International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 66, Pages 4145-4182
Duality models for some nonclassical problems in the calculus of
Department of Mathematics and Computer Science, Northern Michigan University, Marquette 49855, MI, USA
Received 20 March 2003
Copyright © 2003 G. J. Zalmai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Parametric and nonparametric necessary and sufficient optimality
conditions are established for a class of nonconvex variational
problems with generalized fractional objective functions and
nonlinear inequality constraints containing arbitrary norms.
Based on these optimality criteria, ten parametric and
parameter-free dual problems are constructed and appropriate
duality theorems are proved. These optimality and duality results
contain, as special cases, similar results for minmax fractional
variational problems involving square roots of positive
semidefinite quadratic forms as well as for variational problems
with fractional, discrete max, and conventional objective
functions, which are particular cases of the main problem
considered in this paper. The duality models presented here
subsume various existing duality formulations for variational
problems and include variational generalizations of a great
variety of cognate dual problems investigated previously in the
area of finite-dimensional nonlinear programming by an assortment
of ad hoc methods.