Fakultät für Mathematik
Phone: +43 1 4277 506 62
Zimmer 04.120 (4. Stock im Turm)
Global and local optimization
Population dynamics: structured populations
List of Publications
Publications in journals and refereed conference proceedings
W. Huyer, A size-structured population model with dispersion, J.
Math. Anal. Appl. 181 (1994), 716-754
W. Huyer, Semigroup formulation and approximation of a linear age-dependent
population problem with spatial diffusion, Semigroup Forum 49 (1994),
W. Huyer, On periodic cohort solutions of a size-structured population
model, J. Math. Biol. 35 (1997), 908-934
We consider a size-structured population model with discontinuous reproduction
and feedback through the environmental variable `substrate'. The model
admits solutions with finitely many cohorts and in that case the problem
is described by a system of ODEs involving a bifurcation parameter beta.
Existence of nontrivial periodic n-cohort solutions is investigated. Moreover,
we discuss the question whether n cohorts (n >= 2) with small size differences
will tend to a periodic one-cohort solution as t tends to infinity.
W. Huyer, Well-posedness of a linear age-dependent population model
with spatial diffusion in L^2, Advances in Mathematical Population
Dynamics - Molecules, Cells and Man (O. Arino, D. Axelrod and M. Kimmel,
eds.), World Scientific, Singapore, 1997, pp. 713-732
A linear model for an age-structured population with random diffusion
in a bounded domain in R^n is studied in the framework in L^2. Three different
approaches to establishing well-posedness of the problem are presented.
Two of them involve applying known results on the m-dissipativeness of
the sum of two m-dissipative operators to the population operator and the
A. Neumaier, S. Dallwig, W. Huyer, and H. Schichl, New techniques for
the construction of residue potentials for protein folding, Computational
Molecular Dynamics: Challenges, Methods, Ideas (P. Deuflhard et al., eds.),
Lecture Notes Comput. Sci. Eng. 4, Springer, Berlin, 1999, pp. 212-224
A smooth empirical potential is constructed for use in off-lattice
protein folding studies. Our potential is a function of the amino acid
labels and of the distances between the C(alpha) atoms of a protein. The
potential is a sum of smooth surface potential terms that model solvent
interactions and of pair potentials that are functions of a distance, with
a smooth cutoff at 12 Ångström. Techniques include the use of
a fully automatic and reliable estimator for smooth densities, of cluster
analysis to group together amino acid pairs with similar distance distributions,
and of quadratic programming to find appropriate weights with which the
various terms enter the total potential. For nine small test proteins,
the new potential has local minima within 1.3-4.7Å of the PDB geometry,
with one exception that has an error of 8.5Å.
W. Huyer and A. Neumaier, Global optimization by multilevel coordinate
search, J. Global Optimization 14 (1999), 331-355
Inspired by a method by Jones et al., we present a global optimization
algorithm based on multilevel coordinate search. It is guaranteed to converge
if the function is continuous in the neighborhood of a global minimizer.
By starting a local search from certain good points, an improved convergence
result is obtained. We discuss implementation details and give some numerical
You can download a Matlab version of the algorithm here.
- W. Huyer and A. Neumaier, A new exact penalty function, SIAM J. Optim.
13 (2003), 1141-1158
For constrained smooth or nonsmooth optimization problems, new continuously
differentiable penalty functions are derived. They are proved exact in
the sense that under some nondegeneracy assumption, local optimizers of
a nonlinear program are precisely the optimizers of the associated penalty
function. This is achieved by augmenting the dimension of the program by
a variable that controls both the weight of the penalty terms and the regularization
of the nonsmooth terms.
W. Huyer, Approximation of a linear age-dependent population model with
spatial diffusion, Commun. Appl. Anal. 8 (2004), 87-108
In this paper we first address the question under which conditions
it is possible to obtain from Trotter-Kato approximations of two infinitesimal
generators of C0-semigroups a Trotter-Kato approximation of the semigroup
generated by the sum operator. This cannot be true in general since the
sum operator might not even be a generator, and we derive sufficient conditions.
However, when wellposedness is already established, it may be more convenient
to verify the conditions of the Trotter-Kato theorem directly, taking into
account the special structure of the problem. We consider the semigroup
generated by an age-dependent population model with spatial diffusion,
and numerical examples are presented to demonstrate feasibility of the
W. Huyer and A. Neumaier, Integral approximation of rays and verification of
feasibility, Reliable Computing 10 (2004), 195-207
An algorithm is presented that produces an integer vector nearly
parallel to a given vector. The algorithm can be used to discover
exact rational solutions of homogeneous or inhomogeneous
linear systems of equations, given a sufficiently accurate
As an application, we show how to verify rigorously the feasibility
of degenerate vertices of a linear program with integer coefficients,
and how to recognize rigorously certain redundant linear constraints
in a given system of linear equations and inequalities. This is
a first step towards the handling of degeneracies and redundancies
within rigorous global optimization codes.
A. Neumaier, O. Shcherbina, W. Huyer, and T. Vinko, A comparison of complete global optimization
solvers, Math. Program., Ser. B 103 (2005), 335-356
Results are reported of testing a number of existing state of the art solvers for global
constrained optimization and constraint satisfaction on a set of over 1000 test problems in up to
1000 variables, collected from the literature. The test problems are available online in AMPL and
were translated into the input formats of the various solvers using routines from the COCONUT
environment. These translators are available online, too.
W. Huyer and A. Neumaier, SNOBFIT - stable noisy optimization by branch and
fit, ACM Trans. Math. Software 35 (2008), No. 2, Article 9
The software package SNOBFIT for bound constrained noisy optimization of an
expensive objective function is described. It combines global and local search
by branching and local fits. The program is made robust and flexible for
practical use by allowing for soft or hidden constraints, batch function
evaluations, change of search regions, etc.
You can download a Matlab version of the algorithm here.
P. Pošík, W. Huyer, and L. Pál, A comparison of global search algorithms for continuous black-box optimization, Evolutionary Computation 20 (2012), 509-541
Four methods for global numerical black-box optimization with the origins in the mathematical
programming community are described and experimentally compared with
the state-of-the-art evolutionary method, BIPOP-CMA-ES. The methods chosen for the
comparison exhibit various features potentially interesting for the evolutionary computation
community: systematic sampling of the search space (DIRECT, MCS) possibly
combined with a local search method (MCS), or a multistart approach (NEWUOA,
GLOBAL) possibly equipped with a careful selection of points to run a local optimizer
from (GLOBAL). The recently proposed "comparing continuous optimizers" (COCO)
methodology was adopted as the basis for the comparison. Based on the results, we
draw suggestions about which algorithm should be used depending on the available
budget of function evaluations, and we propose several possibilities for hybridizing
evolutionary algorithms with features of the other compared algorithms.
P. Pošík and W. Huyer, Restarted local search algorithms for continuous black-box optimization, Evolutionary Computation 20 (2012), 575-607
Several local search algorithms for real-valued domains (axis-parallel line search,
Nelder-Mead simplex search, Rosenbrock's algorithm, quasi-Newton method,
NEWUOA and VXQR) are described and thoroughly compared in this article, embedding
them in a multistart method. Their comparison aims (1) to help the researchers
from the evolutionary community to choose the right opponent for their algorithm
(to choose an opponent that would constitute a hard-to-beat baseline algorithm), (2)
to describe individual features of these algorithms and show how they influence the
algorithm on different problems, and (3) to provide inspiration for the hybridization
of evolutionary algorithms with these local optimizers. The recently proposed "comparing
continuous optimizers" (COCO) methodology was adopted as the basis for
the comparison. The results show that in low dimensional spaces, the old method of
Nelder and Mead is still the most successful among those compared, while in spaces
of higher dimensions it is better to choose an algorithm based on quadratic modeling,
like NEWUOA or a quasi-Newton method.
A. Neumaier, W. Huyer and E. Bornberg-Bauer,
Hydrophobicity analysis of amino acids
Übungen zu "Einführung in die Analysis"
Übung zu Numerische Mathematik 1
Waltraud Huyer ( Waltraud.Huyer@univie.ac.at),
last change October 7, 2015