Abstract
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.
Highlights
standard quadratic optimization problem (StQP) problems are widely used in game theory, operating research, system control, financial mathematics, and etc
This paper proves that for a given arbitrary possibility distribution, there exists an associated possibility distribution that satisfies the constrained conditions of the standard simplex
Theorem 3.1 indicates that the constrained conditions of variable x being over the standard simplex are encapsulated into the possibility distribution function
Summary
StQP problems are widely used in game theory, operating research, system control, financial mathematics, and etc. Bomze studies gradient projection methods for solving StQP problems (Bomze, 1998). The difficulty with traditional gradient projection methods is to obtain a feasible direction via projection onto the feasible constrained conditions of the standard simplex (Bomze, 2002). Zadeh indicated that” The possibility distribution function associated with a variable is defined to be numerically equal to the membership function of a fuzzy subset” (Zadeh, 1978). He claimed that the membership function of a fuzzy set could serve as a possibility distribution.
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