A new algorithm for generalized fractional programs

A new dual problem for convex generalized fractional programs with no duality gap is presented and it is shown how this dual problem can be efficiently solved using a parametric approach. The resulting algorithm can be seen as “dual” to the Dinkelbach-type algorithm for generalized fractional programs since it approximates the optimal objective value of the dual (primal) problem from below. Convergence results for this algorithm are derived and an easy condition to achieve superlinear convergence is also established. Moreover, under some additional assumptions the algorithm also recovers at the same time an optimal solution of the primal problem. We also consider a variant of this new algorithm, based on scaling the “dual” parametric function. The numerical results, in case of quadratic-linear ratios and linear constraints, show that the performance of the new algorithm and its scaled version is superior to that of the Dinkelbach-type algorithms. From the computational results it also appears that contrary to the primal approach, the “dual” approach is less influenced by scaling

A Theory of the Perturbed Consumer with General Budgets

We consider demand systems for utility-maximizing consumers facing general budget constraints whose utilities are perturbed by additive linear shifts in marginal utilities. Budgets are required to be compact but are not required to be convex. We define demand generating functions (DGF) whose subgradients with respect to these perturbations are convex hulls of the utility-maximizing demands. We give necessary as well as sufficient conditions for DGF to be consistent with utility maximization, and establish under quite general conditions that utility-maximizing demands are almost everywhere single-valued and smooth in their arguments. We also give sufficient conditions for integrability of perturbed demand. Our analysis provides a foundation for applications of consumer theory to problems with nonlinear budget constraints

Introduction to Convex and Quasiconvex Analysis

In the first chapter of this book the basic results within convex and quasiconvex analysis are presented. In Section 2 we consider in detail the algebraic and topological properties of convex sets within Rn together with their primal and dual representations. In Section 3 we apply the results for convex sets to convex and quasiconvex functions and show how these results can be used to give primal and dual representations of the functions considered in this field. As such, most of the results are well-known with the exception of Subsection 3.4 dealing with dual representations of quasiconvex functions. In Section 3 we consider applications of convex analysis to noncooperative game and minimax theory, Lagrangian duality in optimization and the properties of positively homogeneous evenly quasiconvex functions. Among these result an elementary proof of the well-known Sion’s minimax theorem concerningquasiconvex-quasiconcave bifunctions is presented, thereby avoiding the less elementary fixed point arguments. Most of the results are proved in detail and the authors have tried to make these proofs as transparent as possible. Remember that convex analysis deals with the study of convex cones and convex sets and these objects are generalizations of linear subspaces and affine sets, thereby extending the field of linear algebra. Although some of the proofs are technical, it is possible to give a clear geometrical interpretation of the main ideas of convex analysis. Finally in Section 5 we list a short and probably incomplete overview on the history of convex and quasiconvex analysis