Research
Numerical Optimization
home > Research > Design Optimization > Numerical Optimization
General Optimization Tool
B_INOPL is based on Augmented Lagrange Multiplier (ALM) Method[1]. In order to elevate the efficiency, the Projected Linearization Method (PLM) and Variable-Order Polynomial Approximation (VOPA) employed for line search[2-4]. For large scaled structural optimization, Approximate Augmented Lagrange Multiplier (AALM) is supported[5]. In this method, the original optimization problem is approximated using linear, reciprocal or conservative approximations.
Especially, unlike the conventional numerical optimizers, B-INOPL can directly solve the parameter (time or frequency) dependent optimization problems with ALM or AALM[2-7]. Also, in order to avoid the expensive computations of gradients, an approximate gradient scheme[8] such as Broyden¡¯s method is algorithmically implemented. Also, in order to direct solve the min-max formulation of multi-objectives, the approximate generalized gradient values are employed[9].
B_INOPL uses U_INOPL to minimize the transformed Augmented Lagrangian sequentially. Following nonlinear problem[10] has 2000 design variables and 2000 inequality constraints. B_INOPL uses 9 evaluations of gradient and 33 evaluations of function to obtain f(x^* )= -1.99680E+03.
Reference
[1] M-S. Kim, H.-S. Kim, J.-W. Lee, and D.-H. Choi, 'Computational Enhancement to the Augmented Lagrange Multiplier Multiplier Method for the Constrained Nonlinear Optimization problems", KSME (in Korean), Vol. 15, No. 2, pp. 544-556, 1991.

[2] M.-S. Kim and D.-H. Choi, ¡°Multibody Dynamic Response Optimization with ALM and Approximate Line Search¡±, Multibody System Dynamics, Vol. 1, pp. 47-64, 1997.

[3] M.-S. Kim and D.-H. Choi, ¡°Min-Max Dynamic Response Optimization of Mechanical Systems using Approximate Augmented Lagrangian¡±, International Journal for Numerical Method in Engineering, Vol. 43, pp. 549-564, 1998.

[4] M.-S. Kim and D.-H. Choi, ¡°Dynamic Response Optimization Using Approximate Line Search¡±, KSME, Vol. 22, No. 4, pp. 811-825, 1998.

[5] M.-S. Kim and D.-H. Choi, ¡°Dynamic Response Optimization Using Approximate Augmented Lagrange Multiplier Method¡±, KSME, Vol. 22, No. 7, pp. 1135-1147, 1998.

[6] M.-S. Kim and D.-H. Choi, ¡°Direct Treatment of a Max-Value Cost Function in Parametric Optimization¡±, International Journal for Numerical Method in Engineering, Vol. 50, pp.169-180, 2001.

[7] M.-S. Kim and D.-H. Choi, ¡°A New Penalty Parameter Update Rule in the Augmented Lagrange Multiplier Method for Dynamic Response Optimization¡± KSME International Journal, Vol.14, No. 10, pp. 1122-1130, 2000.

[8] M.-S. Kim and D.-H. Choi, ¡°An Efficient Dynamic Response optimization Using design Sensitivity Approximated within Estimated Confidence Radius¡±, KSME International Journal, Vol. 15, No. 8, 2001, pp. 413-421.

[9] M.-S. Kim, D.-H. Choi and Yll Hwang, ¡°Composite Non-Smooth Optimization using the Approximate generalized Gradient¡±, Journal of Optimization Theory and Applications, Vol.112, No. 1, pp. 145-165, 2002.

[10] J. Qin and D.T. Nguyesn, "Generalized Exponential Penalty Function For Nonlinear programming", AIAA-94-1360-C, 1994.