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An Efficient Duality-based Numerical Method for Sparse Optimal Control Problems

2019-05-15
 

Academic Report


Title: An Efficient Duality-based Numerical Method for Sparse Optimal Control Problems

Reporter: SONG Xiaoliang (Hong Kong Polytechnic University, postdoctoral)

Time: May 16, 2019 (Thursday) PM 15:30-16:30

Location: A1101# room, Innovation Park Building

Contact: Prof. LEI Fengchun (tel:84708360)

Abstract: 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.

The brief introduction to the reporter: Song Xiaoliang, Ph. D. graduated from Dalian University of Technology in 2018. He jointly trained at the National University of Singapore from September 2015 to February 2017. He is now a postdoctoral candidate in the Department of Applied Mathematics, Hong Kong University of Technology. Dr. Song Xiaoliang's research direction is numerical optimization and optimal control. The main research contents are numerical discretization and optimization algorithms for PDE constrained optimization problems. At present, 11 academic papers have been published.