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

Abstract

Summary Partial differential equation (PDE)–constrained optimization problems with control or state constraints are challenging from an analytical and numerical perspective. The combination of these constraints with a sparsity‐promoting L1 term within the objective function requires sophisticated optimization methods. We propose the use of an interior‐point scheme applied to a smoothed reformulation of the discretized problem and illustrate that such a scheme exhibits robust performance with respect to parameter changes. To increase the potency of this method, we introduce fast and efficient preconditioners that enable us to solve problems from a number of PDE applications in low iteration numbers and CPU times, even when the parameters involved are altered dramatically.

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

“…2 )] ⊤ , with control bounds u a = −2, u b = 1.5 and free state (e.g. see [51,Section 5.2]). Once again, the problem is discretized using Q1 finite elements, employing the Streamline Upwind Petrov-Galerkin (SUPG) upwinding scheme implemented in [12].…”

“…Now if the primal-dual pair (P)-(D) is feasible, it must remain feasible for any positive semi-definite D, since in this case (P) can be written as a convex quadratic problem by appending appropriate (necessarily feasible) linear equality and inequality constraints (e.g. as in [21,51]). Thus, Assumption 1 suffices to guarantee that the solution set of (P)-(D) is non-empty.…”

“…see [20,24,25,38,39,41,49,59]), or interior point methods (IPMs) applied to a reformulation of (P) (e.g. see [21,28,31,51]). Most globalized Newton-like approaches or proximal point variants studied in the literature are developed for composite programming problems in which either g(x) = 0 (e.g.…”

“…As a result, the associated linear systems solved within SSN have significantly smaller dimensions, compared to linear systems arising within interior point methods suitable for the solution of ℓ 1 -regularized convex quadratic problems (e.g. see [21,31,51]), potentially making the proposed approach a more attractive alternative for large-scale instances, such as those arising from the optimal control of PDEs. Finally, we propose an efficient warm-starting strategy based on a proximal alternating direction method of multipliers to further improve the efficiency of the approach at a low computational cost.…”

A new preconditioning approach for an interior point‐proximal method of multipliers for linear and convex quadratic programming

“…Preconditioners have also been successfully devised for specific classes of programming problems solved using similar optimization methods: applications include those arising from multicommodity network flow problems, 18 stochastic programming problems, 19 formulations within which the constraint matrix has primal block-angular structure, 20 and PDE-constrained optimization problems. 21,22 However, such preconditioners exploit particular structures arising from specific applications; unless there exists such a structure which hints as to the appropriate way to develop a solver, the design of bespoke preconditioners remains a challenge.…”