In this paper, the global optimality conditions and optimization methods for cubic polynomial optimization problems with linear inequality  constraints are considered. Firstly,  we propose a necessary global optimality condition for cubic polynomial optimization problems with linear inequality constraints. Then, a new local optimization method (or called  strongly local optimization methods) is presented by using its necessary global optimality conditions. A global optimization method is proposed for cubic polynomial optimization problems with linear inequality constraints by combining the new local optimization methods together with some auxiliary functions. Finally, some numerical examples are given to illustrate that these approaches are efficient.