A primal dual projection algorithm for efficient constraint preconditioning

TL;DR

This paper introduces a primal-dual projection algorithm for efficiently solving large-scale constrained quadratic problems, especially in PDE-constrained optimization, demonstrating competitive numerical performance.

Contribution

The paper proposes a novel primal-dual projection method that interprets as a gradient approach on a quotient space, enabling efficient constraint preconditioning in large-scale problems.

Findings

Reliable performance in elasticity optimal control

Effective constraint preconditioning with block triangular preconditioner

Competitive numerical results in PDE-based optimization

Abstract

We consider a linear iterative solver for large scale linearly constrained quadratic minimization problems that arise, for example, in optimization with PDEs. By a primal-dual projection (PDP) iteration, which can be interpreted and analysed as a gradient method on a quotient space, the given problem can be solved by computing sulutions for a sequence of constrained surrogate problems, projections onto the feasible subspaces, and Lagrange multiplier updates. As a major application we consider a class of optimization problems with PDEs, where PDP can be applied together with a projected cg method using a block triangular constraint preconditioner. Numerical experiments show reliable and competitive performance for an optimal control problem in elasticity.