On the entropic perturbation and exponential penalty methods for linear programming

This note points out that the recently proposed exponential penalty approach to linear programming is identical to the well-known entropic perturbation approach. The primal and dual trajectories provided by these two approaches are shown to be equivalent.The work of the first author was supported partially by the North Carolina Supercomputing Center and 1995 Cray Research Grant.     

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.     

Deriving an unconstrained convex program for linear programming

Consider a linear programming problem in Karmarkar's standard form. By perturbing its linear objective function with an entropic barrier function and applying generalized geometric programming theory to it, Fang recently proposed an unconstrained convex programming approach to finding an epsilon-optimal solution. In this paper, we show that Fang's derivation of an unconstrained convex dual program can be greatly simplified by using only one simple geometric inequality. In addition, a system of nonlinear equations, which leads to a pair of primal and dual epsilon-optimal solutions, is proposed for further investigation.This work was partially supported by the North Carolina Supercomputing Center and a 1990 Cray Research Grant. The authors are indebted to Professors E. L. Peterson and R. Saigal for stimulating discussions.     

Convex analysis treated by linear programming

The theme of this paper is the application of linear analysis to simplify and extend convex analysis. The central problem treated is the standard convex program — minimize a convex function subject to inequality constraints on other convex functions. The present approach uses the support planes of the constraint region to transform the convex program into an equivalent linear program. Then the duality theory of infinite linear programming shows how to construct a new dual program of bilinear type. When this dual program is transformed back into the convex function formulation it concerns the minimax of an unconstrained Lagrange function. This result is somewhat similar to the Kuhn—Tucker theorem. However, no constraint qualifications are needed and yet perfect duality maintains between the primal and dual programs.Work prepared under Research Grant DA-AROD-31-124-71-G17, Army Research Office (Durham).     

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.     

Towards Strong Duality in Integer Programming

We consider in this paper the Lagrangian dual method for solving general integer programming. New properties of Lagrangian duality are derived by a means of perturbation analysis. In particular, a necessary and sufficient condition for a primal optimal solution to be generated by the Lagrangian relaxation is obtained. The solution properties of Lagrangian relaxation problem are studied systematically. To overcome the difficulties caused by duality gap between the primal problem and the dual problem, we introduce an equivalent reformulation for the primal problem via applying a pth power to the constraints. We prove that this reformulation possesses an asymptotic strong duality property. Primal feasibility and primal optimality of the Lagrangian relaxation problems can be achieved in this reformulation when the parameter p is larger than a threshold value, thus ensuring the existence of an optimal primal-dual pair. We further show that duality gap for this partial pth power reformulation is a strictly decreasing function of p in the case of a single constraint. Dedicated to Professor Alex Rubinov on the occasion of his 65th birthday. Research supported by the Research Grants Council of Hong Kong under Grant CUHK 4214/01E, and the National Natural Science Foundation of China under Grants 79970107 and 10571116.     

Stable strong Fenchel and Lagrange duality for evenly convex optimization problems

By means of a conjugation scheme based on generalized convex conjugation theory instead of Fenchel conjugation, we build an alternative dual problem, using the perturbational approach, for a general optimization one defined on a separated locally convex topological space. Conditions guaranteeing strong duality for primal problems which are perturbed by continuous linear functionals and their respective dual problems, which is named stable strong duality, are established. In these conditions, the fact that the perturbation function is evenly convex will play a fundamental role. Stable strong duality will also be studied in particular for Fenchel and Lagrange primal–dual problems, obtaining a characterization for Fenchel case.     

Controlled perturbations for quadratically constrained quadratic programs

Consider a minimization problem of a convex quadratic function of several variables over a set of inequality constraints of the same type of function. The duel program is a maximization problem with a concave objective function and a set of constrains that are essentially linear. However, the objective function is not differentiable over the constraint region. In this paper, we study a general theory of dual perturbations and derive a fundamental relationship between a perturbed dual program and the original problem. Based on this relationship, we establish a perturbation theory to display that a well-controlled perturbation on the dual program can overcome the nondifferentiability issue and generate an ε-optimal dual solution for an arbitrarily small number ε. A simple linear program is then constructed to make an easy conversion from the dual solution to a corresponding ε-optimal primal solution. Moreover, a numerical example is included to illustrate the potential of this controlled perturbation scheme.     

PRIMAL PERTURBATION SIMPLEX ALGORITHMS FOR LINEAR PROGRAMMING

1. IntroductionExtensive research in linear programming, such as [1,2,9,10,if, 12,13,14,19], hasbeen to improve pivot rules to reduce the number of iterations required. Relatively lesseffort was made on perturbing problem data with pivot rules unaltered (for instance, theself--dual parametric method [7] and perturbation--based methods [3,5]). And, becauseof the papametrization, the latter do not proceed as simply as the conventional simplexalgorithm itself.Recently, Pan [17] proposes new pert…     

On a Modified Subgradient Algorithm for Dual Problems via Sharp Augmented Lagrangian*

We study convergence properties of a modified subgradient algorithm, applied to the dual problem defined by the sharp augmented Lagrangian. The primal problem we consider is nonconvex and nondifferentiable, with equality constraints. We obtain primal and dual convergence results, as well as a condition for existence of a dual solution. Using a practical selection of the step-size parameters, we demonstrate the algorithm and its advantages on test problems, including an integer programming and an optimal control problem. *Partially Supported by 2003 UniSA ITEE Small Research Grant Ero2. Supported by CAPES, Brazil, Grant No. 0664-02/2, during her visit to the School of Mathematics and Statistics, UniSA.     

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.     

A numerically stable dual method for solving strictly convex quadratic programs

An efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point. Its implementation utilizes the Cholesky and QR factorizations and procedures for updating them. The performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code (the principal motivation for the development of the algorithm). These computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available. The algorithm is also compared theoretically to the modified-simplex type dual methods of Lemke and Van de Panne and Whinston and it is illustrated by a numerical example. This research was supported in part by the Army Research Office under Grant No. DAAG 29-77-G-0114 and in part by the National Science Foundation under Grant No. MCS-6006065.     

A Trust Region Target Value Method for Optimizing Nondifferentiable Lagrangian Duals of Linear Programs

In this paper, we design a new variable target value procedure, the trust region target value (TRTV) method, for optimizing nondifferentiable Lagrangian dual formulations of large-scale, ill-conditioned linear programming problems. Such problems typically arise in the context of Lagrangian relaxation approaches and branch-and-bound/cut algorithms for solving linear mixed-integer programs. Subgradient optimization strategies are well-suited for this purpose and are popularly used, particularly in Lagrangian relaxation contexts, because of their simplicity in computation and mild memory requirements. However, they lack robustness and can often stall while yet remote from optimality. With this motivation, we design our proposed TRTV method to retain simplicity in computations, be theoretically convergent, as well as yield an effective and robust performance in practice. Furthermore, we augment this approach with dual refinement and primal recovery procedures based on outer-linearization and trust region strategies to further improve the accuracy of the resulting solutions and to derive primal solutions as well. Our computational study reveals a highly competitive performance of the proposed TRTV algorithm among several implemented nondifferentiable optimization procedures. Moreover, the dual refinement and primal recovery procedures help further reduce the optimality gap and promote attaining a relatively greater degree of primal feasibility as compared with several alternative ergodic primal recovery schemes. Also, the proposed method displays significantly lesser computational requirement than that of a commercial linear programming solver CPLEX.This research has been supported by the National Science Foundation under Grant Number DMI-0094462.     

Post-optimality sensitivity analysis in abstract spaces with applications to continuous-time programming problems

The programming problem under consideration consists in maximizing a concave objective functional, subject to convex operator inequality contraints. The assumptions include the existence of an optimum solution, Fréchet differentiability of all operators involved, and the existence of the topological complement of the null space of the Fréchet derivative of the constraint operator. It is shown that the rate of change of the optimum value of the objective functional due to the perturbation is measured by the dual. The optimum values of the primal variables are locally approximated as linear functions of the perturbation; the theory of generalized inverse operators is used in the approximation. We give an approximation to the primal variables if the problem is perturbed. The results are specialized for some continuous-time and finite-dimensional cases. Two examples for finite-dimensional problems are given. We apply the theory to the continuous-time linear programming problem and prove some continuity results for the optimal primal and dual objective functionals.The authors are indebted to the Natural Sciences and Engineering Research Council of Canada for financial support through Grants A4109 and A7329, respectively. They would also like to thank the referee for his comments.     

A primal—dual infeasible-interior-point algorithm for linear programming

As in many primal—dual interior-point algorithms, a primal—dual infeasible-interior-point algorithm chooses a new point along the Newton direction towards a point on the central trajectory, but it does not confine the iterates within the feasible region. This paper proposes a step length rule with which the algorithm takes large distinct step lengths in the primal and dual spaces and enjoys the global convergence.Part of this research was done when M. Kojima and S. Mizuno visited at the IBM Almaden Research Center. Partial support from the Office of Naval Research under Contract N00014-91-C-0026 is acknowledged.Supported by Grant-in-Aids for Co-operative Research (03832017) of The Japan Ministry of Education, Science and Culture.Supported by Grant-in-Aids for Encouragement of Young Scientist (03740125) and Co-operative Research (03832017) of The Japan Ministry of Education, Science and Culture.     

加权线性支持向量分类机的数据扰动分析

加权线性支持向量分类机是数据挖掘的新方法.它对应于一个优化问题.针对加权线性支持向量分类机优化问题建立了数据扰动分析理论方法.具体地针对加权线性支持向量分类机的原始问题建立了数据扰动分析基本定理,定理可以得到加权线性支持向量分类机问题的解及决策函数对数据参数的偏导数,同时可以定量分析输入数据的误差以及数据各种变化对其解以及决策函数值的定量影响,可以回答加权线性支持向量分类机问题的稳定性问题和灵敏度分析问题.     

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.     

A convergent criss-cross method

Our paper presents a new Criss-Cross method for solving linear programming problems. Starting from a neither primal nor dual feasible solution, we reach an optimal solution in finite number of steps if it exists. If there is no optimal solution, then we show that there is not primal feasible or dual feasible solution, We prove the finiteness of this procedure. Our procedure is not the same as the primal or dual simplex method if we have a primal or dual feasible solution, so we have constructed a quite new procedure for solving linear programming problems.     

Theoretical convergence of large-step primal—dual interior point algorithms for linear programming

This paper proposes two sets of rules, Rule G and Rule P, for controlling step lengths in a generic primal—dual interior point method for solving the linear programming problem in standard form and its dual. Theoretically, Rule G ensures the global convergence, while Rule P, which is a special case of Rule G, ensures the O(nL) iteration polynomial-time computational complexity. Both rules depend only on the lengths of the steps from the current iterates in the primal and dual spaces to the respective boundaries of the primal and dual feasible regions. They rely neither on neighborhoods of the central trajectory nor on potential function. These rules allow large steps without performing any line search. Rule G is especially flexible enough for implementation in practically efficient primal—dual interior point algorithms.Part of the research was done when M. Kojima and S. Mizuno visited at the IBM Almaden Research Center. Partial support from the Office of Naval Research under Contracts N00014-87-C-0820 and N00014-91-C-0026 is acknowledged.     

新的滤子方法(英)

本文定义了一种新的滤子方法,并提出了求解光滑不等式约束最优化问题的滤子QP-free非可行域方法. 通过乘子和分片线性非线性互补函数,构造一个等价于原约束问题一阶KKT条件的非光滑方程组.在此基础上, 通过牛顿-拟牛顿迭代得到满足KKT最优条件的解,在迭代中采用了滤子线搜索方法,证明了该算法是可实现,并具有全局收敛性. 另外,在较弱条件下可以证明该方法具有超线性收敛性.