• Much of the earlier work in generalized convexity was done by researchers in mathematical programming (e.g., Avriel, Fenchel, Mangasarian, Martos) and in economic theory (e.g., Arrow, Diewert, Eichhorn, Kannai). To this day, both disciplines continue to benefit from this kind of research. In addition, generalized convexity has been studied in management science, engineering, applied sciences, mathematics, statistics, for example. We also mention fractional programming (optimization of ratios of functions) with well over one thousand publications which significantly overlaps with generalized convexity. On the other hand, a nontraditional topic such as abstract convexity is gaining interest now too. Another development in generalized convexity has been the study of generalized monotone maps, introduced in such a way that in case of gradient maps generalized monotonicity characterizes generalized convexity of the underlying function. Generalized monotone maps play a role in economics, complementarity problems, variational inequalities and, more generally, equilibrium problems arising in various disciplines.
  • Like in other areas of mathematical programming, contributions in generalized convexity/monotonicity range from theoretical and algorithmic to computational and applied research.
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