The quadratic programming problem with fuzzy relation inequality constraints

The quadratic programming has been widely applied to solve real world problems. The quadratic functions are often applied in the inventory management, portfolio selection, engineering design, molecular study, and economics, etc. Fuzzy relation inequalities (FRI) are important elements of fuzzy mathematics, and they have recently been widely applied in the fuzzy comprehensive evaluation and cybernetics. In view of the importance of quadratic functions and FRI, we present a fuzzy relation quadratic programming model with a quadratic objective function subject to the max-product fuzzy relation inequality constraints. Some sufficient conditions are presented to determine its optimal solution in terms of the maximum solution or the minimal solutions of its feasible domain. Also, some simplification operations have been given to accelerate the resolution of the problem by removing the components having no effect on the solution process. The simplified problem can be converted into a traditional quadratic programming problem. An algorithm is also proposed to solve it. Finally, some numerical examples are given to illustrate the steps of the algorithm.