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Appl Math Optim 2020 6;81(3):1021-1054. Epub 2018 Oct 6.

Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria.

The paper presents new results about convergence of the gradient projection and the conditional gradient methods for abstract minimization problems on strongly convex sets. In particular, linear convergence is proved, although the objective functional does not need to be convex. Such problems arise, in particular, when a recently developed discretization technique is applied to optimal control problems which are affine with respect to the control. This discretization technique has the advantage to provide higher accuracy of discretization (compared with the known discretization schemes) and involves strongly convex constraints and possibly non-convex objective functional. The applicability of the abstract results is proved in the case of linear-quadratic affine optimal control problems. A numerical example is given, confirming the theoretical findings.

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http://dx.doi.org/10.1007/s00245-018-9528-3 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC7319312 | PMC |

October 2018

Comput Optim Appl 2018 6;69(2):403-422. Epub 2017 Oct 6.

2Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria.

This paper considers a linear-quadratic optimal control problem where the control function appears linearly and takes values in a hypercube. It is assumed that the optimal controls are of purely bang-bang type and that the switching function, associated with the problem, exhibits a suitable growth around its zeros. The authors introduce a scheme for the discretization of the problem that doubles the rate of convergence of the Euler's scheme. The proof of the accuracy estimate employs some recently obtained results concerning the stability of the optimal solutions with respect to disturbances.

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http://dx.doi.org/10.1007/s10589-017-9948-z | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC6566299 | PMC |

October 2017

J Math Biol 2017 04 7;74(5):1081-1106. Epub 2016 Sep 7.

ORCOS, Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstraße 8/E105-4, 1040, Vienna, Austria.

The paper presents an approach for set-membership estimation of the state of a heterogeneous population in which an infectious disease is spreading. The population state may consist of susceptible, infected, recovered, etc. groups, where the individuals are heterogeneous with respect to traits, relevant to the particular disease. Set-membership estimations in this context are reasonable, since only vague information about the distribution of the population along the space of heterogeneity is available in practice. The presented approach comprises adapted versions of methods which are known in estimation and control theory, and involve solving parametrized families of optimization problems. Since the models of disease spreading in heterogeneous populations involve distributed systems (with non-local dynamics and endogenous boundary conditions), these problems are non-standard. The paper develops the needed theoretical instruments and a solution scheme. SI and SIR models of epidemic diseases are considered as case studies and the results reveal qualitative properties that may be of interest.

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http://dx.doi.org/10.1007/s00285-016-1050-0 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5388773 | PMC |

April 2017

J Biol Dyn 2016 12;10(1):457-76

a ORCOS , Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology , Vienna , Austria.

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http://dx.doi.org/10.1080/17513758.2016.1221474 | DOI Listing |

December 2016

Math Med Biol 2016 09 26;33(3):295-318. Epub 2015 May 26.

ORCOS, Institute of Mathematical Methods in Economics, Vienna University of Technology, Argentinierstrasse 8, A-1040 Vienna, Austria

The paper investigates a version of a simple epidemiological model involving only susceptible and infected individuals, where the heterogeneity of the population with respect to susceptibility/infectiousness is taken into account. A comprehensive analysis of the asymptotic behaviour of the disease is given, based on an explicit aggregation of the model. The results are compared with those of a homogeneous version of the model to highlight the influence of the heterogeneity on the asymptotics. Moreover, the performed analysis reveals in which cases incomplete information about the heterogeneity of the population is sufficient in order to determine the long-run outcome of the disease. Numerical simulation is used to emphasize that, for a given level of prevalence, the evolution of the disease under the influence of heterogeneity may in the long run qualitatively differ from the one 'predicted' by the homogeneous model. Furthermore, it is shown that, in a closed population, the indicator for the survival of the population is in the presence of heterogeneity distinct from the basic reproduction number.

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http://dx.doi.org/10.1093/imammb/dqv018 | DOI Listing |

September 2016