In this paper, we show that the moving directions of the primal-affine scaling method (with logarithmic barrier function), the dual-affine scaling method (with logarithmic barrier function), and the primal-dual interior point method are merely the Newton directions along three different algebraic "paths" that lead to a solution of the Karush-Kuhn-Tucker conditions of a given linear programming ...

We discuss interior point methods for large-scale linear programming, with an emphasis on methods that are useful for problems arising in telecommunications. We give the basic framework of a primal-dual interior point method, and consider the numerical issues involved in calculating the search direction in each iteration, including the use of factorization methods and/or preconditioned conjugat...

We consider an extension of ordinary linear programming (LP) that adds weighted logarithmic barrier terms for some variables. The resulting problem generalizes both LP and the problem of finding the weighted analytic center of a polytope. We show that the problem has a dual of the same form and give complexity results for several different interior-point algorithms. We obtain an improved comple...

We consider the application of primal-dual interior methods to the optimization of systems arising in the finite-element discretization of a class of elliptic variational inequalities. These problems lead to very large (possibly non-convex) optimization problems with upper and lower bound constraints. When interior methods are applied to the discretized problem, the resulting linear systems hav...

Abstract. In this paper a duality of transformation functions in the interior point method is treated. A dual pair of convex or linear programming problems is considered and the primal problem is transformed by the parametrized transformation function of a more general form than logarithmic is. The construction of the parametrized transformation function for the dual problem is carried out so t...

A correlation lattice is an algebra (B,∧,∨, σ, 0, 1) where (B,∧,∨, 0, 1) is a bounded lattice and σ is a dual endomorphism on B which has the property σ(x) = x for a fixed number n, n ∈ 2N + 1 ([3], [4]). This notion generalizes orthomodular lattices, and in the distributive case, Boolean algebras, De Morgan algebras and some classes of Ockham algebras. Moreover, in [4], D. Schweigert and M. Sz...

In this paper we propose new primal-dual interior point methods (IPMs) for P∗(κ) linear complementarity problems (LCPs) and analyze the iteration complexity of the algorithm. New search directions and proximity measures are defined based on a class of kernel functions, ψ(t) = t 2−1 2 − R t 1 e q “ 1 ξ −1 ” dξ, q ≥ 1. If a strictly feasible starting point is available and the parameter q = log „...

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...