On the Primal-Dual Geometry of Level Sets in Linear and Conic Optimization

On the Primal - Dual Geometry of Levelsets in Linear and Conic Optimization 1

Generalization of Primal-Dual Interior-Point Methods to Convex Optimization Problems in Conic Form

We generalize primal-dual interior-point methods for linear programming problems to the convex optimization problems in conic form. Previously, the most comprehensive theory of symmetric primal-dual interior-point algorithms was given by Nesterov and Todd 8, 9] for the feasible regions expressed as the intersection of a symmetric cone with an aane subspace. In our setting, we allow an arbitrary...

Primal and dual robust counterparts of uncertain linear programs: an application to portfolio selection

This paper proposes a family of robust counterpart for uncertain linear programs (LP) which is obtained for a general definition of the uncertainty region. The relationship between uncertainty sets using norm bod-ies and their corresponding robust counterparts defined by dual norms is presented. Those properties lead us to characterize primal and dual robust counterparts. The researchers show t...

Primal-dual Interior-Point Methods with Asymmetric Barriers

In this paper we develop several polynomial-time interior-point methods (IPM) for solving nonlinear primal-dual conic optimization problem. We assume that the barriers for the primal and the dual cone are not conjugate. This broken symmetry does not allow to apply the standard primal-dual IPM. However, we show that in this situation it is also possible to develop very efficient optimization met...

Exponential membership function and duality gaps for I-fuzzy linear programming problems

Fuzziness is ever presented in real life decision making problems. In this paper, we adapt the pessimistic approach tostudy a pair of linear primal-dual problem under intuitionistic fuzzy (I-fuzzy) environment and prove certain dualityresults. We generate the duality results using exponential membership and non-membership functions to represent thedecision maker’s satisfaction and dissatisfacti...