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.     

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.     

On the convergence rate of Newton interior-point methods in the absence of strict complementarity

In the absence of strict complementarity, Monteiro and Wright [7] proved that the convergence rate for a class of Newton interior-point methods for linear complementarity problems is at best linear. They also established an upper bound of 1/4 for the Q ₁-factor of the duality gap sequence when the steplengths converge to one. In the current paper, we prove that the Q ₁ factor of the duality gap sequence is exactly 1/4. In addition, the convergence of the Tapia indicators is also discussed.This author was supported in part by NSF Coop. Agr. No. CCR-8809615 and AFOSR 89-0363 and the REDI Foundation.This author was supported in part by NSF Coop. Agr. No. CCR-8809615, AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Visiting member of the Center for Research on Parallel Computations, Rice University, Houston, Texas, 77251-1892. This author was supported in part by DOE DE-FG02-93ER25171.     

On quadratic andO\left( {\sqrt {nL} } \right) convergence of a predictor—corrector algorithm for LCP

Recently several new results have been developed for the asymptotic (local) convergence of polynomial-time interior-point algorithms. It has been shown that the predictor—corrector algorithm for linear programming (LP) exhibits asymptotic quadratic convergence of the primal—dual gap to zero, without any assumptions concerning nondegeneracy, or the convergence of the iteration sequence. In this paper we prove a similar result for the monotone linear complementarity problem (LCP), assuming only that a strictly complementary solution exists. We also show by example that the existence of a strictly complementarity solution appears to be necessary to achieve superlinear convergence for the algorithm.Research supported in part by NSF Grants DDM-8922636 and DDM-9207347, and an Interdisciplinary Research Grant of the University of Iowa, Iowa Center for Advanced Studies.     

Finding an interior point in the optimal face of linear programs

We study the problem of finding a point in the relative interior of the optimal face of a linear program. We prove that in the worst case such a point can be obtained in O(n ³ L) arithmetic operations. This complexity is the same as the complexity for solving a linear program. We also show how to find such a point in practice. We report and discuss computational results obtained for the linear programming problems in the NETLIB test set.Research supported in part by NSF Grant CCR-8810107, CCR-9019469 and a grant from GTE Laboratories.Research supported in part by NSF Grant DDM-8922636 and NSF Coop. Agr. No. CCR-8809615 through Rice University.     

On polynomiality of the Mehrotra-type predictor—corrector interior-point algorithms

Recently, Mehrotra [3] proposed a predictor—corrector primal—dual interior-point algorithm for linear programming. At each iteration, this algorithm utilizes a combination of three search directions: the predictor, the corrector and the centering directions, and requires only one matrix factorization. At present, Mehrotra's algorithmic framework is widely regarded as the most practically efficient one and has been implemented in the highly successful interior-point code OB1 [2]. In this paper, we study the theoretical convergence properties of Mehrotra's interior-point algorithmic framework. For generality, we carry out our analysis on a horizontal linear complementarity problem that includes linear and quadratic programming, as well as the standard linear complementarity problem. Under the monotonicity assumption, we establish polynomial complexity bounds for two variants of the Mehrotra-type predictor—corrector interior-point algorithms. These results are summarized in the last section in a table.Research supported in part by NSF DMS-9102761, DOE DE-FG05-91ER25100 and DOE DE-FG02-93ER25171.Corresponding author.     

On the formulation and theory of the Newton interior-point method for nonlinear programming

In this work, we first study in detail the formulation of the primal-dual interior-point method for linear programming. We show that, contrary to popular belief, it cannot be viewed as a damped Newton method applied to the Karush-Kuhn-Tucker conditions for the logarithmic barrier function problem. Next, we extend the formulation to general nonlinear programming, and then validate this extension by demonstrating that this algorithm can be implemented so that it is locally and Q-quadratically convergent under only the standard Newton method assumptions. We also establish a global convergence theory for this algorithm and include promising numerical experimentation.The first two authors were supported in part by NSF Cooperative Agreement No. CCR-8809615, by Grants AFOSR 89-0363, DOE DEFG05-86ER25017, ARO 9DAAL03-90-G-0093, and the REDI Foundation. The fourth author was supported in part by NSF DMS-9102761 and DOE DE-FG02-93ER25171. The authors would like to thank Sandra Santos for painstakingly proofreading an earlier verion of this paper.     

Cubically convergent method for locating a nearby vertex in linear programming

Given a point sufficiently close to a nondegenerate basic feasible solutionx* of a linear program, we show how to generate a sequence {p ^k} that converges to the 0–1 vector sign(x*) at aQ-cubic rate. This extremely fast convergence enables us to determine, with a high degree of certainty, which variables will be zero and which will be nonzero at optimality and then constructx* from this information.This research was supported in part by NSF Cooperative Agreement No. CCR-8809615, by AFOSR Grant No. 89-0363, and by DOE Grant No. DEFG05-86ER 25017. The authors would like to thank Bob Bixby for helpful discussions.     

Modified predictor-corrector algorithm for locating weighted centers in linear programming

In certain applications of linear programming, the determination of a particular solution, the weighted center of the solution set, is often desired, giving rise to the need for algorithms capable of locating such center. In this paper, we modify the Mizuno-Todd-Ye predictor-corrector algorithm so that the modified algorithm is guaranteed to converge to the weighted center for given weights. The key idea is to ensure that iterates remain in a sequence of shrinking neighborhoods of the weighted central path. The modified algorithm also possesses polynomiality and superlinear convergence.The work of the first author was supported in part by NSF Grant DMS-91-02761 and DOE Contract DE-FG05-91-ER25100.The work of the second author was supported in part by NSF Cooperative Agreement CCR-88-09615.     

Local convergence of predictor—corrector infeasible-interior-point algorithms for SDPs and SDLCPs

An example of an SDP (semidefinite program) exhibits a substantial difficulty in proving the superlinear convergence of a direct extension of the Mizuno—Todd—Ye type predictor—corrector primal-dual interior-point method for LPs (linear programs) to SDPs, and suggests that we need to force the generated sequence to converge to a solution tangentially to the central path (or trajectory). A Mizuno—Todd—Ye type predictor—corrector infeasible-interior-point algorithm incorporating this additional restriction for monotone SDLCPs (semidefinite linear complementarity problems) enjoys superlinear convergence under strict complementarity and nondegeneracy conditions. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Interior-Point Gradient Method for Large-Scale Totally Nonnegative Least Squares Problems

We study an interior-point gradient method for solving a class of so-called totally nonnegative least-squares problems. At each iteration, the method decreases the residual norm along a diagonally-scaled negative gradient direction with a special scaling. We establish the global convergence of the method and present some numerical examples to compare the proposed method with a few similar methods including the affine scaling method.This author was supported in part by DOE/LANL Contract 03891-99-23This author was supported in part by NSF Grant DMS-0442065     

On homogeneous and self-dual algorithms for LCP

We present some generalizations of a homogeneous and self-dual linear programming (LP) algorithm to solving the monotone linear complementarity problem (LCP). Again, while it achieves the best known interior-point iteration complexity, the algorithm does not need to use any “big-M” number, and it detects LCP infeasibility by generating a certificate. To our knowledge, this is the first interior-point and infeasible-starting algorithm for the LCP with these desired features. Research supported in part by NSF Grant DDM-9207347, the University of Iowa Oberman Fellowship and the Iowa College of Business Administration Summer Grant. Part of this work is done while the author is visiting the Delft Optimization Center at the University of Technology, Delft, Netherlands, supported by the Dutch Organization for Scientific Research (NWO).     

Inexact Interior-Point Method

In this paper, we introduce an inexact interior-point algorithm for a constrained system of equations. The formulation of the problem is quite general and includes nonlinear complementarity problems of various kinds. In our convergence theory, we interpret the inexact interior-point method as an inexact Newton method. This enables us to establish a global convergence theory for the proposed algorithm. Under the additional assumption of the invertibility of the Jacobian at the solution, the superlinear convergence of the iteration sequence is proved.     

Chapter 6 On the construction of strong complementarity slackness solutions for DEA linear programming problems using a primal-dual interior-point method

A novel approach for solving the DEA linear programming problems using a primaldual interior-point method is presented. The solution found by this method satisfies the Strong Complementarity Slackness Condition (SCSC) and maximizes the product of the positive components among all SCSC solutions. The first property is critical in the use of DEA and the second one contributes significantly to the reliability of the solution.This research was partially supported by NSF Cooperative Agreement No. CCR-88-09615, ARO Grant 9DAAL03-90-G-0093, DOE Grant DEFG05-86-ER25017, and AFOSR Grant 89-0363.Partially supported by Fulbright/LASPAU.     

Superlinear Convergence of Interior-Point Algorithms for Semidefinite Programming

We prove the superlinear convergence of the primal-dual infeasible interior-point path-following algorithm proposed recently by Kojima, Shida, and Shindoh and by the present authors, under two conditions: (i) the semidefinite programming problem has a strictly complementary solution; (ii) the size of the central path neighborhood approaches zero. The nondegeneracy condition suggested by Kojima, Shida, and Shindoh is not used in our analysis. Our result implies that the modified algorithm of Kojima, Shida, and Shindoh, which enforces condition (ii) by using additional corrector steps, has superlinear convergence under the standard assumption of strict complementarity. Finally, we point out that condition (ii) can be made weaker and show the superlinear convergence under the strict complementarity assumption and a weaker condition than (ii).     

A computational study of a gradient-based log-barrier algorithm for a class of large-scale SDPs

 The authors of this paper recently introduced a transformation [4] that converts a class of semidefinite programs (SDPs) into nonlinear optimization problems free of matrix-valued constraints and variables. This transformation enables the application of nonlinear optimization techniques to the solution of certain SDPs that are too large for conventional interior-point methods to handle efficiently. Based on the transformation, we proposed a globally convergent, first-order (i.e., gradient-based) log-barrier algorithm for solving a class of linear SDPs. In this paper, we discuss an efficient implementation of the proposed algorithm and report computational results on semidefinite relaxations of three types of combinatorial optimization problems. Our results demonstrate that the proposed algorithm is indeed capable of solving large-scale SDPs and is particularly effective for problems with a large number of constraints. Received: June 22, 2001 / Accepted: January 20, 2002 Published online: December 9, 2002 RID="†" ID="†"Computational results reported in this paper were obtained on an SGI Origin2000 computer at Rice University acquired in part with support from NSF Grant DMS-9872009. RID="⋆" ID="⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203426 RID="⋆⋆" ID="⋆⋆"This author was supported in part by NSF Grants CCR-9902010, INT-9910084 and CCR-0203113 RID="⋆⋆⋆" ID="⋆⋆⋆"This author was supported in part by DOE Grant DE-FG03-97ER25331, DOE/LANL Contract 03891-99-23 and NSF Grant DMS-9973339. Key Words. semidefinite program – semidefinite relaxation – nonlinear programming – interior-point methods – limited memory quasi-Newton methods. Mathematics Subject Classification (1991): 90C06, 90C27, 90C30.     

Superlinear convergence of affine scaling interior point newton method for linear inequality constrained minimization without strict complementarity

In this paper we extend and improve the classical affine scaling interior-point Newton method for solving nonlinear optimization subject to linear inequality constraints in the absence of the strict complementarity assumption. Introducing a computationally efficient technique and employing an identification function for the definition of the new affine scaling matrix, we propose and analyze a new affine scaling interior-point Newton method which improves the Coleman and Li affine sealing matrix in [2] for solving the linear inequlity constrained optimization. Local superlinear and quadratical convergence of the proposed algorithm is established under the strong second order sufficiency condition without assuming strict complementarity of the solution.     

Primal—dual methods for linear programming

Many interior-point methods for linear programming are based on the properties of the logarithmic barrier function. After a preliminary discussion of the convergence of the (primal) projected Newton barrier method, three types of barrier method are analyzed. These methods may be categorized as primal, dual and primal—dual, and may be derived from the application of Newton's method to different variants of the same system of nonlinear equations. A fourth variant of the same equations leads to a new primal—dual method.In each of the methods discussed, convergence is demonstrated without the need for a nondegeneracy assumption or a transformation that makes the provision of a feasible point trivial. In particular, convergence is established for a primal—dual algorithm that allows a different step in the primal and dual variables and does not require primal and dual feasibility.Finally, a new method for treating free variables is proposed.Presented at the Second Asilomar Workshop on Progress in Mathematical Programming, February 1990, Asilomar, CA, United StatesThe material contained in this paper is based upon research supported by the National Science Foundation Grant DDM-9204208 and the Office of Naval Research Grant N00014-90-J-1242.     

Convergence behavior of interior-point algorithms

We show that most interior-point algorithms for linear programming generate a solution sequence in which every limit point satisfies the strict complementarity condition. These algorithms include all path-following algorithms and some potential reduction algorithms. The result also holds for the monotone complementarity problem if a strict complementarity solution exists. In general, the limit point is a solution that maximizes the number of its nonzero components among all solutions.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.     

Error bounds and convergence analysis of feasible descent methods: a general approach

We survey and extend a general approach to analyzing the convergence and the rate of convergence of feasible descent methods that does not require any nondegeneracy assumption on the problem. This approach is based on a certain error bound for estimating the distance to the solution set and is applicable to a broad class of methods.The research of the first author is supported by the Natural Sciences and Engineering Research Council of Canada, Grant No. OPG0090391, and the research of the second author is supported by the National Science Foundation, Grant No. CCR-9103804.