Cees Roos

A survey of search directions in interior point methods for linear programming

A basic characteristic of an interior point algorithm for linear programming is the search direction. Many papers on interior point algorithms only give an implicit description of the search direction. In this report we derive explicit expressions for the search directions used in many well-known algorithms. Comparing these explicit expressions gives a good insight into the similarities and differences between the various algorithms. Moreover, we give a survey of projected gradient and Newton directions for all potential and barrier functions. This is done both for the affine and projective variants.

An exponential example for Terlaky's pivoting rule for the criss-cross simplex method

Recently T. Terlaky has proposed a new pivoting rule for the criss-cross simplex method for linear programming and he proved that his rule is convergent. In this note we show that the required number of iterations may be exponential in the number of variables and constraints of the problem.

New Complexity Analysis of the Primal—Dual Newton Method for Linear Optimization

We deal with the primal–dual Newton method for linear optimization (LO). Nowadays, this method is the working horse in all efficient interior point algorithms for LO, and its analysis is the basic element in all polynomiality proofs of such algorithms. At present there is still a gap between the practical behavior of the algorithms and the theoretical performance results, in favor of the practical behavior. This is especially true for so-called large-update methods. We present some new analysis tools, based on a proximity measure introduced by Jansen et al., in 1994, that may help to close this gap. This proximity measure has not been used in the analysis of large-update methods before. The new analysis does not improve the known complexity results but provides a unified way for the analysis of both large-update and small-update methods.

A polynomial primal-dual Dikin-type algorithm for linear programming

In this paper we present a new primal-dual affine scaling method for linear programming. The method yields a strictly complementary optimal solution pair, and also allows a polynomial-time convergence proof. The search direction is obtained by using the original idea of Dikin, namely by minimizing the objective function which is the duality gap in the primal-dual case, over some suitable ellipsoid. This gives rise to completely new primal-dual affine scaling directions, having no obvious relation with the search directions proposed in the literature so far. The new directions guarantee a significant decrease in the duality gap in each iteration, and at the same time they drive the iterates to the central path. In the analysis of our algorithm we use a barrier function which is the natural primal-dual generalization of Karmarkars potential function. The iteration bound is OnL, which is a factor OL better than the iteration bound of an earlier primal-dual affine scaling method Monteiro, Adler and Resende [Monteiro, R. D. C., I. Adler, M. G. C. Resende. 1990. A polynomial-time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power series extension. Math. Oper. Res.15 191--214.].

A generalization of the BCH bound for cyclic codes, including the Hartmann-Tzeng bound

Abstract In this paper it is shown that both the BCH bound and the Hartmann-Tzeng bound for the minimum distance of a cyclic code can be obtained quite easily as consequences of an elementary result concerning the defining set of its zeros.

A Full Nesterov–Todd Step Infeasible Interior-Point Method for Second-Order Cone Optimization

After a brief introduction to Jordan algebras, we present a primal–dual interior-point algorithm for second-order conic optimization that uses full Nesterov–Todd steps; no line searches are required. The number of iterations of the algorithm coincides with the currently best iteration bound for second-order conic optimization. We also generalize an infeasible interior-point method for linear optimization to second-order conic optimization. As usual for infeasible interior-point methods, the starting point depends on a positive number. The algorithm either finds a solution in a finite number of iterations or determines that the primal–dual problem pair has no optimal solution with vanishing duality gap.

A logarithmic barrier cutting plane method for convex programming

The paper presents a logarithmic barrier cutting plane algorithm for convex (possibly non-smooth, semi-infinite) programming. Most cutting plane methods, like that of Kelley, and Cheney and Goldstein, solve a linear approximation (localization) of the problem and then generate an additional cut to remove the linear programs optimal point. Other methods, like the “central cutting” plane methods of Elzinga-Moore and Goffin-Vial, calculate a center of the linear approximation and then adjust the level of the objective, or separate the current center from the feasible set. In contrast to these existing techniques, we develop a method which does not solve the linear relaxations to optimality, but rather stays in the interior of the feasible set. The iterates follow the central path of a linear relaxation, until the current iterate either leaves the feasible set or is too close to the boundary. When this occurs, a new cut is generated and the algorithm iterates. We use the tools developed by den Hertog, Roos and Terlaky to analyze the effect of adding and deleting constraints in long-step logarithmic barrier methods for linear programming. Finally, implementation issues and computational results are presented. The test problems come from the class of numerically difficult convex geometric and semi-infinite programming problems.

A note on the existence of perfect constant weight codes

By using an appropriate anticode it is shown that the Johnson graph J(n, @n) can contain an e-perfect subset only if @n =< (n - 1)(2e + 1)e.

An Improved and Simplified Full-Newton Step

We present an improved version of an infeasible interior-point method for linear optimization published in 2006. In the earlier version each iteration consisted of one so-called feasibility step and a few---at most three---centering steps. In this paper each iteration consists of only a feasibility step, whereas the iteration bound improves the earlier bound by a factor

O(n)

2\sqrt{2}