An interactive method for solving optimization problems with an unknown objective function

This paper proposes an interactive method to solve optimization problems with an unknown objective function. In daily life, we as human beings solve practical optimization problems without using numerical information and find a promising solution for them according to our preference. We aim to develop a mathematical method to solve the problems in which only qualitative information about the objective function is available. In the proposed method, (i) the decision maker´s preference of solutions is scored in a simple manner; (ii) support vector regression (SVR) is applied to approximate his/her utility function from the scored data; and (iii) another solution to be presented to him/her is searched for by maximizing the current utility function. Repetitive computation of (i)-(iii) with an additional mechanism of removing useless solutions can bring the one that he/she would like. By numerical experiments, the proposed method is shown to be effective in finding a local optimum of such problems.