Interior-point algorithms for semi-infinite programming

In order to study the behavior of interior-point methods on very large-scale linear programming problems, we consider the application of such methods to continuous semi-infinite linear programming problems in both primal and dual form. By considering different discretizations of such problems we are led to a certain invariance property for (finite-dimensional) interior-point methods. We find that while many methods are invariant, several, including all those with the currently best complexity bound, are not. We then devise natural extensions of invariant methods to the semi-infinite case. Our motivation comes from our belief that for a method to work well on large-scale linear programming problems, it should be effective on fine discretizations of a semi-infinite problem and it should have a natural extension to the limiting semi-infinite case.Research supported in part by NSF, AFORS and ONR through NSF grant DMS-8920550.     

Implementation of nonsymmetric interior-point methods for linear optimization over sparse matrix cones

We describe an implementation of nonsymmetric interior-point methods for linear cone programs defined by two types of matrix cones: the cone of positive semidefinite matrices with a given chordal sparsity pattern and its dual cone, the cone of chordal sparse matrices that have a positive semidefinite completion. The implementation takes advantage of fast recursive algorithms for evaluating the function values and derivatives of the logarithmic barrier functions for these cones. We present experimental results of two implementations, one of which is based on an augmented system approach, and a comparison with publicly available interior-point solvers for semidefinite programming.     

对称锥规划的邻域跟踪算法

本文把艾文宝的邻域跟踪算法推广到对称锥规划, 定义中心路径的宽邻域N(τ, β), 并证明该邻域的一个重要性质, 该性质在算法的复杂性分析中起到关键作用. 取宽邻域N(τ, β) 中一点为初始点并采用Nesterov-Todd (NT) 搜索方向, 则该算法的迭代复杂界为O(√r logε^-1), 其中, r是EuclidJordan 代数的秩, ε是允许误差. 这是对称锥规划的宽邻域内点算法最好的复杂界.     

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.     

A PREDICTOR-CORRECTOR INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interior-point method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.     

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     

A Solution Concept for Fuzzy Multiobjective Programming Problems Based on Convex Cones

A solution concept for fuzzy multiobjective programming problems based on ordering cones (convex cones) is proposed in this paper. The notions of ordering cones and partial orderings on a vector space are essentially equivalent. Therefore, the optimality notions in a real vector space can be elicited naturally by invoking a concept similar to that of the Pareto-optimal solution in vector optimization problems. We introduce a corresponding multiobjective programming problem and a weighting problem of the original fuzzy multiobjective programming problem using linear functionals so that the optimal solution of its corresponding weighting problem is also the Pareto-optimal solution of the original fuzzy multiobjective programming problem.     

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.     

Numerical Comparisons of Path-Following Strategies for a Primal-Dual Interior-Point Method for Nonlinear Programming

An important research activity in primal-dual interior-point methods for general nonlinear programming is to determine effective path-following strategies and their implementations. The objective of this work is to present numerical comparisons of several path-following strategies for the local interior-point Newton method given by El-Bakry, Tapia, Tsuchiya, and Zhang. We conduct numerical experimentation of nine strategies using two central regions, three notions of proximity measures, and three merit functions to obtain an optimal solution. Six of these strategies are implemented for the first time. The numerical results show that the best path-following strategy is that given by Argáez and Tapia.     

Parabolic target space and primal-dual interior-point methods

In this paper we develop new primal-dual interior-point methods for linear programming problems, which are based on the concept of parabolic target space. We show that such schemes work in the infinity-neighborhood of the primal-dual central path. Nevertheless, these methods possess the best known complexity estimate. We demonstrate that the adaptive-step path-following strategies can be naturally incorporated in such schemes.     

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.     

Approximate Farkas lemmas and stopping rules for iterative infeasible-point algorithms for linear programming

In exact arithmetic, the simplex method applied to a particular linear programming problem instance with real data either shows that it is infeasible, shows that its dual is infeasible, or generates optimal solutions to both problems. Most interior-point methods, on the other hand, do not provide such clear-cut information. If the primal and dual problems have bounded nonempty sets of optimal solutions, they usually generate a sequence of primal or primaldual iterates that approach feasibility and optimality. But if the primal or dual instance is infeasible, most methods give less precise diagnostics. There are methods with finite convergence to an exact solution even with real data. Unfortunately, bounds on the required number of iterations for such methods applied to instances with real data are very hard to calculate and often quite large. Our concern is with obtaining information from inexact solutions after a moderate number of iterations. We provide general tools (extensions of the Farkas lemma) for concluding that a problem or its dual is likely (in a certain well-defined sense) to be infeasible, and apply them to develop stopping rules for a homogeneous self-dual algorithm and for a generic infeasible-interior-point method for linear programming. These rules allow precise conclusions to be drawn about the linear programming problem and its dual: either near-optimal solutions are produced, or we obtain certificates that all optimal solutions, or all feasible solutions to the primal or dual, must have large norm. Our rules thus allow more definitive interpretation of the output of such an algorithm than previous termination criteria. We give bounds on the number of iterations required before these rules apply. Our tools may also be useful for other iterative methods for linear programming. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Global Convergence Analysis of Line Search Interior-Point Methods for Nonlinear Programming without Regularity Assumptions

As noted by Wächter and Biegler (Ref. 1), a number of interior-point methods for nonlinear programming based on line-search strategy may generate a sequence converging to an infeasible point. We show that, by adopting a suitable merit function, a modified primal-dual equation, and a proper line-search procedure, a class of interior-point methods of line-search type will generate a sequence such that either all the limit points of the sequence are KKT points, or one of the limit points is a Fritz John point, or one of the limit points is an infeasible point that is a stationary point minimizing a function measuring the extent of violation to the constraint system. The analysis does not depend on the regularity assumptions on the problem. Instead, it uses a set of satisfiable conditions on the algorithm implementation to derive the desired convergence property.Communicated by Z. Q. LuoThis research was partially supported by Grant R-314-000-026/042/057-112 of National University of Singapore and Singapore-MIT Alliance. We thank Professor Khoo Boo Cheong, Cochair of the High Performance Computation Program of Singapore-MIT Alliance, for his support     

Local Convergence of a Primal-Dual Method for Degenerate Nonlinear Programming

In recent work, the local convergence behavior of path-following interior-point methods and sequential quadratic programming methods for nonlinear programming has been investigated for the case in which the assumption of linear independence of the active constraint gradients at the solution is replaced by the weaker Mangasarian–Fromovitz constraint qualification. In this paper, we describe a stabilization of the primal-dual interior-point approach that ensures rapid local convergence under these conditions without enforcing the usual centrality condition associated with path-following methods. The stabilization takes the form of perturbations to the coefficient matrix in the step equations that vanish as the iterates converge to the solution.     

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.     

Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.     

Implementation of warm-start strategies in interior-point methods for linear programming in fixed dimension

We implement several warm-start strategies in interior-point methods for linear programming (LP). We study the situation in which both the original LP instance and the perturbed one have exactly the same dimensions. We consider different types of perturbations of data components of the original instance and different sizes of each type of perturbation. We modify the state-of-the-art interior-point solver PCx in our implementation. We evaluate the effectiveness of each warm-start strategy based on the number of iterations and the computation time in comparison with “cold start” on the NETLIB test suite. Our experiments reveal that each of the warm-start strategies leads to a reduction in the number of interior-point iterations especially for smaller perturbations and for perturbations of fewer data components in comparison with cold start. On the other hand, only one of the warm-start strategies exhibits better performance than cold start in terms of computation time. Based on the insight gained from the computational results, we discuss several potential improvements to enhance the performances of such warm-start strategies. This research was supported in part by NSF through CAREER grant DMI-0237415.     

An Interior-Point Algorithm for Nonconvex Nonlinear Programming

The paper describes an interior-point algorithm for nonconvex nonlinear programming which is a direct extension of interior-point methods for linear and quadratic programming. Major modifications include a merit function and an altered search direction to ensure that a descent direction for the merit function is obtained. Preliminary numerical testing indicates that the method is robust. Further, numerical comparisons with MINOS and LANCELOT show that the method is efficient, and has the promise of greatly reducing solution times on at least some classes of models.     

Polynomial time solvability of non-symmetric semidefinite programming

In this paper, we consider semidefinite programming with non-symmetric matrices, which is called non-symmetric semidefinite programming (NSDP). We convert such a problem into a linear program over symmetric cones, which is polynomial time solvable by interior point methods. Thus, the NSDP problem can be solved in polynomial time. Such a result corrects the corresponding result given in the literature. Similar methods can be applied to nonlinear programming with non-symmetric matrices.     

Combined Entropic Regularization and Path-Following Method for Solving Finite Convex Min-max Problems Subject to Infinitely Many Linear Constraints

In this paper, we study the minimization of the max function of q smooth convex functions on a domain specified by infinitely many linear constraints. The difficulty of such problems arises from the kinks of the max function and it is often suggested that, by imposing certain regularization functions, nondifferentiability will be overcome. We find that the entropic regularization introduced by Li and Fang is closely related to recently developed path-following interior-point methods. Based on their results, we create an interior trajectory in the feasible domain and propose a path-following algorithm with a convergence proof. Our intention here is to show a nice combination of minmax problems, semi-infinite programming, and interior-point methods. Hopefully, this will lead to new applications.