## Contributed Session Mon.1.H 0107

#### Monday, 10:30 - 12:00 h, Room: H 0107

**Cluster 16: Nonlinear programming** [...]

### Methods for nonlinear optimization I

**Chair: Jean-Pierre Dussault**

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

**Xin-Wei Liu**

How does the linear independence assumption affect algorithms of nonlinear constrained optimization

**Abstract:**

The terminology on the global convergence of algorithms for constrained optimization is first defined. Some recent progress in nonlinear equality constrained optimization is then surveyed. The Steihaug's conjugate gradient method is applied to the linearized constraint residual minimization

problem and its convergence result is proved. The discussions are then extended to the optimization with inequality constraints. The local results demonstrate that the algorithm can be of superlinear convergence even though the gradients of constraints are not linearly independent at the solution.

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

**Mario S. Mommer**

A nonlinear preconditioner for experimental design problems

**Coauthors: Hans-Georg Bock, Johannes P. SchlĂ¶der, Andreas Sommer**

**Abstract:**

Optimal experimental design is the task of finding, given an

experimental budget, a setup that reduces as much as possible the

uncertainty in the estimates of a set of parameters associated

with a model. These optimization problems are difficult to solve

numerically, in particular when they are large. Beyond the

technical challenges inherent to the formulation of the problem

itself, which is based on the optimality conditions of a

nonlinear regression problem, it is common to observe slow

convergence of the sequential quadratic programming (SQP) methods

that are used for its solution. We show that the minima of

experimental design problem can have large absolute condition

numbers under generic conditions. We develop a nonlinear

preconditioner that addresses this issue, and show that its use

leads to a drastic reduction in the number of needed SQP

iterations. Our results suggest a role for absolute condition

numbers in the preasymptotic convergence behavior of SQP methods.

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

**Jean-Pierre Dussault**

The behaviour of numerical algorithms without constraint qualifications

**Abstract:**

We consider inequality constrained mathematical optimisation problems. Under suitable constraint qualifications, at *x*^{*} a minimiser of such a problem there exists a KKT multiplier set *\Lambda(x*^{*}) so that for any * λ ∈ \Lambda(x*^{*}) *x*^{*} satisfies the so called KKT necessary conditions. Usually, stronger assumptions are used to study the behaviour of numerical algorithms in the neighbourhood of a solution, such as LICQ and the strict complementarity condition. Recent works weakened such assumptions and studied the behaviour of algorithm close to degenerate solutions. We explore here the case where no CQ is satisfied, so that *\Lambda(x*^{*}) may be the empty set. In such a case, clearly, primal-dual algorithmic forms are ill-defined. Based on our recent high order path following strategy, we obtain a useful algorithmic framework. This context provides a case where Shamanskii-like high order variants are useless while genuine high order extrapolations yield a solution.