线性规划的对偶基线算法

In this paper,we studied the dual form of the basic line algorthm for linear programs.It can be easily implemented in tableau that similar to the primal/dual simplex method.Different from primal simplex method or dual simplex method,the dual basic line algorithm can keep primal feasibility and dual feasibility at the same time in a tableau,which makes it more efficient than the former ones.Principles and convergence of dual basic line algorthm were discussed.Some examplex and computational experience were given to illustrate the efficiency of our method.     

On the computation of weighted analytic centers and dual ellipsoids with the projective algorithm

The primal projective algorithm for linear programs with unknown optimal objective function value is extended to the case where one uses a weighted Karmarkar potential function. This potential is defined with respect to a strict lower bound to the optimum. The minimization of this potential when the lower bound is kept fixed, yields a primal and a dual feasible solution. The dual solution is the weighted analytic center of a certain dual polytope. Finally one exhibits a pair of homothetic dual ellipsoids that extends results obtained by Sonnevend, Todd, Ye, Freund and Anstreicher.This research has been supported by NSERC-Canada, FCAR-Quebec and FNRS-Switzerland.     

一种原始——对偶单纯形算法的枢轴准则选择

Curet曾提出了一种有趣的原始一对偶技术,在优化对偶问题的同时单调减少原始不可行约束的数量,当原始可行性产生时也就产生了原问题的最优解.然而该算法需要一个初始对偶可行解来启动,目标行的选择也是灵活、不确定的.根据Curet的原始一对偶算法原理,提出了两种目标行选择准则,并通过数值试验进行比较和选择.对不存在初始对偶可行解的情形,通过适当改变目标函数的系数来构造一个对偶可行解,以求得一个原始可行解,再应用原始单纯形算法求得原问题的最优解.数值试验对这种算法的计算性能进行验证,通过与经典两阶段单纯形算法比较,结果表明,提出的算法在大部分问题上具有更高的计算效率.     

Augmented Lagrangian algorithms for linear programming

We introduce new augmented Lagrangian algorithms for linear programming which provide faster global convergence rates than the augmented algorithm of Polyak and Treti'akov. Our algorithm shares the same properties as the Polyak-Treti'akov algorithm in that it terminates in finitely many iterations and obtains both primal and dual optimal solutions. We present an implementable version of the algorithm which requires only approximate minimization at each iteration. We provide a global convergence rate for this version of the algorithm and show that the primal and dual points generated by the algorithm converge to the primal and dual optimal set, respectively.     

Finite Convergence of the Partial Inverse Algorithm

In Refs. 1–2, Lefebvre and Michelot proved the finite convergence of the partial inverse algorithm applied to a polyhedral convex function by means of some suitable tools of convex analysis. They obtained their result under some assumptions on the primal and dual solution sets. The aim of this note is to show that the proof can be extended to remove the nasty assumption on the dual solution set. The result is in conformity with the proof given in Ref. 3, which has been obtained using the concept of folding.     

A dual approach to primal degeneracy

The Revised Primal Simplex algorithm, in its simplest form, has no defence against degeneracy. Various forms of the perturbation method are usually effective, but most offer no guarantee of avoiding all degeneracy, and can lead to numerical difficulties. This paper presents a method that avoids cycling and circling by taking a dual approach.The degenerate subproblem consists of all the original variables, but only the degenerate transformed constraints. The current primal objective, which may be mixed, is used. This subproblem may be solved using the dual simplex algorithm, starting from the current dual infeasible solution, and with a zero dual objective. If the dual algorithm terminates optimally then the whole problem is optimal (subject to primal feasibility). Otherwise the final solution provides a non-basic direction which improves the value of the mixed primal objective and moves away from the degenerate vertex. A purification algorithm then renders the solution basic and further improves the mixed objective.     

Application of the alternating direction method of multipliers to separable convex programming problems

This paper presents a decomposition algorithm for solving convex programming problems with separable structure. The algorithm is obtained through application of the alternating direction method of multipliers to the dual of the convex programming problem to be solved. In particular, the algorithm reduces to the ordinary method of multipliers when the problem is regarded as nonseparable. Under the assumption that both primal and dual problems have at least one solution and the solution set of the primal problem is bounded, global convergence of the algorithm is established.     

第一阶段原有单纯形和对偶单纯形算法的计算比较

线性最优化广泛应用于经济与管理的各个领域.在线性规划问题的求解中,如果一个初始基本可行解没有直接给出,则常采用经典的两阶段法求解.对含有"≥"不等式约束的线性规划问题,讨论了第一阶段原有单纯形法和对偶单纯形法两种算法形式,并根据第一阶段问题的特点提出了改进的对偶单纯形枢轴准则.最后,通过大规模数值试验对两种算法进行计算比较,结果表明,改进后的对偶单纯形算法在计算效率上明显优于原有单纯形算法.     

带次模惩罚和随机需求的设施选址问题

考虑带次模惩罚和随机需求的设施选址问题,目的是开设设施集合的一个子集,把客户连接到开设的设施上并对没有连接的客户进行惩罚,使得开设费用、连接费用、库存费用、管理费用和惩罚费用之和达到最小. 根据该问题的特殊结构,给出原始对偶3-近似算法. 在算法的第一步,构造了一组对偶可行解;在第二步中构造了对应的一组原始整数可行解,这组原始整数可行解给出了最后开设的设施集合和被惩罚的客户集合. 最后,证明了算法在多项式时间内可以完成,并且算法所给的整数解不会超过最优解的3倍.     

Linear Convergence Rates for Variants of the Alternating Direction Method of Multipliers in Smooth Cases

In the present paper, we propose a novel convergence analysis of the alternating direction method of multipliers, based on its equivalence with the overrelaxed primal–dual hybrid gradient algorithm. We consider the smooth case, where the objective function can be decomposed into one differentiable with Lipschitz continuous gradient part and one strongly convex part. Under these hypotheses, a convergence proof with an optimal parameter choice is given for the primal–dual method, which leads to convergence results for the alternating direction method of multipliers. An accelerated variant of the latter, based on a parameter relaxation, is also proposed, which is shown to converge linearly with same asymptotic rate as the primal–dual algorithm.     

Primal-dual proximal point algorithm for linearly constrained convex programming problems

A primal-dual version of the proximal point algorithm is developed for linearly constrained convex programming problems. The algorithm is an iterative method to find a saddle point of the Lagrangian of the problem. At each iteration of the algorithm, we compute an approximate saddle point of the Lagrangian function augmented by quadratic proximal terms of both primal and dual variables. Specifically, we first minimize the function with respect to the primal variables and then approximately maximize the resulting function of the dual variables. The merit of this approach exists in the fact that the latter function is differentiable and the maximization of this function is subject to no constraints. We discuss convergence properties of the algorithm and report some numerical results for network flow problems with separable quadratic costs.     

一类单边约束问题的数值方法

本文分别基于原始变分形式与对偶混合变分形式，对一类单边约束问题进行了数值求解，提出了求解离散对偶混合变分问题的Uzawa型算法，并用数值例子验证了算法的有效性．     

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.     

线性不等式组的简单对偶非线性方法

将线性不等式组问题转化为一个形式简单的对偶空间非线性极值问题,本提出了一类新的求解线性不等式组的方法-简单对偶非线性方法,它在理论上是多项式算法,并可以从任意点启动,可以应用共轭梯度方法有效地求解大规模线性不等式组问题。本给出了不同的算法实现,数值实验结果表明,简单对偶非线性方法是有效的。     

全局收敛的凸规划的原始—对偶不可行内点算法

本对一类凸规划提出了一个原始-对偶不可行内点算法，并证明了算法的全局收敛性。     

An infeasible interior-point algorithm for solving primal and dual geometric programs

In this paper an algorithm is presented for solving the classical posynomial geometric programming dual pair of problems simultaneously. The approach is by means of a primal-dual infeasible algorithm developed simultaneously for (i) the dual geometric program after logarithmic transformation of its objective function, and (ii) its Lagrangian dual program. Under rather general assumptions, the mechanism defines a primal-dual infeasible path from a specially constructed, perturbed Karush-Kuhn-Tucker system.Subfeasible solutions, as described by Duffin in 1956, are generated for each program whose primal and dual objective function values converge to the respective primal and dual program values. The basic technique is one of a predictor-corrector type involving Newton’s method applied to the perturbed KKT system, coupled with effective techniques for choosing iterate directions and step lengths. We also discuss implementation issues and some sparse matrix factorizations that take advantage of the very special structure of the Hessian matrix of the logarithmically transformed dual objective function. Our computational results on 19 of the most challenging GP problems found in the literature are encouraging. The performance indicates that the algorithm is effective regardless of thedegree of difficulty, which is a generally accepted measure in geometric programming. Research supported in part by the University of Iowa Obermann Fellowship and by NSF Grant DDM-9207347.     

The primal douglas-rachford splitting algorithm for a class of monotone mappings with application to the traffic equilibrium problem

We apply the Douglas-Rachford splitting algorithm to a class of multi-valued equations consisting of the sum of two monotone mappings. Compared with the dual application of the same algorithm, which is known as the alternating direction method of multipliers, the primal application yields algorithms that seem somewhat involved. However, the resulting algorithms may be applied effectively to problems with certain special structure. In particular we show that they can be used to derive decomposition algorithms for solving the variational inequality formulation of the traffic equilibrium problem. This research was supported in part by the Scientific Research Grant-in-Aid from the Ministry of Education, Science and Culture, Japan.     

Preconditioning and globalizing conjugate gradients in dual space for quadratically penalized nonlinear-least squares problems

When solving nonlinear least-squares problems, it is often useful to regularize the problem using a quadratic term, a practice which is especially common in applications arising in inverse calculations. A solution method derived from a trust-region Gauss-Newton algorithm is analyzed for such applications, where, contrary to the standard algorithm, the least-squares subproblem solved at each iteration of the method is rewritten as a quadratic minimization subject to linear equality constraints. This allows the exploitation of duality properties of the associated linearized problems. This paper considers a recent conjugate-gradient-like method which performs the quadratic minimization in the dual space and produces, in exact arithmetic, the same iterates as those produced by a standard conjugate-gradients method in the primal space. This dual algorithm is computationally interesting whenever the dimension of the dual space is significantly smaller than that of the primal space, yielding gains in terms of both memory usage and computational cost. The relation between this dual space solver and PSAS (Physical-space Statistical Analysis System), another well-known dual space technique used in data assimilation problems, is explained. The use of an effective preconditioning technique is proposed and refined convergence bounds derived, which results in a practical solution method. Finally, stopping rules adequate for a trust-region solver are proposed in the dual space, providing iterates that are equivalent to those obtained with a Steihaug-Toint truncated conjugate-gradient method in the primal space.     

A new algorithm for generalized fractional programs

A new dual problem for convex generalized fractional programs with no duality gap is presented and it is shown how this dual problem can be efficiently solved using a parametric approach. The resulting algorithm can be seen as “dual” to the Dinkelbach-type algorithm for generalized fractional programs since it approximates the optimal objective value of the dual (primal) problem from below. Convergence results for this algorithm are derived and an easy condition to achieve superlinear convergence is also established. Moreover, under some additional assumptions the algorithm also recovers at the same time an optimal solution of the primal problem. We also consider a variant of this new algorithm, based on scaling the “dual” parametric function. The numerical results, in case of quadratic-linear ratios and linear constraints, show that the performance of the new algorithm and its scaled version is superior to that of the Dinkelbach-type algorithms. From the computational results it also appears that contrary to the primal approach, the “dual” approach is less influenced by scaling. This research was carried out at the Econometric Institute, Erasmus University, Rotterdam, the Netherlands and was supported by J.N.I.C.T. (Portugal) under contract BD/707/90-RM.     

Ergodic,primal convergence in dual subgradient schemes for convex programming,II: the case of inconsistent primal problems

Consider the utilization of a Lagrangian dual method which is convergent for consistent convex optimization problems. When it is used to solve an infeasible optimization problem, its inconsistency will then manifest itself through the divergence of the sequence of dual iterates. Will then the sequence of primal subproblem solutions still yield relevant information regarding the primal program? We answer this question in the affirmative for a convex program and an associated subgradient algorithm for its Lagrange dual. We show that the primal–dual pair of programs corresponding to an associated homogeneous dual function is in turn associated with a saddle-point problem, in which—in the inconsistent case—the primal part amounts to finding a solution in the primal space such that the Euclidean norm of the infeasibility in the relaxed constraints is minimized; the dual part amounts to identifying a feasible steepest ascent direction for the Lagrangian dual function. We present convergence results for a conditional \(\varepsilon \)-subgradient optimization algorithm applied to the Lagrangian dual problem, and the construction of an ergodic sequence of primal subproblem solutions; this composite algorithm yields convergence of the primal–dual sequence to the set of saddle-points of the associated homogeneous Lagrangian function; for linear programs, convergence to the subset in which the primal objective is at minimum is also achieved.