Xiaoliang Song

Optimization problems with PDE constraints are widely used in engineering and technical fields. In some practical applications, it is necessary to smooth the control variables and suppress their large fluctuations, especially at the boundary. Therefore, we propose an elliptic PDE-constrained optimization model with a control gradient penalty term. However, introducing this penalty term increases the complexity and difficulty of the problems. To solve the problems numerically, we adopt the strategy of “First discretize, then optimize”. First, the finite element method is employed to discretize the optimization problems. Then, a heterogeneous strategy is introduced to formulate the augmented Lagrangian function for the subproblems. Subsequently, we propose a three-block inexact heterogeneous alternating direction method of multipliers (three-block ihADMM). Theoretically, we provide a global convergence analysis of the three-block ihADMM algorithm and discuss the iteration complexity results. Numerical results are provided to demonstrate the efficiency of the proposed algorithm.

The optimization problem of sparse and low-rank matrix recovery is considered, which involves a least squares problem with a rank constraint and a cardinality constraint. To overcome the challenges posed by these constraints, an asymptotic difference-of-convex (ADC) method that employs a Moreau smoothing approach and an exact penalty approach is proposed to transform this problem into a DC programming format gradually. To solve the gained DC programming, by making full use of its DC structure, an efficient inexact DC algorithm with sieving strategy (siDCA) is introduced. The subproblem of siDCA is solved by an efficient dual-based semismooth Newton method. The convergence of the solution sequence generated by siDCA is proved. To illustrate the effectiveness of ADC-siDCA, matrix recovery experiments on nonnegative and positive semidefinite matrices. The numerical results are compared with those obtained using a successive DC approximation minimization method and a penalty proximal alternating linearized minimization approach. The outcome of the comparison indicates that ADC-siDCA surpasses the other two methods in terms of efficiency and recovery error. Additionally, numerical experiments on sparse phase retrieval demonstrate that ADC-siDCA is a valuable tool for recovering sparse and low-rank Hermitian matrices.