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Immanuel Bomze
Immanuel Bomze is an Austrian mathematician. In his Ph.D. thesis, he completely classified all (more than 100 topologically different) possible flows of the generalized Lotka–Volterra dynamics (generalized Lotka–Volterra equation) on the plane, employing equivalence of this dynamics to the 3-type replicator equation.〔I. Bomze, Lotka–Volterra equation and replicator dynamics: a two-dimensional classification. ''Biological Cybernetics'' 48, 201–211 (1983); I. Bomze, Lotka–Volterra equation and replicator dynamics: new issues in classification. ''Biological Cybernetics'' 72, 447–453 (1995).〕
In “Non-cooperative two-person games in biology: a classification” (1986)〔''International Journal of Game Theory'' 15, 31–57.〕 and his book jointly authored with B. M. Pötscher (''Game theoretic foundations of evolutionary stability'', Springer 1989), he popularized the field of evolutionary game theory which at that time received most attention within Theoretical Biology, among researchers in Economics and Social Sciences.〔E.g. P. F. Stadler, P. Schuster, Dynamics of small autocatalytic reaction networks I: bifurcations, permanence and exclusion. ''Bulletin of mathematical biology'' 52, 485–508 (1990); E. Szatmáry, Natural selection and dynamical coexistence of defective and complementing virus segments. ''Journal of theoretical biology'' 157, 383–406 (1992); P. Hammerstein, R. Selten, Game theory and evolutionary biology, in: R. J. Aumann, S. Hart (ed.), ''Handbook of game theory with economic applications'', vol. 2, Elsevier Science B.V., 1994, 930–993; K. Ritzberger, J. W. Weibull, Evolutionary selection in normal-form games. ''Econometrica'' 63, 1371–1399 (1995).〕
Around the turn of the millennium, he coined, together with his co-authors, the now widely used terms "Standard Quadratic Optimization" and "Copositive Optimization" or "Copositive Programming".〔I. Bomze, On standard quadratic optimization problems. ''Journal of Global Optimization'' 13, 369–387 (1998); I. Bomze, M. Dür, E. de Klerk, A. Quist, C. Roos and T. Terlaky, On copositive programming and standard quadratic optimization problems. ''Journal of Global Optimization'' 18, 301–320 (2000).〕 While the further deals with the simplest problem class in non-linear optimization with an NP-hard complexity, copositive optimization allows a conic reformulation of these hard problems as a linear optimization problem over a closed convex cone of symmetric matrices, a so-called conic optimization problem. In this type of problems, the full extent of complexity is put into the cone constraint, while structural constraints and also the objective function are linear and therefore easy to handle.〔I. Bomze, Copositive optimization – recent developments and applications. ''European Journal of Operational Research'' 216, 509–520 (2012); I. Bomze, W. Schachinger, G. Uchida, Think co(mpletely)positive! Matrix properties, examples and a clustered bibliography on copositive optimization. ''Journal of Global Optimization'' 52, 423–445 (2012).〕
==Research interests==
Bomze's research interests are in the areas of nonlinear optimization, qualitative theory of dynamical systems, game theory, mathematical modeling and statistics, where he has edited one and published four books, as well as over 80 peer-reviewed articles in scientific journals and monographs. The list of his coauthors comprises almost sixty scientists from a dozen countries in four continents.
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