Asymptotic duality over closed convex sets

AbstractThe asymptotic duality theory of linear programming over closed convex cones [4] is extended to closed convex sets, by embedding such sets in appropriate cones. Applications to convex programming and to approximation theory are given

Controlled dual perturbations for central path trajectories in geometric programming

European Journal of Operational Research733524-531EJOR

An infeasible interior-point algorithm for solving primal and dual geometric programs

In this paper an algorithm is presented for solving the classical posynomial geometric programming dual pair of problems simultaneously. The approach is by means of a primal-dual infeasible algorithm developed simultaneously for (i) the dual geometric program after logarithmic transformation of its objective function, and (ii) its Lagrangian dual program. Under rather general assumptions, the mechanism defines a primal-dual infeasible path from a specially constructed, perturbed Karush-Kuhn-Tucker system. Subfeasible solutions, as described by Duffin in 1956, are generated for each program whose primal and dual objective function values converge to the respective primal and dual program values. The basic technique is one of a predictor-corrector type involving Newton's method applied to the perturbed KKT system, coupled with effective techniques for choosing iterate directions and step lengths. We also discuss implementation issues and some sparse matrix factorizations that take advantage of the very special structure of the Hessian matrix of the logarithmically transformed dual objective function. Our computational results on 19 of the most challenging GP problems found in the literature are encouraging. The performance indicates that the algorithm is effective regardless of the degree of difficulty, which is a generally accepted measure in geometric programming

Primal-dual stability in continuous linear optimization

Any linear (ordinary or semi-infinite) optimization problem, and also its dual problem, can be classified as either inconsistent or bounded or unbounded, giving rise to nine duality states, three of them being precluded by the weak duality theorem. The remaining six duality states are possible in linear semi-infinite programming whereas two of them are precluded in linear programming as a consequence of the existence theorem and the non-homogeneous Farkas Lemma. This paper characterizes the linear programs and the continuous linear semi-infinite programs whose duality state is preserved by sufficiently small perturbations of all the data. Moreover, it shows that almost all linear programs satisfy this stability property.This research was supported by DGES and FEDER, Grant MTM2005-08572-C03-01 and partially supported by CONACyT of MX.Grant 44003