Optimality conditions and optimization methods for quartic polynomial optimization problems with mixed variables

Multivariate quartic polynomial optimization problems, as a special case of the general polynomial optimization, have a lot of practical applications in real world and are proved to be NP-hard. In this paper, some necessary local optimality conditions and some necessary global optimality conditions for quartic polynomial optimization problems with mixed variables are established. Then some local optimization methods, including weakly local optimization methods for general problems with mixed variables and strongly local optimization methods for quartic polynomial optimization problems with mixed variables, are proposed by exploiting these necessary local optimality conditions and necessary global optimality conditions. A global optimization method is proposed for quartic polynomial optimization problems by combining these local optimization methods together with some auxiliary functions. Some numerical examples are also given to illustrate that these approaches are very efficient.