Constant potential primal—dual algorithms: A framework

We start with a study of the primal—dual affine-scaling algorithms for linear programs. Using ideas from Kojima et al., Mizuno and Nagasawa, and new potential functions we establish a framework for primal—dual algorithms that keep a potential function value fixed. We show that if the potential function used in the algorithm is compatible with a corresponding neighborhood of the central path then the convergence proofs simplify greatly. Our algorithms have the property that all the iterates can be kept in a neighborhood of the central path without using any centering in the search directions.Research performed while the author was 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 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.     

An Optimal Method for Adjusting the Centering Parameter in the Wide-Neighborhood Primal-Dual Interior-Point Algorithm for Linear Programming

In this paper we present a dynamic optimal method for adjusting the centering parameter in the wide-neighborhood primal-dual interior-point algorithms for linear programming, while the centering pararheter is generally a constant in the classical wideneighborhood primal-dual interior-point algorithms. The computational results show that the new method is more efficient.     

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.     

New parameterized kernel functions for linear optimization

Recent studies on the kernel function-based primal-dual interior-point algorithms indicate that a kernel function not only represents a measure of the distance between the iteration and the central path, but also plays a critical role in improving the computational complexity of an interior-point algorithm. In this paper, we propose a new class of parameterized kernel functions for the development of primal-dual interior-point algorithms for solving linear programming problems. The properties of the proposed kernel functions and corresponding parameters are investigated. The results lead to a complexity bounds of ${O\left(\sqrt{n}\,{\rm log}\,n\,{\rm log}\,\frac{n}{\epsilon}\right)}$ for the large-update primal-dual interior point methods. To the best of our knowledge, this is the best known bound achieved.     

Solving combinatorial optimization problems using Karmarkar's algorithm

We describe a cutting plane algorithm for solving combinatorial optimization problems. The primal projective standard-form variant of Karmarkar's algorithm for linear programming is applied to the duals of a sequence of linear programming relaxations of the combinatorial optimization problem.Computational facilities provided by the Cornell Computational Optimization Project supported by NSF Grant DMS-8706133 and by the Cornell National Supercomputer Facility. The Cornell National Supercomputer Facility is a resource of the Center for Theory and Simulation in Science and Engineering at Cornell Unversity, which is funded in part by the National Science Foundation, New York State, and the IBM Corporation. The research of both authors was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research partially supported by ONR Grant N00014-90-J-1714.Research partially supported by NSF Grant ECS-8602534 and by ONR Contract N00014-87-K-0212.     

A wide neighborhood interior-point algorithm for linear optimization based on a specific kernel function

This paper presents an interior point algorithm for solving linear optimization problems in a wide neighborhood of the central path introduced by Ai and Zhang (SIAM J Optim 16:400–417, 2005). In each iteration, the algorithm computes the new search directions by using a specific kernel function. The convergence of the algorithm is shown and it is proved that the algorithm has the same iteration bound as the best short-step algorithms. We demonstrate the computational efficiency of the proposed algorithm by testing some Netlib problems in standard form. To best our knowledge, this is the first wide neighborhood path-following interior-point method with the same complexity as the best small neighborhood path-following interior-point methods that uses the kernel function.

     

Containing and shrinking ellipsoids in the path-following algorithm

We describe a new potential function and a sequence of ellipsoids in the path-following algorithm for convex quadratic programming. Each ellipsoid in the sequence contains all of the optimal primal and dual slack vectors. Furthermore, the volumes of the ellipsoids shrink at the ratio , in comparison to 2^–(1) in Karmarkar's algorithm and 2^–(1/n) in the ellipsoid method. We also show how to use these ellipsoids to identify the optimal basis in the course of the algorithm for linear programming.Research supported by The U.S. Army Research Office through The Mathematical Sciences Institute of Cornell University when the author was visiting at Cornell.Research supported in part by National Science Foundation Grant ECS-8602534 and Office of Naval Research Contract N00014-87-K-0212.     

Potential-reduction methods in mathematical programming

We provide a survey of interior-point methods for linear programming and its extensions that are based on reducing a suitable potential function at each iteration. We give a fairly complete overview of potential-reduction methods for linear programming, focusing on the possibility of taking long steps and the properties of the barrier function that are necessary for the analysis. We then describe briefly how the methods and results can be extended to certain convex programming problems, following the approach of Nesterov and Todd. We conclude with some open problems. Research supported in part by NSF, AFOSR and ONR through NSF Grant DMS-8920550. Some of this work was done while the author was on a sabbatical leave from Cornell University visiting the Department of Mathematics at the University of Washington.     

A simplified homogeneous and self-dual linear programming algorithm and its implementation

We present a simplification and generalization of the recent homogeneous and self-dual linear programming (LP) algorithm. The algorithm does not use any Big-M initial point and achieves -iteration complexity, wheren andL are the number of variables and the length of data of the LP problem. It also detects LP infeasibility based on a provable criterion. Its preliminary implementation with a simple predictor and corrector technique results in an efficient computer code in practice. In contrast to other interior-point methods, our code solves NETLIB problems, feasible or infeasible, starting simply fromx=e (primal variables),y=0 (dual variables),z=e (dual slack variables), wheree is the vector of all ones. We describe our computational experience in solving these problems, and compare our results with OB1.60, a state-of-the-art implementation of interior-point algorithms.Research supported in part by NSF Grant DDM-9207347 and by an Iowa College of Business Administration Summer Grant.Part of this 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, USA, 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.     

Global convergence in infeasible-interior-point algorithms

This paper presents a wide class of globally convergent interior-point algorithms for the nonlinear complementarity problem with a continuously differentiable monotone mapping in terms of a unified global convergence theory given by Polak in 1971 for general nonlinear programs. The class of algorithms is characterized as: Move in a Newton direction for approximating a point on the path of centers of the complementarity problem at each iteration. Starting from a strictly positive but infeasible initial point, each algorithm in the class either generates an approximate solution with a given accuracy or provides us with information that the complementarity problem has no solution in a given bounded set. We present three typical examples of our interior-point algorithms, a horn neighborhood model, a constrained potential reduction model with the use of the standard potential function, and a pure potential reduction model with the use of a new potential function.Research supported in part by Grant-in-Aids for Co-Operative Research (03832017) of the Japan Ministry of Education, Science and Culture.Corresponding author.     

Insights into the interior-point methods

In this paper, we study the search directions of three important interior-point algorithms, namely, the primal-affine scaling method (with logarithmic barrier function), the dual-affine scaling method (with logarithmic barrier function), and the primal-dual interior point method. From an algebraic point of view, we show that the search directions of these three algorithms are merely Newton directions along three different paths that lead to a solution of the Karush-Kuhn-Tucker conditions of a given linear programming problem. From a geometric point of view, we show that these directions can be obtained by solving certain well-defined subproblems. Both views provide a general platform for studying the existing interior-point methods and deriving new interior-point algorithms. We illustrate the derivation of new interior-point algorithms by replacing the logarithmic barrier function with an entropic barrier function. The results have been generalized and discussed.This work is partially supported by the North Carolina Supercomputing Center 1990 Cray Grant Program sponsored by Cray Research.     

Infeasible interior-point method for symmetric optimization using a positive-asymptotic barrier

We propose a new primal-dual infeasible interior-point method for symmetric optimization by using Euclidean Jordan algebras. Different kinds of interior-point methods can be obtained by using search directions based on kernel functions. Some search directions can be also determined by applying an algebraic equivalent transformation on the centering equation of the central path. Using this method we introduce a new search direction, which can not be derived from a usual kernel function. For this reason, we use the new notion of positive-asymptotic kernel function which induces the class of corresponding barriers. In general, the main iterations of the infeasible interior-point methods are composed of one feasibility and several centering steps. We prove that in our algorithm it is enough to take only one centering step in a main iteration in order to obtain a well-defined algorithm. Moreover, we conclude that the algorithm finds solution in polynomial time and has the same complexity as the currently best known infeasible interior-point methods. Finally, we give some numerical results.     

Exploiting special structure in Karmarkar's linear programming algorithm

We propose methods to take advantage of specially-structured constraints in a variant of Karmarkar's projective algorithm for standard form linear programming problems. We can use these constraints to generate improved bounds on the optimal value of the problem and also to compute the necessary projections more efficiently, while maintaining the theoretical bound on the algorithm's performance. It is shown how various upper-bounding constraints can be handled implicitly in this way. Unfortunately, the situation for network constraints appears less favorable.Research supported in part by National Science Foundation Grant ECS-8602534, ONR Contract N00014-87-K-0212 and the US Army Research Office through the Mathematical Sciences Institute of Cornell University.     

On the Interplay among Entropy, Variable Metrics and Potential Functions in Interior-Point Algorithms

We are motivated by the problem of constructing aprimal-dual barrier function whose Hessian induces the (theoreticallyand practically) popular symmetric primal and dual scalings forlinear programming problems. Although this goal is impossible toattain, we show that the primal-dual entropy function may provide asatisfactory alternative. We study primal-dual interior-pointalgorithms whose search directions are obtained from a potentialfunction based on this primal-dual entropy barrier. We providepolynomial iteration bounds for these interior-point algorithms. Thenwe illustrate the connections between the barrier function and areparametrization of the central path equations. Finally, we considerthe possible effects of more general reparametrizations oninfeasible-interior-point algorithms.     

Quadratic convergence of potential-reduction methods for degenerate problems

Global and local convergence properties of a primal-dual interior-point pure potential-reduction algorithm for linear programming problems is analyzed. This algorithm is a primal-dual variant of the Iri-Imai method and uses modified Newton search directions to minimize the Tanabe-Todd-Ye (TTY) potential function. A polynomial time complexity for the method is demonstrated. Furthermore, this method is shown to have a unique accumulation point even for degenerate problems and to have Q-quadratic convergence to this point by an appropriate choice of the step-sizes. This is, to the best of our knowledge, the first superlinear convergence result on degenerate linear programs for primal-dual interior-point algorithms that do not follow the central path. Received: February 12, 1998 / Accepted: March 3, 2000?Published online January 17, 2001     

Random walks in varying dimensions

We establish recurrence criteria for sums of independent random variables which take values in Euclidean lattices of varying dimension. In particular, we describe transient inhomogeneous random walks in the plane which interlace two symmetric step distributions of bounded support.Research partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.Research supported in part by National Science Foundation Grant No. DMS 9300191, by a Sloan Foundation Fellowship, and by a Presidential Faculty Fellowship.     

一个求解半正定规划问题的新原始一对偶内点算法

在原始对偶内点算法的设计和分析中,障碍函数对算法的搜索方法和复杂性起着重要的作用。本文由核函数来确定障碍函数,设计了一个求解半正定规划问题的原始。对偶内点算法。这个障碍函数即可以定义算法新的搜索方向,又度量迭代点与中心路径的距离,同时对算法的复杂性分析起着关键的作用。我们计算了算法的迭代界,得出了关于大步校正法和小步校正法的迭代界,它们分别是O（√n log n log n/c）和O（√n log n/ε）,这里n是半正定规划问题的维数。最后,我们根据一个算例,说明了算法的有效性以及对核函数的参数的敏感性。     

Combining search directions using gradient flows

 The efficient combination of directions is a significant problem in line search methods that either use negative curvature, or wish to include additional information such as the gradient or different approximations to the Newton direction. In this paper we describe a new procedure to combine several of these directions within an interior-point primal-dual algorithm. Basically, we combine in an efficient manner a modified Newton direction with the gradient of a merit function and a direction of negative curvature, if it exists. We also show that the procedure is well-defined, and it has reasonable theoretical properties regarding the rate of convergence of the method. We also present numerical results from an implementation of the proposed algorithm on a set of small test problems from the CUTE collection. Received: November 2000 / Accepted: October 2002 Published online: February 14, 2003 Key Words. negative curvature – primal-dual methods – interior-point methods – nonconvex optimization – line searches Mathematics Subject Classification (1991): 49M37, 65K05, 90C30     

A globally and quadratically convergent affine scaling method for linearℓ 1 problems

Recently, various interior point algorithms related to the Karmarkar algorithm have been developed for linear programming. In this paper, we first show how this interior point philosophy can be adapted to the linear ₁ problem (in which there are no feasibility constraints) to yield a globally and linearly convergent algorithm. We then show that the linear algorithm can be modified to provide aglobally and ultimatelyquadratically convergent algorithm. This modified algorithm appears to be significantly more efficient in practise than a more straightforward interior point approach via a linear programming formulation: we present numerical results to support this claim.This paper was presented at the Third SIAM Conference on Optimization, in Boston, April 1989.Research partially supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013.A000, by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University, and by the Computational Mathematics Program of the National Science Foundation under grant DMS-8706133.Research partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University and by the Computational Mathematics Program of the National Science Foundation under grant DMS-8706133.