Optimality conditions and optimization methods for quartic polynomial optimization

In this paper multivariate quartic polynomial optimization program (QPOP) is considered. Quartic optimization problems arise in various practical applications and are proved to be NP hard. We discuss necessary global optimality conditions for quartic problem (QPOP). And then we present a new (strongly or εε-strongly) local optimization method according to necessary global optimality conditions, which may escape and improve some KKT points. Finally we design a global optimization method for problem (QPOP) by combining the new (strongly or εε-strongly) local optimization method and an auxiliary function. Numerical examples show that our algorithms are efficient and stable.