In this paper, elliptic control problems with the integral constraint on the gradient of the state and the box constraint on the control are considered. The optimality conditions for the problem are proved. To numerically solve the problem, a finite element duality-based inexact majorized accelerated block coordinate descent (FE-dABCD) algorithm is proposed. Specifically, both the state and the control are discretized by piecewise linear functions. An inexact majorized ABCD algorithm is employed to solve the discretized problem via its dual, which is a multi-block unconstrained convex optimization problem, but the primal variables are also generated in each iteration. Thanks to the inexactness of the FE-dABCD algorithm, the subproblems at each iteration are allowed to be solved inexactly. For the smooth subproblem, we use the preconditioned generalized minimal residual (GMRES) method to solve it. For the two nonsmooth subproblems, one of them has a closed form solution through introducing an appropriate proximal term, and another one is solved by the line search Newton's method. Based on these efficient strategies, we prove that our proposed FE-dABCD algorithm enjoys iteration complexity. Moreover, to make the algorithm more efficient and further reduce its computation cost, based on the mesh-independence of ABCD method, we propose an FE-dABCD algorithm with a warm-start strategy (wFE-dABCD). Some numerical experiments are done and the numerical results show the efficiency of the FE-dABCD algorithm and wFE-dABCD algorithm.

中文翻译：

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一种针对控制和状态梯度约束的椭圆最优控制问题的热启动FE-dABCD算法

本文考虑了状态梯度积分约束和控制盒约束的椭圆控制问题。证明了问题的最优性条件。为了数值求解该问题，提出了一种基于有限元对偶性的不精确主化加速块坐标下降（FE-dABCD）算法。具体来说，状态和控制都由分段线性函数离散化。采用不精确的大化ABCD算法通过其对偶来求解离散化问题，这是一个多块无约束凸优化问题，但在每次迭代中也会生成原始变量。由于 FE-dABCD 算法的不精确性，每次迭代的子问题都被允许不精确地求解。对于平滑子问题，我们使用预处理广义最小残差（GMRES）方法来求解。对于两个非光滑子问题，其中一个通过引入适当的近端项得到封闭式解，另一个通过线搜索牛顿法求解。基于这些有效的策略，我们证明了我们提出的 FE-dABCD 算法具有迭代复杂性。此外，为了使算法更加高效并进一步降低计算成本，基于ABCD方法的网格无关性，我们提出了一种带有热启动策略的FE-dABCD算法（wFE-dABCD）。进行了数值实验，数值结果表明了FE-dABCD算法和wFE-dABCD算法的有效性。