## 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**

**Abstract:**

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**

**Abstract:**

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

form

A D^{2} \!A^{T} 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.

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**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**

**Abstract:**

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.