  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.
