Invited Session Thu.1.H 0107

Thursday, 10:30 - 12:00 h, Room: H 0107

Cluster 16: Nonlinear programming [...]

Linear algebra for optimization


Chair: Dominique Orban



Thursday, 10:30 - 10:55 h, Room: H 0107, Talk 1

Martin Stoll
Preconditioning for time-dependent PDE-constrained optimization problems

Coauthors: John W. Pearson, Tyrone Rees, Andrew J. Wathen


In this talk, we motivate and test effective preconditioners to be used within a Krylov subspace algorithm for solving a number of saddle point systems, which arise in PDE-constrained optimization problems. We consider a variety of setups for different time-dependent PDEs such as the distributed control problem involving the heat equation, the Neumann boundary control problem subject to the heat equation and a distributed control problem with Stokes equations as the PDE-constraint. Crucial to the performance of our preconditioners in each case is an effective approximation of the Schur complement of the matrix system. In each case, we propose the preconditioning approach and provide numerical results, which demonstrate that our solvers are effective for a range of regularization parameter values, as well as mesh sizes and time-steps.



Thursday, 11:00 - 11:25 h, Room: H 0107, Talk 2

Santiago Akle
Preconditioning for iterative computation of search directions within interior methods for constrained optimization

Coauthor: Michael Saunders


Our primal-dual interior-point optimizer PDCO has found many applications
for optimization problems of the form

min, \varphi(x)  st   Ax=b,   l ≤ x ≤ u,

in which \varphi(x) is convex and A is a sparse matrix or a linear
operator. We focus on the latter case and the need for iterative
methods to compute dual search directions from linear systems of the

A D2 \!AT Δ y = r, \qquad
D diagonal and positive definite.

Although the systems are positive definite, they do not need to be
solved accurately and there is reason to use MINRES rather than
CG (see PhD thesis of David Fong (2011)). When the original problem
is regularized, the systems can be converted to least-squares
problems and there is similar reason to use LSMR rather than LSQR.
Since D becomes increasingly ill-conditioned as the interior method
proceeds, there is need for some kind of preconditioning. We examine
the partial Cholesky approach of Bellavia, Gondzio and Morini (2011)
and explore some alternatives that are better suited to applications
in which A is a linear operator.



Thursday, 11:30 - 11:55 h, Room: H 0107, Talk 3

Dominique Orban
Spectral analysis of matrices arising in regularized interior-point methods

Coauthors: Chen Greif, Erin Moulding


Interior-point methods feature prominently in the solution of
inequality-constrained optimization problems, and involve the need to solve a
sequence of 3 × 3 block indefinite systems that become increasingly
ill-conditioned with the iterations. To solve those systems, it is common
practice to perform a block Gaussian elimination, and either solve the
resulting reduced 2 × 2 block indefinite system that has a typical
saddle-point form, or further reduce the system to the normal equations and
apply a symmetric positive definite solver. In this paper we explore whether
the step of reducing the system from a 3 × 3 block matrix to a 2
× 2
block matrix necessarily pays off. We use energy estimates to
obtain bounds on the eigenvalues of the coefficient matrices, and conclude
that, at least in terms of spectral structure, it may be better to
keep the matrix in its original unreduced form rather than perform a partial
elimination before solving it.


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