2 research outputs found
An FE-dABCD algorithm for elliptic optimal control problems with constraints on the gradient of the state and control
In this paper, elliptic control problems with integral constraint on the
gradient of the state and box constraints on the control are considered. The
optimal conditions of the problem are proved. To numerically solve the problem,
we use the 'First discretize, then optimize' approach. Specifically, we
discretize both the state and the control by piecewise linear functions. To
solve the discretized problem efficiently, we first transform it into a
multi-block unconstrained convex optimization problem via its dual, then we
extend the inexact majorized accelerating block coordinate descent (imABCD)
algorithm to solve it. The entire algorithm framework is called finite element
duality-based inexact majorized accelerating block coordinate descent
(FE-dABCD) algorithm. Thanks to the inexactness of the FE-dABCD algorithm, each
subproblems are allowed to be solved inexactly. For the smooth subproblem, we
use the generalized minimal residual (GMRES) method with preconditioner to
slove it. For the nonsmooth subproblems, one of them has a closed form solution
through introducing appropriate proximal term, another is solved combining
semi-smooth Newton (SSN) method. Based on these efficient strategies, we prove
that our proposed FE-dABCD algorithm enjoys iteration
complexity. Some numerical experiments are done and the numerical results show
the efficiency of the FE-dABCD algorithm.Comment: 24 page
OPTIMAL CONTROL FOR AN ELLIPTIC SYSTEM WITH POLYGONAL STATE CONSTRAINTS
Abstract. This work is devoted to stationary optimal control problems with polygonal constraints on the components of the state. Existence of Lagrange multipliers, of different regularity, is verified for the cases with and without Slater condition holding. For the numerical realization a semi-smooth Newton method is proposed for an appropriately chosen family of regularized problems. The asymptotic behavior of the regularized problem class is studied, and numerical feasibility of the method is shown. 1