Duality results on countably infinite linear programs are scarce. Subspaces that admit an interior point, which is a sufficient condition for a zero duality gap, yield a dual where the constraints cannot be expressed using the ordinary transpose of the primal constraint matrix. Subspaces that permit a dual with this transpose do not admit an interior point. This difficulty has stumped researche...

In this paper we continue the development of a theoretical foundation for efficient primal-dual interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled (see [NT97]). The class of problems under consideration includes linear programming, semidefinite programming and convex quadratically constrained quadratic prog...

Interior point methods were widely used in the past in the form of barrier methods. In linear programming, the simplex method dominated, mainly due to inefficiencies of barrier methods. Interior point methods became quit popular again after 1984, when Karmarkar announced a fast polynomial-time interior method for nonlinear programming [Karmarkar, 1984]. In this section we present primal-dual in...

In this paper, we present a new primal-dual interior-point algorithm for linear optimization based on a trigonometric kernel function. By simple analysis, we derive the worst case complexity for a large-update primal-dual interior-point method based on this kernel function. This complexity estimate improves a result from [1] and matches the one obtained in [2].

We observe a curious property of dual versus primal-dual path-following interior-point methods when applied to unbounded linear or conic programming problems in dual form. While primal-dual methods can be viewed as implicitly following a central path to detect primal infeasibility and dual unboundedness, dual methods can sometimes implicitly move away from the analytic center of the set of infe...

In this paper we present a generic primal-dual interior point methods (IPMs) for linear optimization in which the search direction depends on a univariate kernel function which is also used as proximity measure in the analysis of the algorithm. The proposed kernel function does not satisfy all the conditions proposed in [2]. We show that the corresponding large-update algorithm improves the ite...

In this paper, we consider a primal-dual interior point method for solving nonlinear semidefinite programming problems. By combining the primal barrier penalty function and the primal-dual barrier function, a new primal-dual merit function is proposed within the framework of the line search strategy. We show the global convergence property of our method.

Nonlinear Primal-Dual Interior Point methods have been recognized as a numerical tool of great potential to solve constrained optimization problems in electric power systems. Recently, a number of versions of the primal-dual Interior Point algorithm have been proposed in the area of mathematical programming. In this work, the application of the Largest-Step Central-Path algorithm to the Optimal...