An FE-dABCD algorithm for elliptic optimal control problems with constraints on the gradient of the state and control

TL;DR

This paper introduces the FE-dABCD algorithm, a novel numerical method for efficiently solving elliptic optimal control problems with gradient and control constraints, demonstrating high convergence speed and practical effectiveness.

Contribution

The paper develops the FE-dABCD algorithm, extending inexact majorized accelerated block coordinate descent to elliptic control problems with complex constraints, and proves its optimal iteration complexity.

Findings

The FE-dABCD algorithm achieves $O(1/k^2)$ convergence rate.

Numerical experiments confirm the algorithm's efficiency.

The method effectively handles inexact subproblem solutions.

Abstract

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 $O(\frac{1}{k^2})$ iteration complexity. Some numerical experiments are done and the numerical results show the efficiency of the FE-dABCD algorithm.