Elliptic PDE-constrained optimal control problems with L 1 -control cost (L 1 -EOCP) are considered. To solve L 1 -EOCP, the primal-dual active set (PDAS) method, which is a special semismooth Newton (SSN) method, used to be a priority. However, in general solving Newton equations is expensive. Motivated by the success of alternating direction method of multipliers (ADMM), we consider extending the ADMM to L 1 -EOCP. To discretize L 1 -EOCP, the piecewise linear finite element (FE) is considered. However, different from the finite dimensional l 1 -norm, the discretized L 1 -norm does not have a decoupled form. To overcome this difficulty, an effective approach is utilizing nodal quadrature formulas to approximately discretize the L 1 -norm and L 2norm. It is proved that these approximation steps will not change the order of error estimates. To solve the discretized problem, an inexact heterogeneous ADMM (ihADMM) is proposed. Different from the classical ADMM, the ihADMM adopts two different weighted inner product to define the augmented Lagrangian function in two subproblems, respectively. Benefiting from such different weighted techniques, two subproblems of ihADMM can be efficiently implemented. Furthermore, theoretical results on the global convergence as well as the iteration complexity results o(1/k) for ihADMM are given. In order to obtain more accurate solution, a two-phase strategy is also presented, in which the primal-dual active set (PDAS) method is used as a postprocessor of the ihADMM. Numerical results not only confirm error estimates, but also show that the ihADMM and the two-phase strategy are highly efficient.
In this paper, elliptic optimal control problems involving the L 1 -control cost (L 1 -EOCP) is considered. To numerically discretize L 1 -EOCP, the standard piecewise linear finite element is employed. However, different from the finite dimensional l 1 -regularization optimization, the resulting discrete L 1 -norm does not have a decoupled form. A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the L 1 -norm. It is clear that this technique will incur an additional error. To avoid the additional error, solving L 1 -EOCP via its dual, which can be reformulated as a multi-block unconstrained convex composite minimization problem, is considered. Motivated by the success of the accelerated block coordinate descent (ABCD) method for solving large scale convex minimization problems in finite dimensional space, we consider extending this method to L 1 -EOCP. Hence, an efficient inexact ABCD method is introduced for solving L 1 -EOCP. The design of this method combines an inexact 2-block majorized ABCD and the recent advances in the inexact symmetric Gauss-Seidel (sGS) technique for solving a multi-block convex composite quadratic programming whose objective contains a nonsmooth term involving only the first block. The proposed algorithm (called sGS-imABCD) is illustrated at two numerical examples. Numerical results not only confirm the finite element error estimates, but also show that our proposed algorithm is more efficient than (a) the ihADMM (inexact heterogeneous alternating direction method of multipliers), (b) the APG (accelerated proximal gradient) method.Keywords optimal control · sparsity · finite element · duality approach · accelerated block coordinate descent Mathematics Subject Classification (2000) 49N05 · 65N30 · 49M25 · 68W15 Xiaoliang Song ( )
Summary Elliptic optimal control problems with pointwise box constraints on the control are considered. To numerically solve elliptic optimal control problems with pointwise box constraints on the control, an inexact alternating direction method of multipliers (iADMM) is first proposed on the continuous level with the aim of solving discretized problems with moderate accuracy. Then, the standard piecewise linear finite element is employed to discretize the related subproblems appearing in each iteration of the iADMM algorithm. Such approach will give us the freedom to discretize two inner subproblems of the iADMM algorithm by different discretized scheme, respectively. More importantly, it should be emphasized that the discretized version of the iADMM algorithm can be regarded as a modification of the inexact semiproximal ADMM (isPADMM) algorithm. In order to obtain more accurate solution, the primal‐dual active set method is used as a postprocessor of the isPADMM. Numerical results not only show that the isPADMM and the two‐phase strategy are highly efficient but also show the mesh independence of the isPADMM.
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