Abstract In this paper, a prediction-correction algorithm, built on the method proposed in [1], uses the adjoint method to trace the Pareto front. The method is initialized by a point on the Pareto front obtained by considering one of the objectives only. During the prediction and correction steps, different systems of equations are derived and solved by treating the Karush-Kuhn-Tucker (KKT) optimality conditions in two different ways. The computation of second derivatives of the objective functions (Hessian matrix) which appear in the equations solved to update the design variables is avoided. Instead, two approaches are used: (a) the computation of Hessian-vector products driving a Krylov subspace solver and (b) the approximations of the Hessian via Quasi-Newton methods. Three different variants of the prediction-correction method are developed, applied to 2D aerodynamic shape optimization problems with geometrical constraints and compared in terms of computational cost. It is shown that the inclusion of the prediction step in the algorithm and the use of Quasi-Newton methods with Hessian approximations in both steps has the lowest computational cost. This method is, then, used to compute the Pareto front in a 3D conjugate heat transfer shape optimization problem, with the total pressure losses and max. solid temperature as the two contradicting objectives.

中文翻译：

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空气动力学和共轭传热形状优化中的伴随辅助帕累托前沿追踪

摘要 在本文中，基于[1] 中提出的方法构建的预测校正算法使用伴随方法来追踪帕累托前沿。该方法由帕累托前沿上的一个点初始化，该点通过仅考虑其中一个目标而获得。在预测和校正步骤中，通过以两种不同的方式处理 Karush-Kuhn-Tucker (KKT) 最优条件，推导出和求解不同的方程组。避免计算出现在求解方程中以更新设计变量的目标函数（Hessian 矩阵）的二阶导数。相反，使用了两种方法：(a) 计算驱动 Krylov 子空间求解器的 Hessian 向量乘积和 (b) 通过拟牛顿方法对 Hessian 的近似。开发了预测校正方法的三种不同变体，应用于具有几何约束的二维空气动力学形状优化问题，并在计算成本方面进行比较。结果表明，在算法中包含预测步骤以及在两个步骤中使用具有 Hessian 近似的拟牛顿方法具有最低的计算成本。然后，该方法用于计算 3D 共轭传热形状优化问题中的帕累托前沿，总压力损失和最大值。固体温度作为两个相互矛盾的目标。用于计算 3D 共轭传热形状优化问题中的帕累托前沿，总压力损失和最大值。固体温度作为两个相互矛盾的目标。用于计算 3D 共轭传热形状优化问题中的帕累托前沿，总压力损失和最大值。固体温度作为两个相互矛盾的目标。