Performance and energy are the two most important objectives for optimization on modern heterogeneous HPC platforms. In this work, we study a mathematical problem motivated by the bi-objective optimization of a matrix multiplication application on such platforms for performance and energy. We formulate the problem and propose an algorithm of polynomial complexity solving the problem for the case where all the application profiles of objective type one are continuous and strictly increasing, and all the application profiles of objective type two are linear increasing. We solve the problem for the matrix multiplication application employing five heterogeneous processors that include two Intel multicore CPUs, an Nvidia K40c GPU, an Nvidia P100 PCIe GPU, and an Intel Xeon Phi. Based on our experiments, a dynamic energy saving of 17% is gained while tolerating a performance degradation of 5% (a saving of 106 Joules for an execution time increase of 0.05 seconds).