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  • 1
    Electronic Resource
    Electronic Resource
    Using copositivity for global optimality criteria in concave quadratic programming problems (1993)
    Springer
    Mathematical programming 62 (1993), S. 575-580 
    Details
    ISSN: 1436-4646
    Keywords: Non-convex non-linear programming ; global optimization ; second-order optimality conditions ; copositive matrices ; quadratic programming ; convex maximization problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this note we specify a necessary and sufficient condition for global optimality in concave quadratic minimization problems. Using this condition, it follows that, from the perspective of worst-case complexity of concave quadratic problems, the difference between local and global optimality conditions is not as large as in general. As an essential ingredient, we here use theε-subdifferential calculus via an approach of Hiriart-Urruty and Lemarechal (1990).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585185
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  • 2
    Electronic Resource
    Electronic Resource
    Dynamical aspects of evolutionary stability (1990)
    Springer
    Monatshefte für Mathematik 110 (1990), S. 189-206 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Selection is often viewed as a process that maximizes the average fitness of a population. However, there are often constraints even on the phenotypic level which may prevent fitness optimization. Consequently, in evolutionary game theory, models of frequency dependent selection are investigated, which focus on equilibrium states that are characterized by stability (or uninvadability) rather than by optimality. The aim of this article is to relate these stability notions with asymptotic stability in the so-called “replicator dynamics”, by generalizing results, which are well-known for elementary situations, to a fairly general setting applicable, e.g. to complex populations. Moreover, a purely dynamical characterization of evolutionary stability and uninvadability is presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01301675
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  • 3
    Electronic Resource
    Electronic Resource
    A dynamical characterization of evolutionarily stable states (1992)
    Springer
    Annals of operations research 37 (1992), S. 229-244 
    Details
    ISSN: 1572-9338
    Keywords: Asymptotic stability ; evolutionary games ; mixed strategies ; polymorphisms ; replicator dynamics ; stable sets
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Evolutionary stability, the central solution concept in evolutionary game theory, is closely related to local asymptotic stability in a certain nonlinear dynamical system operating on the state space, the so-called "replicator dynamics". However, a purely dynamical characterization of evolutionary stability is not available in an elementary manner. This characterization can be achieved by investigating so-called "derived games" which consist of mixed strategies corresponding to successful states in the original game. Using well-known facts, several characterization results are obtained within this context. These also may shed light on the extremality properties of evolutionary stability.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02071058
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  • 4
    Electronic Resource
    Electronic Resource
    Measurable supports, reducible spaces and the structure of the optimal σ-field in unbiased estimation (1986)
    Springer
    Monatshefte für Mathematik 101 (1986), S. 27-38 
    Details
    ISSN: 1436-5081
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper generalizes a result bySapozhnikov relating the structure of the optimal σ-field in unbiased estimation to the reducibility of certain linear subspaces of estimators. For this purpose, the concept of a measurable support of (uncountably infinite) sets consisting of random variables is introduced, which could be of general interest. It turns out, that for a wide class of measure spaces, measurable supports of any subsets of measurable functions exist.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01326844
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  • 5
    Electronic Resource
    Electronic Resource
    Lotka-Volterra equation and replicator dynamics: new issues in classification (1995)
    Springer
    Biological cybernetics 72 (1995), S. 447-453 
    Details
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract.  Replicator dynamics serves for modelling many biological processes, e.g. evolution of animal behaviour, but also selection in population genetics, and even prebiotic evolution. The Lotka-Volterra system is used in mathematical ecology to describe the interaction of two populations over time. Here, predator/prey situations can be modelled as well as competition for a resource. After a short account on applications and ramifications of planar classification results, a lacuna is closed which appeared in an earlier publication on classification (Biol Cybern 48: 201–211, 1983). The now complete list of possible phase portraits under the replicator dynamics as well as under the Lotka-Volterra system is specified and contains, up to flow reversal, 49 qualitatively different cases for the former, and 110 or 67 for the latter dynamics, depending on whether or not one discriminates between different asymptotic slope behaviour. Furthermore, a systematic investigation of the flow under the replicator dynamics exhibits a variety of non-robust models which illustrate dynamic aspects of some solution concepts in evolutionary game theory, a field that is receiving widespread interest in the recent literature.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s004220050147
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  • 6
    Electronic Resource
    Electronic Resource
    Lotka-Volterra equation and replicator dynamics: new issues in classification (1995)
    Springer
    Biological cybernetics 72 (1995), S. 447-453 
    Details
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract Replicator dynamics serves for modelling many biological processes, e.g. evolution of animal behaviour, but also selection in population genetics, and even prebiotic evolution. The Lotka-Volterra system is used in mathematical ecology to describe the interaction of two populations over time. Here, predator/prey situations can be modelled as well as competition for a resource. After a short account on applications and ramifications of planar classification results, a lacuna is closed which appeared in an earlier publication on classification (Biol Cybern 48:201–211, 1983). The now complete list of possible phase portraits under the replicator dynamics as well as under the Lotka-Volterra system is specified and contains, up to flow reversal, 49 qualitatively different cases for the former, and 110 or 67 for the latter dynamics, depending on whether or not one discriminates between different asymptotic slope behaviour. Furthermore, a systematic investigation of the flow under the replicator dynamics exhibits a variety of non-robust models which illustrate dynamic aspects of some solution concepts in evolutionary game theory, a field that is receiving widespread interest in the recent literature.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00201420
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    Lotka-Volterra equation and replicator dynamics: A two-dimensional classification (1983)
    Springer
    Biological cybernetics 48 (1983), S. 201-211 
    Details
    ISSN: 1432-0770
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Computer Science , Physics
    Notes: Abstract The replicator equation arises if one equips a certain game theoretical model for the evolution of behaviour in animal conflicts with dynamics. It serves to model many biological processes not only in sociobiology but also in population genetics, mathematical ecology and even in prebiotic evolution. After a short survey of these applications, a complete classification of the two-dimensional phase flows is presented. The methods are also used to obtain a classification of phase portraits of the well-known generalized Lotka-Volterra equation in the plane.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00318088
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  • 8
    Electronic Resource
    Electronic Resource
    Non-cooperative two-person games in biology: A classification (1986)
    Springer
    International journal of game theory 15 (1986), S. 31-57 
    Details
    ISSN: 1432-1270
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract This article compares evolutionary equilibrium notions with solution concepts in rational game theory. Both static and dynamic evolutionary game theory are treated. The methods employed by dynamic theory, so-called “game dynamics”, could be discovered to be relevant for rational game theory also.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01769275
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  • 9
    Electronic Resource
    Electronic Resource
    Cross entropy minimization in uninvadable states of complex populations (1991)
    Springer
    Journal of mathematical biology 30 (1991), S. 73-87 
    Details
    ISSN: 1432-1416
    Keywords: Frequency dependent selection ; Evolutionary game theory ; Replicator dynamics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Selection is often. viewed as a process that maximizes the average fitness of a population. However, there are often constraints even on the phenotypic level which may prevent fitness optimization. Consequently, in evolutionary game theory, models of frequency dependent selection are investigated, which focus on equilibrium states that are characterized by stability (or uninvadability) rather than by optimality. The aim of this article is to show that nevertheless there is a biologically meaningful quantity, namely cross (fitness) entropy, which is optimized during the course of evolution: a dynamical model adapted to evolutionary games is presented which has the property that relative entropy decreases monotonically, if the state of a (complex) population is close to an uninvadable state. This result may be interpreted as if evolution has an “order stabilizing” effect.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00168008
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  • 10
    Electronic Resource
    Electronic Resource
    Evolution towards the Maximum Clique (1997)
    Springer
    Journal of global optimization 10 (1997), S. 143-164 
    Details
    ISSN: 1573-2916
    Keywords: indefinite quadratic programming; replicator dynamics;evolutionary game; independent set
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract As is well known, the problem of finding a maximum clique in a graph isNP-hard. Nevertheless, NP-hard problems may have easy instances. This paperproposes a new, global optimization algorithm which tries to exploit favourabledata constellations, focussing on the continuous problem formulation: maximizea quadratic form over the standard simplex. Some general connections of thelatter problem with dynamic principles of evolutionary game theory areestablished. As an immediate consequence, one obtains a procedure whichconsists (a) of an iterative part similar to interior-path methods based on theso-called replicator dynamics; and (b) a routine to escape from inefficient,locally optimal solutions. For the special case of finding a maximum clique ina graph where the quadratic form arises from a regularization of the adjacencematrix, part (b), i.e. escaping from maximal cliques not of maximal size, isaccomplished with block pivoting methods based on (large) independent sets,i.e. cliques of the complementary graph. A simulation study is included whichindicates that the resulting procedure indeed has some merits.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008230200610
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