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 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.     

Interior-point methods for nonlinear complementarity problems

We present a potential reduction interior-point algorithm for monotone nonlinear complementarity problems. At each iteration, one has to compute an approximate solution of a nonlinear system such that a certain accuracy requirement is satisfied. For problems satisfying a scaled Lipschitz condition, this requirement is satisfied by the approximate solution obtained by applying one Newton step to that nonlinear system. We discuss the global and local convergence rates of the algorithm, convergence toward a maximal complementarity solution, a criterion for switching from the interior-point algorithm to a pure Newton method, and the complexity of the resulting hybrid algorithm.This research was supported in part by NSF Grant DDM-89-22636.The authors would like to thank Rongqin Sheng and three anonymous referees for their comments leading to a better presentation of the results.     

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.     

A quadratically convergent O(\sqrt n L)-iteration algorithm for linear programming

Recently, Ye, Tapia and Zhang (1991) demonstrated that Mizuno—Todd—Ye's predictor—corrector interior-point algorithm for linear programming maintains the O( L)-iteration complexity while exhibiting superlinear convergence of the duality gap to zero under the assumption that the iteration sequence converges, and quadratic convergence of the duality gap to zero under the assumption of nondegeneracy. In this paper we establish the quadratic convergence result without any assumption concerning the convergence of the iteration sequence or nondegeneracy. This surprising result, to our knowledge, is the first instance of a demonstration of polynomiality and superlinear (or quadratic) convergence for an interior-point algorithm which does not assume the convergence of the iteration sequence or nondegeneracy.Supported in part by NSF Grant DDM-8922636 and NSF Coop. Agr. No. CCR-8809615, the Iowa Business School Summer Grant, and the Interdisciplinary Research Grant of the University of Iowa Center for Advanced Studies.Supported in part by NSF Coop. Agr. No. CCR-8809615, AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Supported in part by NSF Grant DMS-9102761 and DOE Grant DE-FG05-91ER25100.     

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.     

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.     

On computing the center of a convex quadratically constrained set

We propose a method for finding analytic center of a convex feasible region whose boundaries are defined by quadratic functions. The algorithm starts from an arbitrary initial point and approaches to the desired center by simultaneously reducing infeasibility or slackness of all constraints. A partial Newton step is taken at each iteration.Research supported in part by the ONR under grant N00014-87-K-0214 and by the NSF under grant CCR-8810107.Research supported in part by the NSF under grant ECS-8721709.     

Enhancing energy conserving methods

Recent observations [5] indicate that energy-momentum methods might be better suited for the numerical integration of highly oscillatory Hamiltonian systems than implicit symplectic methods. However, the popular energy-momentum method, suggested in [3], achieves conservation of energy by a global scaling of the force field. This leads to an undesirable coupling of all degrees of freedom that is not present in the original problem formulation. We suggest enhancing this energy-momentum method by splitting the force field and using separate adjustment factors for each force. In case that the potential energy function can be split into a strong and a weak part, we also show how to combine an energy conserving discretization of the strong forces with a symplectic discretization of the weak contributions. We demonstrate the numerical properties of our method by simulating particles that interact through Lennard-Jones potentials and by integrating the Sine-Gordon equation.This work was partly supported by NIH Grant P41RR05969, DOE/NSF Grant DE-FG02-91-ER25099/DMS-9304268, and NSF GCAG/HPCC ASC-9318159.     

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.