An efficient duality-based approach for PDE-constrained sparse optimization

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.Keywords optimal control · sparsity · finite element · duality approach · accelerated block coordinate descent Mathematics Subject Classification (2000) 49N05 · 65N30 · 49M25 · 68W15 Xiaoliang Song ( )

An interior point-proximal method of multipliers for convex quadratic programming

“…We note that the discretization of the smooth parts of problem (5.1) follows a standarad Galekrin approach (e.g. see [67]), while the L 1 term is discretized by the nodal quadrature rule as in [64,70] (an approximation that achieves a first-order convergence-see [70]). In what follows, we consider two classes of state equations (i.e.…”

Interior‐point methods and preconditioning for PDE‐constrained optimization problems involving sparsity terms

“…The discretization without the additional sparsity term follows a standard Galerkin approach . For the discretization of the L 1 term, we here follow and apply the nodal quadrature rule, as follows: where { ϕ i } are the finite element basis functions used and u i are the components of u . It is shown in the work of Wachsmuth and Wachsmuth that first‐order convergence with respect to mesh size may be achieved using this approximation with piecewise linear discretizations of the control.…”

“…We follow the convention of recent numerical studies and investigate the case where the lower (upper) bounds of the box constraints are nonpositive (nonnegative). Here, the functions y d ,f,g,u a ,u b ,y a ,y b ∈ L 2 (Ω) are provided in the problem statement, with α , β >0 given problem‐specific regularization parameters .…”

“…To implement an interior‐point scheme for this problem, we utilize two key ingredients that will be described in detail in Section 3: an appropriate discretization of the L 1 ‐norm that allows us to write the discretized problem in a matrix–vector form and a suitable smoothing of the resulting vector ℓ 1 ‐norm that yields a final quadratic programming form of the discretized problem. The first ingredient is based on the discretization described in the work of Wachsmuth and Wachsmuth and has recently been applied to problem in the works of Song et al, where block‐coordinate–like methods are then introduced. The second ingredient has been widely used for solving the ubiquitous L 1 ‐norm–regularized quadratic problem as, for example, when computing sparse solutions in wavelet‐based deconvolution problems and compressed sensing .…”

A Unified Algorithmic Framework of Symmetric Gauss-Seidel Decomposition Based Proximal Admms for Convex Composite Programming

“…The latter has long been served as a benchmark for comparing new ADMM-type methods since its impressive numerical performance has been well recognized in extensive numerical experiments, despite its lack of theoretical convergence guarantee. Currently, this line of ADMMs has been applied to many concrete instances of problem (1.1), e.g., [1,7,11,16,21,24,[26][27][28][29], to name just a few. Motivated by the above exposition, in this paper, we plan to propose a unified multi-block ADMM for solving problem (1.1).…”