Preference Optimization Methods for Multi-objective Problems by Using Integrated Optimization Methods with Active Learning

Abstract

This paper considers a preference optimization problem which requires the best Pareto solution for a decision maker's preference among the Pareto solution set of a multi-objective optimization problem. Even if the decision maker's preference is expressed by formula explicitly, it is difficult to solve the this type of problems, because Pareto solution set as their constraint cannot be described analytically. In our approach, choosing one of Pareto solutions is considered as a map from weighting coefficient parameter in a problem to minimize the maximal component of the multi-objective functions, and the preference optimization problem is transformed into a bi-level optimization problem in which the best weighting coefficient is chosen so as to optimize the decision maker's preference with the minimization problem of maximal component of the multi-objective functions. Then, the map is generated approximately with linear combination of radial basis functions on the weighting coefficient space by using optimization procedure with an active learning method presented by the authors, in which effective weighting coefficient data are generated for searching the Pareto solution with the best preference successively. Results of computer simulation for simple examples show effectiveness of the presented integrated optimization method for preference optimization problem.