Global InformationImmanuel Bomze information
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.[1]
In “Non-cooperative two-person games in biology: a classification” (1986)[2] 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.[3]
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".[4] 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.[5]
- ^ 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).
- ^ International Journal of Game Theory 15, 31–57.
- ^ 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).
- ^ 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).
- ^ 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).