[Paper Review] Mesh Independence of an Accelerated Block Coordinate Descent Method for Sparse Optimal Control Problems

Xiaoliang Song, Bo Chen+1 more|arXiv (Cornell University)|Aug 31, 2017

Matrix Theory and Algorithms23 references3 citations

TL;DR

This paper analyzes an accelerated block coordinate descent (ABCD) method in Hilbert space for solving sparse optimal control problems via duality. It establishes $O(1/k)$ iteration complexity for both dual and primal problems and proves two types of mesh independence, showing that convergence behavior remains asymptotically invariant under mesh refinement.

ABSTRACT

An accelerated block coordinate descent (ABCD) method in Hilbert space is analyzed to solve the sparse optimal control problem via its dual. The finite element approximation of this method is investigated and convergence results are presents. Based on the second order growth condition of the dual objective function, we show that iteration sequence of dual variables has the iteration complexity of $O(1/k)$. Moreover, we also prove iteration complexity for the primal problem. Two types of mesh-independence for ABCD method are proved, which asserts that asymptotically the infinite dimensional ABCD method and finite dimensional discretizations have the same convergence property, and the iterations of ABCD method remain nearly constant as the discretization is refined.

Motivation & Objective

• To analyze the convergence properties of an accelerated block coordinate descent (ABCD) method in Hilbert space for sparse optimal control problems.
• To investigate the finite element approximation of the ABCD method and establish convergence results.
• To prove mesh independence, showing that infinite-dimensional and finite-dimensional formulations exhibit equivalent convergence behavior under refinement.
• To establish iteration complexity bounds for both the dual and primal problems under second-order growth conditions.

Proposed method

• The ABCD method is applied to the dual formulation of the sparse optimal control problem in Hilbert space.
• Finite element discretization is used to approximate the infinite-dimensional problem, enabling numerical computation.
• Convergence analysis relies on the second-order growth condition of the dual objective function.
• The method employs block updates to coordinate descent steps, accelerating convergence via momentum-like mechanisms.
• Mesh independence is proven by comparing iteration counts and convergence rates between continuous and discrete formulations.
• Theoretical bounds on iteration complexity are derived using variational analysis and properties of the dual objective.

Experimental results

Research questions

• RQ1 Does the ABCD method achieve $O(1/k)$ convergence rate for the dual and primal problems under second-order growth?
• RQ2 How does the convergence behavior of the ABCD method change as the finite element mesh is refined?
• RQ3 Can the infinite-dimensional ABCD method be shown to be asymptotically equivalent to its finite-dimensional discretizations in terms of convergence?
• RQ4 What conditions ensure that the number of iterations remains nearly constant across mesh refinements?
• RQ5 How do the dual and primal iteration complexities relate under the same assumptions?

Key findings

• The dual problem iteration sequence achieves an $O(1/k)$ convergence rate under the second-order growth condition.
• The primal problem also exhibits $O(1/k)$ iteration complexity, confirming accelerated convergence.
• Two types of mesh independence are proven: one showing asymptotic equivalence in convergence behavior between infinite and finite-dimensional formulations.
• The other mesh independence result asserts that the number of iterations required remains nearly constant as the mesh is refined.
• The ABCD method maintains consistent convergence performance across different discretization levels, indicating robustness to mesh refinement.
• The theoretical results confirm that the ABCD method is both efficient and scalable in solving sparse optimal control problems via duality.