Benjamin Jansen

SENSITIVITY ANALYSIS IN LINEAR PROGRAMMING: JUST BE CAREFUL

In this paper we review the topic of sensitivity analysis in linear programming. We describe the problems that may occur when using standard software and advocate a framework for performing complete sensitivity analysis. Three approaches can be incorporated within it: one using bases, an approach using the optimal partition and one using optimal values. We elucidate problems and solutions with an academic example and give results from an implementation of these approaches to a large practical linear programming model of an oil refinery. This shows that the approaches are viable and useful in practice.

Primal-dual target-following algorithms for linear programming

In this paper, we propose a method for linear programming with the property that, starting from an initial non-central point, it generates iterates that simultaneously get closer to optimality and closer to centrality. The iterates follow paths that in the limit are tangential to the central path. Together with the convergence analysis, we provide a general framework which enables us to analyze various primal-dual algorithms in the literature in a short and uniform way.

The theory of linear programming:skew symmetric self-dual problems and the central path *

The literature in the field of interior point methods for linear programming has been almost exclusively algorithm oriented. Recently Guler, Roos, Terlaky and Vial presented a complete duality theory for linear programming based on the interior point approach. In this paper we present a more simple approach which is based on an embedding of the primal problem and its dual into a skew symmetric self-dual problem. This embedding is essentially due Ye, Todd and Mizuno First we consider a skew symmetric self-dual linear program. We show that the strong duality theorem trivally holds in this case. Then, using the logarithmic barrier problem and the central path, the existence of a strictly complementary optimal solution is proved. Using the embedding just described, we easily obtain the strong duality theorem and the existence of strictly complementary optimal solutions for general linear programming problems

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.].

Primal-dual algorithms for linear programming based on the logarithmic barrier method

AbstractIn this paper, we deal with primal-dual interior point methods for solving the linear programming problem. We present a short-step and a long-step path-following primal-dual method and derive polynomial-time bounds for both methods. The iteration bounds are as usual in the existing literature, namely

A potential reduction approach to the frequency assignment problem

O(\sqrt n L)

Polynomiality of primal-dual affine scaling algorithms for nonlinear complementarity problems

iterations for the short-step variant andO(nL) for the long-step variant. In the analysis of both variants, we use a new proximity measure, which is closely related to the Euclidean norm of the scaled search direction vectors. The analysis of the long-step method depends strongly on the fact that the usual search directions form a descent direction for the so-called primal-dual logarithmic barrier function.

Target-Following Methods for Linear Programming

In this paper the new polynomial affine scaling algorithm of Jansen, Roos, and Terlaky for linear programming (LP) is extended to positive semidefinite (PSD) linear complementarity problems. The algorithm is immediately further generalized to allow higher order scaling. These algorithms are also new for the LP case. The analysis is based on Lings proof for the LP case; hence, it allows an arbitrary interior feasible pair to start with. With the scaling of Jansen, Roos, and Terlaky, the complexity of the algorithm is