We formulate a pair of multiobjective nonlinear generalized symmetric dual variational problems involving vector-valued functions which unify the Wolfe and Mond–Weir models. For a special case, our problems become the symmetric dual pair of D. S. Kim and W. J. Lee (1998,J. Math. Anal. Appl.218, 34–48). Under invexity assumptions, we establish weak, strong, and converse duality theorems for our variational problems by using the concept of efficiency. Also, we give the static case of our multiobjective generalized symmetric duality results. Several known results are obtained as special cases.
We present some equivalences between some symmetric dualities in nonlinear programming and some matrix games. We study three pairs of dual problems: Wolfe, Mond–Weir and fractional types. For each pair, the paper presents two different zero-sum games whose Nash equilibria correspond to the solutions of the pair of nonlinear symmetric dual problems. And certain conclusions about symmetric dual fractional programming and its special cases are also treated.
We prove that groups acting geometrically on delta-quasiconvex spaces contain no essential Baumslag-Solitar quotients as subgroups. This implies that they are translation discrete, meaning that the translation numbers of their nontorsion elements are bounded away from zero.