Abstract
A new sequential method based on multi-resolution approximation is proposed for solving computationally expensive multi-objective optimization problems. A traditional strategy is to decompose a multi-objective optimization problem into a number of single-objective optimization problems, whereby the PF can be regarded as a function of weights. Therefore, it is very natural to use wavelet multi-resolution approximation techniques for setting weight vectors. In our framework, the sequential approach starts with sampling aggressive functions on the initial coarsest grid with a few collocation points; once a rough PF is obtained, new points are automatically added on the basis of an adaptive wavelet collocation method. Therefore, the PF can be approximated with a relatively small number of weights. The efficiency of our method is demonstrated on two examples: a typical multi-objective optimization problem and an expensive multi-objective control optimal problem.
Highlights
Multi-objective optimization is becoming increasingly important in many engineering fields
The optimal solution of each problem is Pareto optimal to the multi-objective optimization problem (MOP). All these optimal solutions form an approximation to the Pareto front (PF)
A set of evenly distributed weight vectors may not results in a set of evenly distributed Pareto optimal solutions, and they may not approximate the PF very well [2, 3]
Summary
Multi-objective optimization is becoming increasingly important in many engineering fields. A multi-objective optimization problem (MOP) can have many, even an infinite number of Pareto optimal vectors. To the best of our knowledge, our work in this paper is the first attempt to adopt a wavelet multi-resolution method for approximating the PF of expensive multi-objective optimal control problems. The basic idea of our proposed method is to embed multi-resolution approximation techniques into the MOP decomposition strategy using an adaptive wavelet collocation method [9,10]. To illustrate the approximation ability of the wavelet multi-resolution decomposition, the following function is used as an example. A function can be approximated by deleting wavelets whose coefficients are lower than the given threshold
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