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 variations

G. J. Zalmai

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.