We present a flexible trust region descend algorithm for unconstrained and convexly constrained multiobjective optimization problems. It is targeted at heterogeneous and expensive problems, i.e., problems that have at least one objective function that is computationally expensive. The method is derivative-free in the sense that neither need derivative information be available for the expensive objectives nor are gradients approximated using repeated function evaluations as is the case in finite-difference methods. Instead, a multiobjective trust region approach is used that works similarly to its well-known scalar pendants. Local surrogate models constructed from evaluation data of the true objective functions are employed to compute possible descent directions. In contrast to existing multiobjective trust region algorithms, these surrogates are not polynomial but carefully constructed radial basis function networks. This has the important advantage that the number of data points scales linearly with the parameter space dimension. The local models qualify as fully linear and the corresponding general scalar framework is adapted for problems with multiple objectives. Convergence to Pareto critical points is proven and numerical examples illustrate our findings.

中文翻译：

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使用径向基函数替代模型的无导数多目标信任域下降方法

针对无约束和凸约束的多目标优化问题，我们提出了一种灵活的信任区域下降算法。它针对异构和昂贵的问题，即具有至少一个目标函数的问题，这些目标函数在计算上是昂贵的。该方法是无导数的，即既不需要导数信息可用于昂贵的目标，也不需要像有限差分法那样使用重复函数评估来近似梯度。取而代之的是，使用了一种多目标信任域方法，该方法的工作方式与其众所周知的标量垂饰相同。根据真实目标函数的评估数据构建的局部替代模型用于计算可能的下降方向。与现有的多目标信任域算法相比，这些代理不是多项式，而是精心构建的径向基函数网络。这具有重要的优点，即数据点的数量与参数空间尺寸成线性比例。局部模型符合完全线性的条件，相应的通用标量框架适用于具有多个目标的问题。证明了收敛至帕累托临界点，并且通过数值示例说明了我们的发现。