Novel bi-objective optimization algorithms minimizing the max and sum of vectors of functions

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.