Superlinear and quadratic convergence of primal-dual interior-point methods for linear programming revisited

Recently, Zhang, Tapia, and Dennis (Ref. 1) produced a superlinear and quadratic convergence theory for the duality gap sequence in primal-dual interior-point methods for linear programming. In this theory, a basic assumption for superlinear convergence is the convergence of the iteration sequence; and a basic assumption for quadratic convergence is nondegeneracy. Several recent research projects have either used or built on this theory under one or both of the above-mentioned assumptions. In this paper, we remove both assumptions from the Zhang-Tapia-Dennis theory.Dedicated to the Memory of Magnus R. Hestenes, 1906–1991This research was supported in part by NSF Cooperative Agreement CCR-88-09615 and was initiated while the first author was at Rice University as a Visiting Member of the Center for Research in Parallel Computation.The authors thank Yinyu Ye for constructive comments and discussions concerning this material.This author was supported in part by NSF Grant DMS-91-02761 and DOE Grant DE-FG05-91-ER25100.This author was supported in part by AFOSR Grant 89-0363, DOE Grant DE-FG05-86-ER25017, and ARO Grant 9DAAL03-90-G-0093.     

Feasibility issues in a primal-dual interior-point method for linear programming

A new method for obtaining an initial feasible interior-point solution to a linear program is presented. This method avoids the use of a big-M, and is shown to work well on a standard set of test problems. Conditions are developed for obtaining a near-optimal solution that is feasible for an associated problem, and details of the computational testing are presented. Other issues related to obtaining and maintaining accurate feasible solutions to linear programs with an interior-point method are discussed. These issues are important to consider when solving problems that have no primal or dual interior-point feasible solutions.     

A lower bound on the number of iterations of long-step primal-dual linear programming algorithms

Recently, Todd has analyzed in detail the primal-dual affine-scaling method for linear programming, which is close to what is implemented in practice, and proved that it may take at leastn ^1/3 iterations to improve the initial duality gap by a constant factor. He also showed that this lower bound holds for some polynomial variants of primal-dual interior-point methods, which restrict all iterates to certain neighborhoods of the central path. In this paper, we further extend his result to long-step primal-dual variants that restrict the iterates to a wider neighborhood. This neigh-borhood seems the least restrictive one to guarantee polynomiality for primal-dual path-following methods, and the variants are also even closer to what is implemented in practice.Research supported in part by NSF, AFOSR and ONR through NSF Grant DMS-8920550.This author is supported in part by NSF Grant DDM-9207347. Part of thiw work was done while the author was on a sabbatical leave from the University of Iowa and visiting the Cornell Theory Center, Cornell University, Ithaca, NY 14853, supported in part by the Cornell Center for Applied Mathematics and by the Advanced Computing Research Institute, a unit of the Cornell Theory Center, which receives major funding from the National Science Foundation and IBM Corporation, with additional support from New York State and members of its Corporate Research Institute.     

A primal-dual interior-point method for linear programming based on a weighted barrier function

One motivation for the standard primal-dual direction used in interior-point methods is that it can be obtained by solving a least-squares problem. In this paper, we propose a primal-dual interior-point method derived through a modified least-squares problem. The direction used is equivalent to the Newton direction for a weighted barrier function method with the weights determined by the current primal-dual iterate. We demonstrate that the Newton direction for the usual, unweighted barrier function method can be derived through a weighted modified least-squares problem. The algorithm requires a polynomial number of iterations. It enjoys quadratic convergence if the optimal vertex is nondegenerate.The research of the second author was supported in part by ONR Grants N00014-90-J-1714 and N00014-94-1-0391.     

On the convergence of the iteration sequence in primal-dual interior-point methods

Recently, numerous research efforts, most of them concerned with superlinear convergence of the duality gap sequence to zero in the Kojima—Mizuno—Yoshise primal-dual interior-point method for linear programming, have as a primary assumption the convergence of the iteration sequence. Yet, except for the case of nondegeneracy (uniqueness of solution), the convergence of the iteration sequence has been an important open question now for some time. In this work we demonstrate that for general problems, under slightly stronger assumptions than those needed for superlinear convergence of the duality gap sequence (except of course the assumption that the iteration sequence converges), the iteration sequence converges. Hence, we have not only established convergence of the iteration sequence for an important class of problems, but have demonstrated that the assumption that the iteration sequence converges is redundant in many of the above mentioned works.This research was supported in part by NSF Coop. Agr. No. CCR-8809615. A part of this research was performed in June, 1991 while the second and the third authors were at Rice University as visiting members of the Center for Research in Parallel Computation.Corresponding author. Research supported in part by AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Research supported in part by NSF DMS-9102761 and DOE DE-FG05-91ER25100.Research supported in part by NSF DDM-8922636.     

A primal-dual interior-point algorithm for quadratic programming

In this paper we propose a primal-dual interior-point method for large, sparse, quadratic programming problems. The method is based on a reduction presented by Gonzalez-Lima, Wei, and Wolkowicz [14] in order to solve the linear systems arising in the primal-dual methods for linear programming. The main features of this reduction is that it is well defined at the solution set and it preserves sparsity. These properties add robustness and stability to the algorithm and very accurate solutions can be obtained. We describe the method and we consider different reductions using the same framework. We discuss the relationship of our proposals and the one used in the LOQO code. We compare and study the different approaches by performing numerical experimentation using problems from the Maros and Meszaros collection. We also include a brief discussion on the meaning and effect of ill-conditioning when solving linear systems.This work was partially supported by DID-USB (GID-001).     

A modified layered-step interior-point algorithm for linear programming

The layered-step interior-point algorithm was introduced by Vavasis and Ye. The algorithm accelerates the path following interior-point algorithm and its arithmetic complexity depends only on the coefficient matrixA. The main drawback of the algorithm is the use of an unknown big constant in computing the search direction and to initiate the algorithm. We propose a modified layered-step interior-point algorithm which does not use the big constant in computing the search direction. The constant is required only for initialization when a well-centered feasible solution is not available, and it is not required if an upper bound on the norm of a primal—dual optimal solution is known in advance. The complexity of the simplified algorithm is the same as that of Vavasis and Ye. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Research supported in part by ONR contract N00014-94-C-0007 and the Grant-in-Aid for Scientific Research (C) 08680478 and the Grant-in-Aid for Encouragement of Young Scientists (A) 08780227 of the Ministry of Science, Education and Culture of Japan. This research was partially done while S. Mizuno and T. Tsuchiya were visiting IBM Almaden Research Center in the summer of 1995.     

Solving symmetric indefinite systems in an interior-point method for linear programming

We describe an implementation of a primal—dual path following method for linear programming that solves symmetric indefinite augmented systems directly by Bunch—Parlett factorization, rather than reducing these systems to the positive definite normal equations that are solved by Cholesky factorization in many existing implementations. The augmented system approach is seen to avoid difficulties of numerical instability and inefficiency associated with free variables and with dense columns in the normal equations approach. Solving the indefinite systems does incur an extra overhead, whose median is about 40% in our tests; but the augmented system approach proves to be faster for a minority of cases in which the normal equations have relatively dense Cholesky factors. A detailed analysis shows that the augmented system factorization is reliable over a fairly large range of the parameter settings that control the tradeoff between sparsity and numerical stability.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.This work has been supported in part by National Science Foundation grants DDM-8908818 (Fourer) and CCR-8810107 (Mehrotra), and by a grant from GTE Laboratories (Mehrotra).     

Primal interior-point decomposition algorithms for two-stage stochastic extended second-order cone programming

ABSTRACT

We study and solve the two-stage stochastic extended second-order cone programming problem. We show that the barrier recourse functions and the composite barrier functions for this optimization problem are self-concordant families with respect to barrier parameters. These results are used to develop primal decomposition-based interior-point algorithms. The worst case iteration complexity of the developed algorithms is shown to be the same as that for the short- and long-step primal interior algorithms applied to the extensive formulation of our problem.     

On the finite convergence of interior-point algorithms for linear programming

It has been shown [8] that numerous interior-point algorithms for linear programming (LP) generate solution sequences that converge to strict complementarity solutions, or interior solutions on the optimal face. In this note we further establish a theoretical base for Gay's test (Gay, 1989) to identify the optimal face, and develop a new termination procedure to obtain an exact solution on the optimal face. We also report some numerical results for solving a set of LP test problems, each of which has a highly degenerate and unbounded optimal face.Research supported in part by NSF Grant DDM-8922636, The Iowa Business School Summer Grant, and the Interdisciplinary Research Grant of the University of Iowa Center for Advanced Studies.     

Polynomial affine algorithms for linear programming

The method of steepest descent with scaling (affine scaling) applied to the potential functionq logcx — _i=1 ⁿ logx _i solves the linear programming problem in polynomial time forq n. Ifq = n, then the algorithm terminates in no more than O(n ² L) iterations; if q n + withq = O(n) then it takes no more than O(nL) iterations. A modified algorithm using rank-1 updates for matrix inversions achieves respectively O(n ⁴ L) and O(n ^3.5 L) arithmetic computions.     

一个新的求解半正定规划问题的原始对偶内点算法

选择合适的核函数对设计求解线性规划与半正定规划的原始对偶内点算法以及复杂性分析都十分重要.Bai等针对线性规划提出三种核函数,并给出求解线性规划的大步迭代复杂界,但未给出数值算例验证算法的实际效果(Bai Y Q,Xie W,Zhang J.New parameterized kernel functions for linear optimization.J Global Optim,2012.DOI 10.1007/s10898-012-9934-z).基于这三种核函数设计了新的求解半正定规划问题的原始对内点算法.进一步分析了算法关于大步方法的计算复杂性界,同时通过数值算例验证了算法的有效性和核函数所带参数对计算复杂性的影响.     

A globally convergent primal-dual interior-point filter method for nonlinear programming

In this paper, the filter technique of Fletcher and Leyffer (1997) is used to globalize the primal-dual interior-point algorithm for nonlinear programming, avoiding the use of merit functions and the updating of penalty parameters.The new algorithm decomposes the primal-dual step obtained from the perturbed first-order necessary conditions into a normal and a tangential step, whose sizes are controlled by a trust-region type parameter. Each entry in the filter is a pair of coordinates: one resulting from feasibility and centrality, and associated with the normal step; the other resulting from optimality (complementarity and duality), and related with the tangential step.Global convergence to first-order critical points is proved for the new primal-dual interior-point filter algorithm.Mathematics Subject Classification (1991): 65K05, 90C06, 90C29, 90C30Support for this author was provided by CRPC grant CCR–9120008.Support for this author was provided by CRPC grant CCR–9120008.Support for this author was provided by Centro de Matemática da Universidade de Coimbra, by FCT under grant POCTI/35059/MAT/2000, by the European Union under grant IST-2000-26063, and by Fundaç\ ao Calouste Gulbenkian. The author would also like to thank the IBM T.J. Watson Research Center and the Institute for Mathematics and Its Applications for their local support.     

Duality theorems and algorithms for linear programming in measure spaces

The purpose of this paper is to present some results on linear programming in measure spaces (LPM). We prove that, under certain conditions, the optimal value of an LPM is equal to the optimal value of the dual problem (DLPM). We also present two algorithms for solving various LPM problems and prove the convergence properties of these algorithms.     

Finite convergence of algorithms for nonlinear programs and variational inequalities

Algorithms for nonlinear programming and variational inequality problems are, in general, only guaranteed to converge in the limit to a Karush-Kuhn-Tucker point, in the case of nonlinear programs, or to a solution in the case of variational inequalities. In this paper, we derive sufficient conditions for nonlinear programs with convex feasible sets such that any convergent algorithm can be modified, by adding a convex subproblem with a linear objective function, to guarantee finite convergence in a generalized sense. When the feasible set is polyhedral, the subproblem is a linear program and finite convergence is obtained. Similar results are also developed for variational inequalities.The research of the first author was supported in part by the Office of Naval Research under Contract No. N00014-86-K-0173.The authors are indebted to Professors Olvi Mangasarian, Garth McCormick, Jong-Shi Pang, Hanif Sherali, and Hoang Tuy for helpful comments and suggestions and to two anonymous referees for constructive remarks and for bringing to their attention the results in Refs. 13 and 14.     

Scaling,shifting and weighting in interior-point methods

We examine certain questions related to the choice of scaling, shifting and weighting strategies for interior-point methods for linear programming. One theme is the desire to make trajectories to be followed by algorithms into straight lines if possible to encourage fast convergence. While interior-point methods in general follow curves, this occurrence of straight lines seems appropriate to honor George Dantzig's contributions to linear programming, since his simplex method can be seen as following either a piecewise-linear path inn-space or a straight line inm-space (the simplex interpretation).Dedicated to Professor George B. Dantzig on the occasion of his eightieth birthday.Research supported in part by NSF, AFOSR, and ONR through NSF Grant DMS-8920550.     

基于指数型核函数的线性规划原始对偶内点算法

给出线性规划原始对偶内点算法的一个单变量指数型核函数.首先研究了这个指数型核函数的性质以及其对应的障碍函数.其次,基于这个指数型核函数,设计了求解线性规划问题的原始对偶内点算法,得到了目前小步算法最好的理论迭代界.最后,通过数值算例比较了基于指数型核函数的原始对偶内点算法和基于对数型核函数的原始对偶内点算法的计算效果.     

On the convergence of primal-dual interior-point methods with wide neighborhoods

We study primal-dual interior-point methods for linear programs. After proposing a new primaldual potential function we describe a new potential reduction algorithm. We make connections between the new potential function and primal-dual interior-point algorithms with wide neighborhoods. Then we describe an algorithm that is a slightly modified version of existing primal-dual algorithms using wide neighborhoods. Assuming the optimal solution is non-degenerate, the algorithm is 1-step Q-quadratically convergent. We also study the degenerate case and show that the neighborhoods of the central path stay large as the iterates approach the optimal solutions.Research performed while the author was a Ph.D. student at Cornell University and was supported in part by the United States Army Research Office through the Army Center of Excellence for Symbolic Methods in Algorithmic Mathematics (ACSyAM), Mathematical Sciences Institute of Cornell University, Contract DAAL03-91-C-0027 and also by NSF, AFOSR and ONR through NSF Grant DMS-8920550.     

A relaxed primal-dual path-following algorithm for linear programming

In this paper, we provide an easily satisfied relaxation condition for the primaldual interior path-following algorithm to solve linear programming problems. It is shown that the relaxed algorithm preserves the property of polynomial-time convergence. The computational results obtained by implementing two versions of the relaxed algorithm with slight modifications clearly demonstrate the potential in reducing computational efforts.Partially supported by the North Carolina Supercomputing Center, the 1993 Cray Research Award, and a National Science Council Research Grant of the Republic of China.     

Sensitivity analysis in linear programming and semidefinite programming using interior-point methods

We analyze perturbations of the right-hand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the perturbations that allow interior-point methods to recover feasible and near-optimal solutions in a single interior-point iteration. For the unique, nondegenerate solution case in LP, we show that the bounds obtained using interior-point methods compare nicely with the bounds arising from using the optimal basis. We also present explicit bounds for SDP using the Monteiro-Zhang family of search directions and specialize them to the AHO, H..K..M, and NT directions. Received: December 1999 / Accepted: January 2001?Published online March 22, 2001