عنوان مقاله [English]
In this paper, a method is presented based on approximating the objective function and constrains in optimization problems in conjunction with Lagrange multiplier method. Besides, an algorithm is developed in this relation. Instead of linear or parabola terms employed in Taylor expansion to proceed cautiously with short step lengths, in the method presented here, an arc with constant curvature is used that makes it possible to proceed with relatively longer step lengths. For an n-dimensional optimization problem, the spheres are n-dimensional too. The radius of curvature and center of spheres can be determined at the tangent point between each function and its corresponding sphere. For the objective function, the parameters of sphere are determined at the reference point obtained by Lagrange equations, but for the constraints, first the reference point is returned to the surface of all the active constraints, then at the points on the constraints, the approximate parameters are calculated. Hence every computational step includes two parts: the determination of the reference point and returning it to the surface of active constraints. The criterion for returning to the active constraint is based on the shortest distance of the reference point from each of the active constraint, because the reference point is the output of optimization represented by Lagrange equations and so is the basis of the calculations. For returning the reference point to the active constraint, only one scalar variable is involved in the calculations. The introduction of the n-dimensional spheres both reduces the number of and simplifies the form of equations that need to be solved simultaneously to determine the optimum point and Lagrange multipliers at each optimization step, because the unknowns are now the Lagrange multipliers. This results in a significant reduction in computation time. Separating the design variables from Lagrange equations, the time of calculations may be saved for the loops of time-history analysis in the optimal seismic design. The method is applied to the optimization of two major parts of the lateral resistance systems, and the results are compared with those from penalty method. Considerable reduction of solution time is observed.
The following remarks and conclusions are pertinent with regard to the formulation and the results presented in the paper:
(1) The structural examples solved by the method presented here have also been solved by the exterior penalty method where both methods have provided exactly the same optimum solutions.
(2) The proposed method does not depend on the convexity or the concavity of the constraints or the objective function, because the radius of sphere that indicates the curvature is directly utilized at each computational step.
(3) Similar to the other optimization methods, the convergence behaviour and success of the proposed method depends on the starting point.
(4) Separating the design variables from Lagrange equations, the time of calculations may be saved for the loops of time-history analysis in the optimal seismic design.
(5) In this method, the criterion for the returning to the active constraint was utilized that is based on the shortest distance of the reference point from each of the active constraint (residual error); however, the methods based on Taylor expansion do not consider the minimization of residual error in every computational step.
(6) Lagrange multipliers related to the constraints of lower and upper bound in the structural optimization problems can be decoupled from the others by the proposal method.
(7) Though of the time of computation to converge, the final solution is of great importance and should be discussed in detail. The space limitation does not let a proper comparison of convergence behaviour between the presented method and the exterior penalty method. Hence this issue has been postponed to a follow-up paper, but just qualitatively, the presented method has shown the convergence faster.