It is known that the optimization algorithms of the excitations can be considered as one of the most difficult parts from a code excited linear prediction (CELP) based encoder. This paper presents an original detailed, rigorous and gradual analysis of the optimization algorithm for the MP-MLQ excitation, which characterizes the high-rate working mode of the ITU-T G723.1 encoder. The complete derivation of the algorithms presented here with various comments and accompanying explanations, allow an ordinary DSP programmer to efficiently implement and debug his encoding software.