• Title/Summary/Keyword: Efficient Quadratic Approximation

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Efficient Approximation Method for Constructing Quadratic Response Surface Model

  • Park, Dong-Hoon;Hong, Kyung-Jin;Kim, Min-Soo
    • Journal of Mechanical Science and Technology
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    • v.15 no.7
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    • pp.876-888
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    • 2001
  • For a large scaled optimization based on response surface methods, an efficient quadratic approximation method is presented in the context of the trust region model management strategy. If the number of design variables is η, the proposed method requires only 2η+1 design points for one approximation, which are a center point and tow additional axial points within a systematically adjusted trust region. These design points are used to uniquely determine the main effect terms such as the linear and quadratic regression coefficients. A quasi-Newton formula then uses these linear and quadratic coefficients to progressively update the two-factor interaction effect terms as the sequential approximate optimization progresses. In order to show the numerical performance of the proposed method, a typical unconstrained optimization problem and two dynamic response optimization problems with multiple objective are solved. Finally, their optimization results compared with those of the central composite designs (CCD) or the over-determined D-optimality criterion show that the proposed method gives more efficient results than others.

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An efficient algorithm for the non-convex penalized multinomial logistic regression

  • Kwon, Sunghoon;Kim, Dongshin;Lee, Sangin
    • Communications for Statistical Applications and Methods
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    • v.27 no.1
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    • pp.129-140
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    • 2020
  • In this paper, we introduce an efficient algorithm for the non-convex penalized multinomial logistic regression that can be uniformly applied to a class of non-convex penalties. The class includes most non-convex penalties such as the smoothly clipped absolute deviation, minimax concave and bridge penalties. The algorithm is developed based on the concave-convex procedure and modified local quadratic approximation algorithm. However, usual quadratic approximation may slow down computational speed since the dimension of the Hessian matrix depends on the number of categories of the output variable. For this issue, we use a uniform bound of the Hessian matrix in the quadratic approximation. The algorithm is available from the R package ncpen developed by the authors. Numerical studies via simulations and real data sets are provided for illustration.

Efficient Mechanical System Optimization Using Two-Point Diagonal Quadratic Approximation in the Nonlinear Intervening Variable Space

  • Park, Dong-Hoon;Kim, Min-Soo;Kim, Jong-Rip;Jeon, Jae-Young
    • Journal of Mechanical Science and Technology
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    • v.15 no.9
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    • pp.1257-1267
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    • 2001
  • For efficient mechanical system optimization, a new two-point approximation method is presented. Unlike the conventional two-point approximation methods such as TPEA, TANA, TANA-1, TANA-2 and TANA-3, this introduces the shifting level into each exponential intervening variable to avoid the lack of definition of the conventional exponential intervening variables due to zero-or negative-valued design variables. Then a new quadratic approximation whose Hessian matrix has only diagonal elements of different values is proposed in terms of these shifted exponential intervening variables. These diagonal elements are determined in a closed form that corrects the typical error in the approximate gradient of the TANA series due to the lack of definition of exponential type intervening variables and their incomplete second-order terms. Also, a correction coefficient is multiplied to the pre-determined quadratic term to match the value of approximate function with that of the previous point. Finally, in order to show the numerical performance of the proposed method, a sequential approximate optimizer is developed and applied to solve six typical design problems. These optimization results are compared with those of TANA-3. These comparisons show that the proposed method gives more efficient and reliable results than TANA-3.

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Optimal Design of Helicopter Tailer Boom (헬리곱터 꼬리 날개의 최적 설계)

  • 한석영
    • Proceedings of the Korean Society of Machine Tool Engineers Conference
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    • 1999.10a
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    • pp.419-424
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    • 1999
  • In this paper, the comparison of the first order approximation schemes such as SLP (sequential linear programming), CONLIN(convex linearization), MMA(method of moving asymptotes) and the second order approximation scheme, SQP(sequential quadratic programming) was accomplished for optimization of and nonlinear structures. It was found that MMA and SQP(sequential quadratic programming) was accomplished for optimization of and nonlinear structures. It was found that MMA and SQP are the most efficient methods for optimization. But the number of function call of SQP is much more than that of MMA. Therefore, when it is considered with the expense of computation, MMA is more efficient than SQP. In order to examine the efficiency of MMA for complex optimization problem, it was applied to the helicopter tail boom considering column buckling and local wall buckling constraints. It is concluded that MMA can be a very efficient approximation scheme from simple problems to complex problems.

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Progressive Quadratic Approximation Method for Effective Constructing the Second-Order Response Surface Models in the Large Scaled System Design (대형 설계 시스템의 효율적 반응표면 근사화를 위한 점진적 이차 근사화 기법)

  • Hong, Gyeong-Jin;Kim, Min-Su;Choe, Dong-Hun
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.24 no.12
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    • pp.3040-3052
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    • 2000
  • For effective construction of second-order response surface models, an efficient quad ratic approximation method is proposed in the context of trust region model management strategy. In the proposed method, although only the linear and quadratic terms are uniquely determined using 2n+1 design points, the two-factor interaction terms are mathematically updated by normalized quasi-Newton formula. In order to show the numerical performance of the proposed approximation method, a sequential approximate optimizer is developed and solves a typical unconstrained optimization problem having 2, 6, 10, 15, 30 and 50 design variables, a gear reducer system design problem and two dynamic response optimization problems with multiple objectives, five objectives for one and two objectives for the other. Finally, their optimization results are compared with those of the CCD or the 50% over-determined D-optimal design combined with the same trust region sequential approximate optimizer. These comparisons show that the proposed method gives more efficient than others.

Robust Optimal Design Method Using Two-Point Diagonal Quadratic Approximation and Statistical Constraints (이점 대각 이차 근사화 기법과 통계적 제한조건을 적용한 강건 최적설계 기법)

  • Kwon, Yong-Sam;Kim, Min-Soo;Kim, Jong-Rip;Choi, Dong-Hoon
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.26 no.12
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    • pp.2483-2491
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    • 2002
  • This study presents an efficient method for robust optimal design. In order to avoid the excessive evaluations of the exact performance functions, two-point diagonal quadratic approximation method is employed for approximating them during optimization process. This approximation method is one of the two point approximation methods. Therefore, the second order sensitivity information of the approximated performance functions are calculated by an analytical method. As a result, this enables one to avoid the expensive evaluations of the exact $2^{nd}$ derivatives of the performance functions unlike the conventional robust optimal design methods based on the gradient information. Finally, in order to show the numerical performance of the proposed method, one mathematical problem and two mechanical design problems are solved and their results are compared with those of the conventional methods.

Design Optimization Using Two-Point Diagonal Quadratic Approximation(TDQA) (이점 대각 이차 근사화(TDQA) 기법을 적용한 최적설계)

  • Kim, Min-Soo;Kim, Jong-Rip;Choi, Dong-Hoon
    • Proceedings of the KSME Conference
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    • 2001.06c
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    • pp.386-391
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    • 2001
  • This paper presents a new two-point approximation method based on the exponential intervening variable. To avoid the lack of definition of the conventional exponential intervening variables due to zero- or negative-valued design variables the shifting level into each exponential intervening variable is introduced. Then a new quadratic approximation, whose Hessian matrix has only diagonal elements of different values, is proposed in terms of these intervening variables. These diagonal elements are computed in a closed form, which correct the typical error in the approximate gradient of the TANA series due to the lack of definition of exponential type intervening variables and their incomplete second-order terms. Also, a correction coefficient is multiplied to the pre-determined quadratic term to match the value of approximate function with that of the original function at the previous point. Finally, the authors developed a sequential approximate optimizer, solved several typical design problems used in the literature and compared these optimization results with those of TANA-3. These comparisons show that the proposed method gives more efficient and reliable results than TANA-3.

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Design Optimization Using Two-Point Diagonal Quadratic Approximation (이점 대각 이차 근사화 기법을 적용한 최적설계)

  • Choe, Dong-Hun;Kim, Min-Su;Kim, Jong-Rip;Jeon, Jae-Yeong
    • Transactions of the Korean Society of Mechanical Engineers A
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    • v.25 no.9
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    • pp.1423-1431
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    • 2001
  • Based on the exponential intervening variable, a new two-point approximation method is presented. This introduces the shifting level into each exponential intervening variable to avoid the lack of def inition of the conventional exponential intervening variables due to zero-or negative-valued design variables. Then a new quadratic approximation whose Hessian matrix has only diagonal elements of different values is proposed in terms of these intervening variables. These diagonal elements are determined in a closed form that corrects the typical error in the approximate gradient of the TANA series due to the lack of definition of exponential type intervening variables and their incomplete second-order terms. Also, a correction coefficient is multiplied to the pre-determined quadratic term to match the value of approximate function with that of the previous point. Finally, in order to show the numerical performance of the proposed method, a sequential approximate optimizer is developed and applied to solve six typical design problems. These optimization results are compared with those of TANA-3. These comparisons show that the proposed method gives more efficient and reliable results than TANA-3.

A SUCCESSIVE QUADRATIC PROGRAMMING ALGORITHM FOR SDP RELAXATION OF THE BINARY QUADRATIC PROGRAMMING

  • MU XUEWEN;LID SANYANG;ZHANG YALING
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.4
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    • pp.837-849
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    • 2005
  • In this paper, we obtain a successive quadratic programming algorithm for solving the semidefinite programming (SDP) relaxation of the binary quadratic programming. Combining with a randomized method of Goemans and Williamson, it provides an efficient approximation for the binary quadratic programming. Furthermore, its convergence result is given. At last, We report some numerical examples to compare our method with the interior-point method on Maxcut problem.

Development of an Efficient Line Search Method by Using the Sequential Polynomial Approximation (순차적 다항식 근사화를 적용한 효율적 선탐색기법의 개발)

  • 김민수;최동훈
    • Transactions of the Korean Society of Mechanical Engineers
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    • v.19 no.2
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    • pp.433-442
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    • 1995
  • For the line search of a multi-variable optimization, an efficient algorithm is presented. The algorithm sequentially employs several polynomial approximations such as 2-point quadratic interpolation, 3-point cubic interpolation/extrapolation and 4-point cubic interpolation/extrapolation. The order of polynomial function is automatically increased for improving the accuracy of approximation. The method of approximation (interpolation or extrapolation) is automatically switched by checking the slope information of the sample points. Also, for selecting the initial step length along the descent vector, a new approach is presented. The performance of the proposed method is examined by solving typical test problems such as mathematical problems, mechanical design problems and dynamic response problems.