中文摘要：

利用目标函数值和近似次梯度，构建了非光滑无约束优化问题目标函数的一个下近似模型，通过对该近似模型取极小寻找下一个可能使目标函数值下降的试探点。利用Lagrange函数写出了原近似问题的对偶问题，揭示了原近似问题的最优解与对偶问题最优解之间的关系，并进一步分析了相应的近似次梯度的某种凸组合与目标函数在当前迭代点的次微分以及目标函数的近似模型在当前迭代点的近似次微分之间的所属关系。所得结果为原近似问题的求解开辟了新思路，也使整个外层束方法的执行变得简单易行。

英文摘要：

We construct a lower approximate model for the ohiective function of nonsmooth unconstrained optimization problem by using the values of the objective function and its approximate subgradients. And by minimizing the lower approximate model, we expect to find out a candidate point which can decrease the value of the objective function. After that,the dual problem of the primal ap- proximate problem is given by utilizing Lagrange function,and at the same time the relation between the solutions of the primal approximate problem and the dual problem is also presented. Furthermore,we conclude that some convex combination of previous approximate subgradients belongs to both the subdifferential of the objective function at the current iterate point and the approximate sub- differential of the lower approximate model at the current iterate point. The results obtained in this paper provide a new way to solve the primal problem,and also make the overall outer bundle method easier to implement.