We study monotonicity of primal and dual objective values in the framework of primal-dual interior-point methods. The primal-dual aane-scaling algorithm is monotone in both objectives. We derive a condition under which a primal-dualinterior-point algorithm with a centering component is monotone. Then we propose primal-dual algorithms that are monotone in both primal and dual objective values an...

We present a new strategy for choosing primal and dual steplengths in a primal-dual interior-point algorithm for convex quadratic programming. Current implementations often scale steps equally to avoid increases in dual infeasibility between iterations. We propose that this method can be too conservative, while safeguarding an unequally-scaled steplength approach will often require fewer steps ...

We present primal-dual interior-point algorithms with polynomial iteration bounds to nd approximate solutions of semidenite programming problems. Our algorithms achieve the current best iteration bounds and, in every iteration of our algorithms, primal and dual objective values are strictly improved.

We propose a build-up interior-point method for solving an m equation n variable linear program which has the same convergence properties as its well known analogue Dikin's algorithm in dual aane form but may require less computational eeort. It diiers from Dikin's algorithm in that the ellipsoid chosen to generate the improving direction in dual space is constructed from only a subset of the d...

We introduce a new barrier function which has a linear growth term in its kernel function. So far all existing kernel functions have a quadratic (or higher degree) growth term. Despite this, a large-update primal-dual interior-point method based on this kernel function has the same iteration bound as the classical primal-dual method, which is based on the logarithmic barrier method.

In this paper, we are concerned with nonlinear minimization problems with second order cone constraints. A primal-dual interior point method is proposed for solving the problems. We also propose a new primal-dual merit function by combining the barrier penalty function and the potential function within the framework of the line search strategy, and show the global convergence property of our me...

The classical column generation is based on optimal solutions of the restricted master problems. This strategy frequently results in an unstable behaviour and may require an unnecessarily large number of iterations. To overcome this weakness, variations of the classical approach use interior points of the dual feasible set, instead of optimal solutions. In this paper, we address the primal-dual...

In [2] we investigated the lattice Λ(Df2) of all subvarieties of the variety Df2 of two-dimensional diagonal free cylindric algebras. In the present paper we investigate the lattice Λ(CA2) of all subvarieties of the variety CA2 of two-dimensional cylindric algebras. We give a dual characterization of representable two-dimensional cylindric algebras, prove that the cardinality of Λ(CA2) is that ...

Cluster algebras are a remarkable discovery of S. Fomin and A. Zelevinsky [FZI]. Cluster algebras are certain commutative algebras defined by a very simple and general data. Below we consider only cluster algebras of geometric origin, which are quite general and probably the most important examples of cluster algebras. We show that a cluster algebra of geometric origin is part of a richer struc...

A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the inne...