Abstract
In this paper, two novel algorithms are designed for solving biobjective optimization engineering problems. In order to obtain the optimal solutions of the biobjective optimization problems in a fast and accurate manner, the algorithms, which have combined Newton's method with Neumann series expansion as well as the weighted sum method, are applied to deal with two objectives, and the Pareto optimal front is achieved through adjusting weighted factors. Theoretical analysis and numerical examples demonstrate the validity and effectiveness of the proposed algorithms. Moreover, an effective biobjective optimization strategy, which is based upon the two algorithms and the surrogate model method, is developed for engineering problems. The effectiveness of the optimization strategy is proved by its application to the optimal design of the dummy head structure in the car crash experiments.
Highlights
It is very important to research on the multiobjective optimization problems in the engineering designs
In multiobjective optimization problem, Newton method combined with weighted sum method is chosen as the main calculation algorithm
When there are only two objective functions, a biobjective optimization algorithm is established in this paper by introducing the technique of Neumann Series Expansion (NSE) [21] to Newton weighted sum method
Summary
It is very important to research on the multiobjective optimization problems in the engineering designs. By using the well-known BFGS method and the idea of [8], the authors had proven that quasi-Newton’s method for multiobjective optimization converges superlinearly to the solution of the given problem, if all functions involved are twice continuously differentiable and strongly convex The advantage of this method, compared to Newton’s approach, is that the approximation of Hessian matrices is usually reasonably faster than their actual evaluation. The early usual weighted sum method transforms multiple objectives into an aggregated objective function by multiplying each objective function by a weighted factor and adding them up It has two drawbacks: difficulty to obtain Pareto optimal solutions uniformly and failure to solve nonconvex problems [16,17,18].
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