Chapter 4 studies the techniques for numerical realization of the dual algorithms and generalization of the dual algorithms to general unconstrained nonlinear programming . at first, we construct modified dual algorithms to overcome the drawback that it needs to resolve a sequential unconstrained minimization problems exactly in the step 2 of the dual algorithms in chapter 2 and chapter 3 首先针对前两章的对偶算法由于需要精确求解一系列无约束极小化问题,因而实际计算中很难实现这一缺点,构造修正的对偶算法,即,关于势函数的无约束极小化问题无需精确求解的对偶算法。