This paper presents a novel numerical strategy based on combination of an adaptive semismooth Newton (ASN) method and the Lavrentiev regularization technique for the solution of elliptic optimal control problems with state constraints. Using the global convergence proof for a nonmonotone semismooth Newton method, we will exploit an adaptive nonmonotone line search method such that the nonmonotonicity degree of this method can be increased when the results are far from the optimum solution and it can be reduced when they are close to the optimizer. In this strategy, the role of the Lavrentiev regularization technique is converting the original optimal control problem to a regularized optimal control problem. Using the finite difference discretization scheme and a Newton–Cotes rule, the regularized optimal control problem is converted to a bound constrained optimization problem (BCOP). Then the ASN method is implemented to solve the resulting BCOP. Numerical results show the efficiency of the proposed procedure.

中文翻译：

---

状态约束椭圆最优控制问题的有效非单调方法

本文提出了一种基于自适应半光滑牛顿（ASN）方法和Lavrentiev正则化技术相结合的新型数值策略，用于求解具有状态约束的椭圆最优控制问题。使用非单调半光滑牛顿法的全局收敛性证明，我们将利用自适应非单调线搜索方法，使得当结果离最佳解很远时可以提高该方法的非单调度，而在接近最优解时可以降低该方法的非单调度。给优化器。在这种策略中，Lavrentiev正则化技术的作用是将原始的最优控制问题转换为正则化的最优控制问题。使用有限差分离散化方案和牛顿-科茨规则，将正则化的最优控制问题转换为约束约束最优化问题（BCOP）。然后实施ASN方法以解决生成的BCOP。数值结果表明了该方法的有效性。