Document Type : Article
Authors
School of Mechanical Engineering Iran Univers
Abstract
The aim of this paper is to enhance the capability of a robust optimization method, which has so far been only employed for cases involving inequality constraints, in order to consider nonlinear equalities as well. In extending this algorithm, the main useful features of the original, as outlined below, have been maintained to the best extent possible.Categorizing optimization methods (those based on the generation of a trajectory of points converging to an optimum in the feasible part of the design space, as defined by the constraints) are known as methods of centers. One of the robust methods of centers is known as the inscribed hyperspheres method. In this method, the objective function and constraints are linearized about the current design point using Taylor series expansion. The useable side of the linearized objective function and the constraints provide a closed polytope in usable-feasible design space. Then, the center of the largest hypersphere inscribed inside this polytope is found by the linear programming (LP) algorithm. The center of this hypersphere is selected as the new design point and the process continues until desired convergence to the optimum is achieved. Thus, the sequence of generated design points, all in the feasible space defined by the constraints and objective function, lie in nearly equidistant positions from them, as a series of center points.Since, in this manner, the trajectory of improving design points funnels down the middle of the design space, some important features of the procedure can be discussed, as given below. Firstly, from an engineering practical point of view, it is highly desirable to converge to the optimum through the feasible design space with few or no constraint violations. Secondly, since the definition of a closed polytope in an n dimensional space can be defined with as few as (n+1) independent constraints, only the near active constraints that adequately define the interior of the feasible-useable design polytop need be considered in each cycle; which, in other words, enhances efficiency in practical applications where a large multitude of constraints must be satisfied. Thirdly, taking the center of the hypersphere as the new design point in each cycle results in maintaining a uniform margin of safety (i.e. generated points being approximately equidistant, with respect to nearly active constraints) in the sequence of points generated in the optimization cycles. Fourthly, as the movement towards the optimum takes place in the middle of feasible space, it is less sensitive to the errors in linearization of constraints, with respect to their exact definition, while only one complete analysis and constraint evaluation need be made in each cycle of advancement towards the optimum, which greatly enhances the efficiency.The inscribed hyperspheres method has so far suffered from the drawback of not being able to handle equalities directly in its formulation. The work presented here is a direct approach to remedy the situation and extend the applicability of this useful method to a larger gamut of engineering problems, where equality constraints are inherently part of the problem. This paper deals with extension and generalization of the method for inclusion of linear and/or nonlinear equality constraints, in addition to inequality constraints, for general applications. Therefore, three techniques are presented and discussed, which, while achieving this goal, also maintain the useful characteristics of the original inscribed hypersphere method outlined above. In the first technique presented, at the end of each cycle of the optimization process, the center of the hypersphere is projected onto the intersection of linearized equality constraint surfaces. In the second technique, the center of the hypersphere is forced to lie in the subspace formed by the intersection of the equalities. Therefore, the linearization of nonlinear equality constraints necessitates a restoration movement at the end of each cycle. In the third technique, in inscribing the hyper-sphere in feasible-usable design space, it is forced to be tangent to all the linearized surfaces of nonlinear equality constraints. Whereupon, as the optimization steps proceed, the radii of hyperspheres shrink to near zero while the center points maintain their position on the intersection of equality constraints. Results of applied numerical examples from mathematical and engineering optimization literature show the efficacy and advantages of the proposed method.
Keywords
)