validity was ascertained through two example problems. However, a function regularization requires much computational cost due to mainly sampling cost for building regularization filters. Reducing the computational cost is our future work. It is expected that a cumulative function approximation [5] is a great help for reducing the computational cost [6]. 7. References [1] Parkinson, A., Robust Mechanical Design Using Engineering Models, Transactions of the ASME, Journal of Mechanical Design, 1995, 117, [2] Zhu, J. and Ting K.-L., Performance Distribution Analysis and Robust Design, Transactions of the ASME, Journal of Mechanical Design, 2001, 123, [3] Du, X. and Chen, W., Towards a Better Understanding of Modeling Feasibility Robustness in Engineering Design, Transactions of the ASME, Journal of Mechanical Design, 2000, 122, [4] Hirokawa, N. and Fujita, K., Mini-max Type Formulation of Strict Robust Design Optimization under Correlative Variation, Proceedings of the 2002 ASME Design Engineering Technical Conferences, 2002, Paper Number DETC2002/DAC [5] Hirokawa, N., Fujita, K. and Iwase, T., Voronoi Diagram Based Blending of Quadratic Response Surfaces for Cumulative Global Optimization, Proceedings of 9th AIAA/ISSMO Symposium on Multi-Disciplinary Analysis and Optimization, 2002, Paper Number AIAA [6] Hirokawa, N., Fujita, K. and Ushiro, T., Computation Cost Saving of Robust Optimal Design by Cumulative Function Approximation, Proceedings of the Third China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems (CJK-OSM3), 2004,