基于指数型核函数的线性规划原始对偶内点算法

给出线性规划原始对偶内点算法的一个单变量指数型核函数.首先研究了这个指数型核函数的性质以及其对应的障碍函数.其次,基于这个指数型核函数,设计了求解线性规划问题的原始对偶内点算法,得到了目前小步算法最好的理论迭代界.最后,通过数值算例比较了基于指数型核函数的原始对偶内点算法和基于对数型核函数的原始对偶内点算法的计算效果.     

一个解半正定规划问题的基于广义对数障碍函数的原始对偶内点算法

本文对经典对数障碍函数推广,给出了一个广义对数障碍函数.基于这个广义对数障碍函数设计了解半正定规划问题的原始-对偶内点算法.分析了该算法的复杂性,得到了一个理论迭代界,它与已有的基于经典对数障碍函数的算法的理论迭代界一致.同时,并给出了一个数值算例,阐明了函数的参数对算法运行时间的影响.     

半无限规划的一个非线性Lagrangian对偶模型

本文给出半无限规划的一个对偶罚函数模型,该模型能处理目标函数不是凸函数的情形,从而凸(SIP)对偶为该模型的一个特例.并且,作为罚函数,本模型的罚因子比l1-罚函数要小,这使得算法更可行,最后,给出零对偶间隙证明.     

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.     

最小化三个凸函数之和的一个简单原始-对偶算法

提出一个简单的原始-对偶算法求解三个凸函数之和的最小化问题, 其中目标函数包含有梯度李普希兹连续的光滑函数, 非光滑函数和含有复合算子的非光滑函数. 在新方法中, 对偶变量迭代使用预估-矫正的方案. 分析了算法的收敛性和收敛速率. 最后, 数值实验说明了算法的有效性.     

一种改进的求解矩阵补全问题的原始-对偶算法

低秩矩阵补全问题作为一类在机器学习和图像处理等信息科学领域中都十分重要的问题已被广泛研究.一阶原始-对偶算法是求解该问题的经典算法之一.然而实际应用中处理的数据往往是大规模的.针对大规模矩阵补全问题,本文在原始-对偶算法的框架下,应用变步长校正技术,提出了一种改进的求解矩阵补全问题的原始-对偶算法.该算法在每一步迭代过程中,首先利用原始-对偶算法对原始变量和对偶变量进行更新,然后采用变步长校正技术对这两块变量进行进一步的校正更新.在一定的假设条件下,证明了新算法的全局收敛性.最后通过求解随机低秩矩阵补全问题及图像修复的实例验证新算法的有效性.     

一个新的求解半正定规划问题的原始对偶内点算法

选择合适的核函数对设计求解线性规划与半正定规划的原始对偶内点算法以及复杂性分析都十分重要.Bai等针对线性规划提出三种核函数,并给出求解线性规划的大步迭代复杂界,但未给出数值算例验证算法的实际效果(Bai Y Q,Xie W,Zhang J.New parameterized kernel functions for linear optimization.J Global Optim,2012.DOI 10.1007/s10898-012-9934-z).基于这三种核函数设计了新的求解半正定规划问题的原始对内点算法.进一步分析了算法关于大步方法的计算复杂性界,同时通过数值算例验证了算法的有效性和核函数所带参数对计算复杂性的影响.     

分片线性NCP函数滤子QP-free算法

本文定义了分片线性NCP函数,并对非线性约束优化问题,提出了带有这分片NCP函数的QP-free非可行域算法.利用优化问题的一阶KKT条件,乘子和NCP函数,得到对应的非光滑方程组.本文给出解这非光滑方程组算法,它包含原始-对偶变量,在局部意义下,可看成关扰动牛顿-拟牛顿迭代算法.在线性搜索时,这算法采用滤子方法.本文给出的算法是可实现的并具有全局收敛性,在适当假设下算法具有超线性收敛性.     

一个求解半正定规划问题的新原始-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 10g n/ε)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.     

3-分片线性NCP函数的滤子QP-free算法

本文定义一个3-分片线性的NCP函数,并对非线性约束优化问题,提出了带有这分片NCP函数的QP-free非可行域算法.根据优化问题的一阶KKT条件,利用乘子和NCP函数,得到非光滑方程,本文给出一个非光滑方程的迭代算法.这算法包含原始-对偶变量,在局部意义下,可看成关于一阶KKT最优条件的的扰动拟牛顿迭代算法.在线性搜索时,这算法采用滤子方法.本文给出的算法是可实现的并具有全局收敛性,且在适当假设下具有超线性收敛性.     

A Dual Approach for Solving Nonlinear Infinity-Norm Minimization Problems with Applications in Separable Cases

In this paper,we consider nonlinear infinity-norm minimization problems.We device a reliable Lagrangian dual approach for solving this kind of problems and based on this method we propose an algorithm for the mixed linear and nonlinear infinity- norm minimization problems.Numerical results are presented.     

Nonlinear Lagrangians for Nonlinear Programming Based on Modified Fischer-Burmeister NCP Functions

This paper proposes nonlinear Lagrangians based on modified Fischer-Burmeister NCP functions for solving nonlinear programming problems with inequality constraints. The convergence theorem shows that the sequence of points generated by this nonlinear Lagrange algorithm is locally convergent when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions, and the error bound of solution, depending on the penalty parameter, is also established. It is shown that the condition number of the nonlinear Lagrangian Hessian at the optimal solution is proportional to the controlling penalty parameter. Moreover, the paper develops the dual algorithm associated with the proposed nonlinear Lagrangians. Numerical results reported suggest that the dual algorithm based on proposed nonlinear Lagrangians is effective for solving some nonlinear optimization problems.     

On the implementation of a log-barrier progressive hedging method for multistage stochastic programs

A progressive hedging method incorporated with self-concordant barrier for solving multistage stochastic programs is proposed recently by Zhao [G. Zhao, A Lagrangian dual method with self-concordant barrier for multistage stochastic convex nonlinear programming, Math. Program. 102 (2005) 1-24]. The method relaxes the nonanticipativity constraints by the Lagrangian dual approach and smoothes the Lagrangian dual function by self-concordant barrier functions. The convergence and polynomial-time complexity of the method have been established. Although the analysis is done on stochastic convex programming, the method can be applied to the nonconvex situation. We discuss some details on the implementation of this method in this paper, including when to terminate the solution of unconstrained subproblems with special structure and how to perform a line search procedure for a new dual estimate effectively. In particular, the method is used to solve some multistage stochastic nonlinear test problems. The collection of test problems also contains two practical examples from the literature. We report the results of our preliminary numerical experiments. As a comparison, we also solve all test problems by the well-known progressive hedging method.     

一类Lagrangian对偶问题的零对偶间隙性质及其最优路径的收敛性

针对一般的非线性规划问题,利用某些Lagrange型函数给出了一类Lagrangian对偶问题的一般模型,并证明它与原问题之间存在零对偶间隙．针对具体的一类增广La- grangian对偶问题以及几类由非线性卷积函数构成的Lagrangian对偶问题,详细讨论了零对偶间隙的存在性．进一步,讨论了在最优路径存在的前提下,最优路径的收敛性质．     

A sparse sequential quadratic programming algorithm

Described here is the structure and theory for a sequential quadratic programming algorithm for solving sparse nonlinear optimization problems. Also provided are the details of a computer implementation of the algorithm along with test results. The algorithm maintains a sparse approximation to the Cholesky factor of the Hessian of the Lagrangian. The solution to the quadratic program generated at each step is obtained by solving a dual quadratic program using a projected conjugate gradient algorithm. An updating procedure is employed that does not destroy sparsity.     

A nonlinear Lagrangian for constrained optimization problems

A novel nonlinear Lagrangian is presented for constrained optimization problems with both inequality and equality constraints, which is nonlinear with respect to both functions in problem and Lagrange multipliers. The nonlinear Lagrangian inherits the smoothness of the objective and constraint functions and has positive properties. The algorithm on the nonlinear Lagrangian is demonstrated to possess local and linear convergence when the penalty parameter is less than a threshold (the penalty parameter in the penalty method has to approximate zero) under a set of suitable conditions, and be super-linearly convergent when the penalty parameter is decreased following Lagrange multiplier update. Furthermore, the dual problem based on the nonlinear Lagrangian is discussed and some important properties are proposed, which fail to hold for the dual problem based on the classical Lagrangian. At last, the preliminary and comparing numerical results for several typical test problems by using the new nonlinear Lagrangian algorithm and the other two related nonlinear Lagrangian algorithms, are reported, which show that the given nonlinear Lagrangian is promising.     

Convergent Lagrangian heuristics for nonlinear minimum cost network flows

We consider the separable nonlinear and strictly convex single-commodity network flow problem (SSCNFP). We develop a computational scheme for generating a primal feasible solution from any Lagrangian dual vector; this is referred to as “early primal recovery”. It is motivated by the desire to obtain a primal feasible vector before convergence of a Lagrangian scheme; such a vector is not available from a Lagrangian dual vector unless it is optimal. The scheme is constructed such that if we apply it from a sequence of Lagrangian dual vectors that converge to an optimal one, then the resulting primal (feasible) vectors converge to the unique optimal primal flow vector. It is therefore also a convergent Lagrangian heuristic, akin to those primarily devised within the field of combinatorial optimization but with the contrasting and striking advantage that it is guaranteed to yield a primal optimal solution in the limit. Thereby we also gain access to a new stopping criterion for any Lagrangian dual algorithm for the problem, which is of interest in particular if the SSCNFP arises as a subproblem in a more complex model. We construct instances of convergent Lagrangian heuristics that are based on graph searches within the residual graph, and therefore are efficiently implementable; in particular we consider two shortest path based heuristics that are based on the optimality conditions of the original problem. Numerical experiments report on the relative efficiency and accuracy of the various schemes.     

A Lagrangian Dual Method with Self-Concordant Barriers for Multi-Stage Stochastic Convex Programming

This paper presents an algorithm for solving multi-stage stochastic convex nonlinear programs. The algorithm is based on the Lagrangian dual method which relaxes the nonanticipativity constraints, and the barrier function method which enhances the smoothness of the dual objective function so that the Newton search directions can be used. The algorithm is shown to be of global convergence and of polynomial-time complexity.Mathematics Subject Classification (2000): 90C15, 90C51, 90C06, 90C25, 90C60Research is partially supported by NUS Academic Research Grant R-146-000-006-112     

About Lagrangian Methods in Integer Optimization

It is well-known that the Lagrangian dual of an Integer Linear Program (ILP) provides the same bound as a continuous relaxation involving the convex hull of all the optimal solutions of the Lagrangian relaxation. It is less often realized that this equivalence is effective, in that basically all known algorithms for solving the Lagrangian dual either naturally compute an (approximate) optimal solution of the “convexified relaxation”, or can be modified to do so. After recalling these results we elaborate on the importance of the availability of primal information produced by the Lagrangian dual within both exact and approximate approaches to the original (ILP), using three optimization problems with different structure to illustrate some of the main points.     

求解随机凸规划概率约束问题的对偶算法

1引言随机规划中的概率约束问题在工程和管理中有广泛的应用.因为问题中包含非线性的概率约束,它们的求解非常困难.如果目标函数是线性的,问题的求解就比较容易.给出了一个求解随机线性规划概率约束问题的综述.原-对偶算法和切平面算法是比较有效的.在本文中,我们讨论随机凸规划概率约束问题: