We study a bi-objective optimization problem, which for a given positive real number $n$ aims to find a vector $X = \{x_0,\cdots,x_{k-1}\} \in \mathbb{R}^{k}_{\ge 0}$ such that $\sum_{i=0}^{k-1} x_i = n$, minimizing the maximum of $k$ functions of objective type one, $\max_{i=0}^{k-1} f_i(x_i)$, and the sum of $k$ functions of objective type two, $\sum_{i=0}^{k-1} g_i(x_i)$. This problem arises in the optimization of applications for performance and energy on high performance computing platforms. We first propose an algorithm solving the problem for the case where all the functions of objective type one are continuous and strictly increasing, and all the functions of objective type two are linear increasing. We then propose an algorithm solving a version of the problem where $n$ is a positive integer and all the functions are discrete and represented by finite sets with no assumption on their shapes. Both algorithms are of polynomial complexity.
翻译：我们研究的是双目标优化问题,对于一个给定的正数实际美元,它的目标是找到一个矢量 $X = {x_0,\\cdockts,x ⁇ k-1 ⁇ {R ⁇ k ⁇ ge0} $,这样,美元=sum ⁇ i=0 ⁇ k-1}x_i=n美元,最大限度地减少目标类型一中美元函数的上限,$max ⁇ i=0 ⁇ k-1} f_i(x_i),以及目标类型二中美元函数的总额,$\sum ⁇ i=0 ⁇ k-1} g_i_i(x_i) 。在优化高性能计算平台上对性能和能量的应用时,出现这一问题。我们首先提出一种算法来解决问题,即目标类型一的所有功能都在持续和严格增加,而目标类型二的所有功能都是线性增长。我们然后建议一种算法来解决问题,即$是正数整数,所有功能都是离散的,并且由不假定其形状复杂性的定数组合代表。
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