A simple proof of a primal affine scaling method

In this paper, we present a simpler proof of the result of Tsuchiya and Muramatsu on the convergence of the primal affine scaling method. We show that the primal sequence generated by the method converges to the interior of the optimum face and the dual sequence to the analytic center of the optimal dual face, when the step size implemented in the procedure is bounded by 2/3. We also prove the optimality of the limit of the primal sequence for a slightly larger step size of 2q/(3q–1), whereq is the number of zero variables in the limit. We show this by proving the dual feasibility of a cluster point of the dual sequence.Partially supported by the grant CCR-9321550 from NSF.     

Convergence analysis of the projective scaling algorithm based on a long-step homogeneous affine scaling algorithm

In this paper we give a new convergence analysis of a projective scaling algorithm. We consider a long-step affine scaling algorithm applied to a homogeneous linear programming problem obtained from the original linear programming problem. This algorithm takes a fixed fraction λ≤2/3 of the way towards the boundary of the nonnegative orthant at each iteration. The iteration sequence for the original problem is obtained by pulling back the homogeneous iterates onto the original feasible region with a conical projection, which generates the same search direction as the original projective scaling algorithm at each iterate. The recent convergence results for the long-step affine scaling algorithm by the authors are applied to this algorithm to obtain some convergence results on the projective scaling algorithm. Specifically, we will show (i) polynomiality of the algorithm with complexities of O(nL) and O(n ² L) iterations for λ<2/3 and λ=2/3, respectively; (ii) global covnergence of the algorithm when the optimal face is unbounded; (iii) convergence of the primal iterates to a relative interior point of the optimal face; (iv) convergence of the dual estimates to the analytic center of the dual optimal face; and (v) convergence of the reduction rate of the objective function value to 1−λ.     

Global convergence of the affine scaling methods for degenerate linear programming problems

In this paper we show the global convergence of the affine scaling methods without assuming any condition on degeneracy. The behavior of the method near degenerate faces is analyzed in detail on the basis of the equivalence between the affine scaling methods for homogeneous LP problems and Karmarkar's method. It is shown that the step-size 1/8, where the displacement vector is normalized with respect to the distance in the scaled space, is sufficient to guarantee the global convergence of the affine scaling methods.This paper was presented at the International Symposium Interior Point Methods for Linear Programming: Theory and Practice, held on January 18–19, 1990, at the Europa Hotel, Scheveningen, the Netherlands.     

Superlinear primal-dual affine scaling algorithms for LCP

We describe an interior-point algorithm for monotone linear complementarity problems in which primal-dual affine scaling is used to generate the search directions. The algorithm is shown to have global and superlinear convergence with Q-order up to (but not including) two. The technique is shown to be consistent with a potential-reduction algorithm, yielding the first potential-reduction algorithm that is both globally and superlinearly convergent.Corresponding author. The work of this author was based on research supported by the Office of Scientific Computing, U.S. Department of Energy, under Contract W-31-109-Eng-38.The work of this author was based on research supported by the National Science Foundation under grant DDM-9109404 and the Office of Naval Research under grant N00014-93-1-0234. This work was done while the author was a faculty member of the Systems and Industrial Engineering Department at the University of Arizona.     

The primal power affine scaling method

In this paper, we present a variant of the primal affine scaling method, which we call the primal power affine scaling method. This method is defined by choosing a realr>0.5, and is similar to the power barrier variant of the primal-dual homotopy methods considered by den Hertog, Roos and Terlaky and Sheu and Fang. Here, we analyze the methods forr>1. The analysis for 0.50<r<1 is similar, and can be readily carried out with minor modifications. Under the non-degeneracy assumption, we show that the method converges for any choice of the step size . To analyze the convergence without the non-degeneracy assumption, we define a power center of a polytope. We use the connection of the computation of the power center by Newton's method and the steps of the method to generalize the 2/3rd result of Tsuchiya and Muramatsu. We show that with a constant step size such that /(1-)^2r > 2/(2r-1) and with a variable asymptotic step size k uniformly bounded away from 2/(2r+1), the primal sequence converges to the relative interior of the optimal primal face, and the dual sequence converges to the power center of the optimal dual face. We also present an accelerated version of the method. We show that the two-step superlieear convergence rate of the method is 1+r/(r+1), while the three-step convergence rate is 1+ 3r/(r+2). Using the measure of Ostrowski, we note thet the three-step method forr=4 is more efficient than the two-step quadratically convergent method, which is the limit of the two-step method asr approaches infinity.Partially supported by the grant CCR-9321550 from NSF.     

Comparative analysis of affine scaling algorithms based on simplifying assumptions

We analyze several affine potential reduction algorithms for linear programming based on simplifying assumptions. We show that, under a strong probabilistic assumption regarding the distribution of the data in an iteration, the decrease in the primal potential function will be with high probability, compared to the guaranteed(1). ( 2n is a parameter in the potential function andn is the number of variables.) Under the same assumption, we further show that the objective reduction rate of Dikin's affine scaling algorithm is with high probability, compared to no guaranteed convergence rate.Research supported in part by NSF Grant DDM-8922636.     

Superlinear convergence of the affine scaling algorithm

In this paper we show that a variant of the long-step affine scaling algorithm (with variable stepsizes) is two-step superlinearly convergent when applied to general linear programming (LP) problems. Superlinear convergence of the sequence of dual estimates is also established. For homogeneous LP problems having the origin as the unique optimal solution, we also show that 2/3 is a sharp upper bound on the (fixed) stepsize that provably guarantees that the sequence of primal iterates converge to the optimal solution along a unique direction of approach. Since the point to which the sequence of dual estimates converge depend on the direction of approach of the sequence of primal iterates, this result gives a plausible (but not accurate) theoretical explanation for why 2/3 is a sharp upper bound on the (fixed) stepsize that guarantees the convergence of the dual estimates. The work of this author was based on research supported by the Overseas Research Scholars of the Ministry of Education, Science and Culture of Japan, 1992. The work of this author was based on research supported by the National Science Foundation (NSF) under grant DDM-9109404 and the Office of Naval Research (ONR) under grant N00014-93-1-0234. This work was done while the second author was a faculty member of the Systems and Industrial Engineering Department at the University of Arizona.     

Limiting behavior of the affine scaling continuous trajectories for linear programming problems

We consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form. By characterizing these trajectories as solutions of certain parametrized logarithmic barrier families of problems, we show that these trajectories tend to an optimal solution which in general depends on the starting point. By considering the trajectories that arise from the Lagrangian multipliers of the above mentioned logarithmic barrier families of problems, we show that the trajectories of the dual estimates associated with the affine scaling trajectories converge to the so called centered optimal solution of the dual problem. We also present results related to asymptotic direction of the affine scaling trajectories. We briefly discuss how to apply our results to linear programs formulated in formats different from the standard form. Finally, we extend the results to the primal-dual affine scaling algorithm.     

Solving nested-constraint resource allocation problems with an interior point method

Efficient methods for convex resource allocation problems usually exploit algebraic properties of the objective function. For problems with nested constraints, we show that constraint sparsity structure alone allows rapid solution with a general interior point method. The key is a special-purpose linear system solver requiring only linear time in the problem dimensions. Computational tests show that this approach outperforms the previous best algebraically specialized methods.     

An interior point Newton‐like method for non‐negative least‐squares problems with degenerate solution

An interior point approach for medium and large non‐negative linear least‐squares problems is proposed. Global and locally quadratic convergence is shown even if a degenerate solution is approached. Viable approaches for implementation are discussed and numerical results are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

Degeneracy in interior point methods for linear programming: a survey

The publication of Karmarkar's paper has resulted in intense research activity into Interior Point Methods (IPMs) for linear programming. Degeneracy is present in most real-life problems and has always been an important issue in linear programming, especially in the Simplex method. Degeneracy is also an important issue in IPMs. However, the difficulties are different in the two methods. In this paper, we survey the various theoretical and practical issues related to degeneracy in IPMs for linear programming.We survey results, which, for the most part, have already appeared in the literature. Roughly speaking, we shall deal with the effect of degeneracy on the following: the convergence of IPMs, the trajectories followed by the algorithms, numerical performance, and finding basic solutions.Partially supported by a research fund of SHELL.On leave from the Eötvös University, Budapest, and partially supported by OTKA No. 2116.     

A unified approach to the feasible point method type for nonlinear programming with linear constraints under degeneracy and the convergence properties

In this paper, we introduce a general class of feasible point algorithms for solving nonlinear programming problems with linear constraints under degeneracy. Many wellknown algorithms are its special cases. Also, many new algorithms can be derived from this general class. Under rather simple conditions, we prove the global convergence property for this class of algorithms.This research was partially supported by the National Natural Science Foundation of China. The first author's research was partially also supported by the National Science Foundation of the United States of America under Grant DCR-86-96135.     

A Note on Mascarenhas' Counterexample about Global Convergence of the Affine Scaling Algorithm

Mascarenhas gave an instance of linear programming problems to show that the long-step affine scaling algorithm can fail to converge to an optimal solution with the step-size λ=0.999 . In this note, we give a simple and clear geometrical explanation for this phenomenon in terms of the Newton barrier flow induced by projecting the homogeneous affine scaling vector field conically onto a hyperplane where the objective function is constant. Based on this interpretation, we show that the algorithm can fail for "any" λ greater than about 0.91 (a more precise value is 0.91071), which is considerably shorter than λ = 0.95 and 0.99 recommended for efficient implementations. Accepted 17 February 1998