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


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. Schlöder, Andreas Sommer


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


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.


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