The Application of Possibility Distribution for Solving Standard Quadratic Optimization Problems

A standard quadratic optimization problem (StQP) is to find optimal values of a quadratic form over the standard simplex. The concept of possibility distribution was proposed by L. A. Zadeh. This paper applies the concept of possibility distribution function to solving StQP. The application of possibility distribution function establishes that it encapsulates the constrained conditions of the standard simplex into the possibility distribution function, and the derivative of the StQP formula becomes a linear function. As a result, the computational complexity of StQP problems is reduced, and the solutions of the proposed algorithm are always over the standard simplex. This paper proves that NP-hard StQP problems are in P. Numerical examples demonstrate that StQP problems can be solved by solving a set of linear equations. Comparing with Lagrangian function method, the solutions of the new algorithm are reliable when the symmetric matrix is indefinite.