Exact Algorithm for the Surrogate Dual of an Integer Programming Problem: Subgradient Method Approach

One of the critical issues in the effective use of surrogate relaxation for an integer programming problem is how to solve the surrogate dual within a reasonable amount of computational time. In this paper, we present an exact and efficient algorithm for solving the surrogate dual of an integer programming problem. Our algorithm follows the approach which Sarin et al. (Ref. 8) introduced in their surrogate dual multiplier search algorithms. The algorithms of Sarin et al. adopt an ad-hoc stopping rule in solving subproblems and cannot guarantee the optimality of the solutions obtained. Our work shows that this heuristic nature can actually be eliminated. Convergence proof for our algorithm is provided. Computational results show the practical applicability of our algorithm.     

无约束多目标规划的信赖域方法

本文将信赖域方法应用于多目标规划,提出了一类解多目标问题的新算法,并证明了全局收敛性。     

A Nonmonotone Trust Region Method for Nonlinear Programming with Simple Bound Constraints

In this paper we propose a nonmonotone trust region algorithm for optimization with simple bound constraints. Under mild conditions, we prove the global convergence of the algorithm. For the monotone case it is also proved that the correct active set can be identified in a finite number of iterations if the strict complementarity slackness condition holds, and so the proposed algorithm reduces finally to an unconstrained minimization method in a finite number of iterations, allowing a fast asymptotic rate of convergence. Numerical experiments show that the method is efficient. Accepted 5 September 2000. Online publication 4 December 2000.     

Success Guarantee of Dual Search in Integer Programming: p-th Power Lagrangian Method

Although the Lagrangian method is a powerful dual search approach in integer programming, it often fails to identify an optimal solution of the primal problem. The p-th power Lagrangian method developed in this paper offers a success guarantee for the dual search in generating an optimal solution of the primal integer programming problem in an equivalent setting via two key transformations. One other prominent feature of the p-th power Lagrangian method is that the dual search only involves a one-dimensional search within [0,1]. Some potential applications of the method as well as the issue of its implementation are discussed.     

pth Power Lagrangian Method for Integer Programming

When does there exist an optimal generating Lagrangian multiplier vector (that generates an optimal solution of an integer programming problem in a Lagrangian relaxation formulation), and in cases of nonexistence, can we produce the existence in some other equivalent representation space? Under what conditions does there exist an optimal primal-dual pair in integer programming? This paper considers both questions. A theoretical characterization of the perturbation function in integer programming yields a new insight on the existence of an optimal generating Lagrangian multiplier vector, the existence of an optimal primal-dual pair, and the duality gap. The proposed pth power Lagrangian method convexifies the perturbation function and guarantees the existence of an optimal generating Lagrangian multiplier vector. A condition for the existence of an optimal primal-dual pair is given for the Lagrangian relaxation method to be successful in identifying an optimal solution of the primal problem via the maximization of the Lagrangian dual. The existence of an optimal primal-dual pair is assured for cases with a single Lagrangian constraint, while adopting the pth power Lagrangian method. This paper then shows that an integer programming problem with multiple constraints can be always converted into an equivalent form with a single surrogate constraint. Therefore, success of a dual search is guaranteed for a general class of finite integer programming problems with a prominent feature of a one-dimensional dual search.     

求解整数规划的单参数填充函数

本文给出了一个新的求解离散全局最优化问题的单参数填充函数,并给出了一个新的算法,同时给出了对几个测试问题的数据计算结果.     

广义几何规划的压缩信赖域算法

1 压缩信赖域子问题本文讨论的广义几何规划问题形式如下:其中。αij∈R,σj=±1,cj>0.此类问题在工程优化设计中应用甚广.在(UGGP)中,令t(i)=:ex(i),则(UGGP)等价于下述问题:     

A Dual Parametrization Method for Convex Semi-Infinite Programming

We formulate convex semi-infinite programming problems in a functional analytic setting and derive optimality conditions and several duality results, based on which we develop a computational framework for solving convex semi-infinite programs.     

一种求解序列二次规划结合信赖域的多维滤子算法

求解非线性规划问题最有效的方法之一为序列二次规划。但是,由于序列二次规划结合信赖域时,会出现可能无解的情况(即不相容性)。而本文针对不相容性提出了一类序列二次规划结合信赖域的多维相容滤子算法。首先,本文根据一般文献中提及的方法对其约束条件引进参数变量,对其目标函数加以惩罚,即实行了可行化处理(也就是无需可行性恢复阶段),从而克服了不相容性。其次,本文提出了多维滤子条件来对迭代步进行选择性的接受,从而避免了传统二维滤子算法的严格条件,使得对迭代步的接受程度大大的放松。最后针对可能出现的maratos效应,我们通过二阶校正策略提出了一种修改后的多维滤子算法。同时,在一定的假设条件下算法具有全局收敛性。     

邻域整点搜索法求解整数规划

从剖析线性规划的优化机理入手,将纯整数规划分为标准型和非标型两类.首先以标准型纯整数规划为突破口,提出一种新的解法,并在理论上加以证明,然后将其拓广延伸,用于求解非标准型纯整数规划和混合整数规划.这种新解法命名为松驰最优解邻域整点搜索法,属于常规解法,但在简捷高效方面,远胜过现有的两种常规解法—分枝定界法和割平面法.     

无约束优化的自适应信赖域方法

本文对无约束优化问题提出一个自适应信赖域方法,每次迭代都充分利用前迭代点的信息自动产生一个恰当的信赖域半径,在此区域内,二次模型与原目标函数尽可能一致,避免盲目的尝试,提高了计算效率。文中在通常条件下证明了全局收敛性及局部超线性收敛结果,给出了新算法与传统信赖域方法的数值结果,证实了新方法的有效性。     

整数规划的广义填充函数算法

文[9,10]设计了直接求整数规划问题近似解的填充函数算法,但其所利用的文[2,3]的填充函数均带有参数,需要在算法过程中逐步调节。本文建立整数规划的广义填充函数的定义,说明了文[9,10]所利用的填充函数是整数规划问题的广义填充函数,并构造了一类不带参数的广义填充函数。进而本文设计了整数规划的一类不带参数的广义填充函数算法,数值试验表明算法是有效的。     

SICOpt: Solution Approach for Nonlinear Integer Stochastic Programming Problems

We propose a novel solution approach for the class of two-stage nonlinear integer stochastic programming models. These problems are characterized by large scale dimensions, as the number of constraints and variables depend on the number of realizations (scenarios) used to capture the underlying distributions of the random data. In addition, the integrality constraints on the decision variables make the solution process even much more difficult preventing the application of general purpose solvers. The proposed solution approach integrates the branch-and-bound framework with the interior point method. The main advantage of this choice is the effective exploitation of the specific structure exhibited by the different subproblems at each node of the search tree. A specifically designed warm start procedure and an early branching technique improve the overall efficiency. Our contribution is well founded from a theoretical point of view and is characterized by good computational efficiency, without any loss in terms of effectiveness. Some preliminary numerical results, obtained by solving a challenging real-life problem, prove the robustness and the efficiency of the proposed approach.     

一个基于离散填充函数的整数线性规划方法

本文提出了一个有效的解决整数线性规划的新算法.如果离散化的局部搜索过程陷入局部最优解,则构造相应的离散填充函数,引导搜索过程跳出局部最优解并得到更好的解.该方法是在离散空间中进行优化的,无需增加新的约束,且一直保持整数可行性,收敛的速度非常快.该方法也为一般整数规划提出了一种新的途径.数值实例表明,与现有的方法相比,该算法能够较快的找到最优解.     

无约束多目标规划的非单调信赖域算法

本提出了无约束多目标规划的一类非单调信赖域算法，并证明了算法的全局收敛性。     

凹整数规划的分枝定界解法

凹整数规划是一类重要的非线性整数规划问题，也是在经济和管理中有着广泛应用的最优化问题．本文主要研究用分枝定界方法求解凹整数规划问题，这一方法的基本思想是对目标函数进行线性下逼近，然后用乘子搜索法求解连续松弛问题．数值结果表明，用这种分枝定界方法求解凹整数规划是有效的．     

求解线性不等式约束优化问题的一类信赖域算法

本文对线性不等式约束的非线性规划问题提出了一类信赖域算法,证明了算法所产生的序列的任一聚点为Ｋｕｈｎ－Ｔｕｃｋｅｒ点,并讨论了子问题求解的有效集方法．     

整数规划的渐进强对偶方法

虽然整数规划中经典的Lagrange对偶方法是一个有效的方法，但是由于对偶缝隙的原因它经常不能求出原问题的最优解。该文提出一个用于有界整数规划的指数对偶公式。此公式具有渐进强对偶的特性并且可以保证找到原问题的最优解。它的另一个特性是当参数选择的合适时不需要进行实际的对偶搜索。     

求线性规划对偶问题最优解的一种方法

线性规划对偶问题的最优解有重要的经济意义,中给出了一种较为简捷的求对偶问题最优解的方法。     

整数规划中割平面法的研究

割平面法是求解整数规划问题常用方法之一.用割平面法求解整数规划的基本思路是:先用单纯形表格方法去求解不考虑整数约束条件的松弛问题的最优解,如果获得的最优解的值都是整数,即为所求,运算停止.如果所得最优解不完全是整数,即松弛问题最优解中存在某个基变量为非整数值时,就从最优表中提取出关于这个基变量的约束等式,再从这个约束式出发构造一个割平面方程加入最优表中,再求出新的最优解,这样不断重复的构造割平面方程,直到找到整数解为止.主要研究以下四个关键点:一是研究从最优表中提取出的、关于基变量的约束等式出发,通过将式中的系数进行整数和非负真分数的分解,从而得到一个小于等于0的另外一个不等式的推导过程;二是总结出从小于等于0的那个约束不等式出发构造割平面方程的四种方法;三是分析构造割平面方程的这四种方法相互之间的区别和联系;四是探讨割平面法的几何意义.通过对这四个方面的分析和研究,对割平面法进行透彻的剖析,使读者能够全面把握割平面法.