图的最大二等分问题的非线性规划算法

基于图的最大二等分问题的半定规划松驰模型 ,本文提出一个非线性规划算法求解该模型 ,得到该半定规划松驰模型的一个次优解 ,并且给出算法的收敛性证明 .数值试验表明该方法可以有效地求解图的最大二等分问题的松驰模型     

单背包问题的半定松弛算法

首先给出了单背包问题的秩1半定松驰规划，然后在此基础上提出了求解该问题的半定松驰随机算法KSSD。分析结果表明：（1）当σ＞0.19时，算法KSSD的近似比就会超过0.27。（2）算法KSSD中的参数θ对某种大规模情形将不起作用。     

半定优化与半光滑牛顿算法

本文讨论半光滑牛顿算法的基本概念与其在求解半定优化问题中的应用．特别地，该算法可用于求解线性或非线性半定互补问题．本文同时综述最近在矩阵方程，增广拉格朗日公式和半定优化稳定性方面的、源于半光滑牛顿算法的理论成果．     

电路二等分问题的一个强化半定松驰模型

由于电路二等分问题在超大规模集成电路 (VLSI)设计中的基础地位 ,电路二等分半定松驰问题一直引人关注 .能否找到更好的半定规划模型 ,使其为电路二等分问题提供一个更好的下界 ,成为一个重要的研究方向 ;本文在已有半定规划松驰模型的基础上 ,通过增加非线性约束 ,得出电路二等分问题的等价模型 ,再利用提升技巧 ,得到一个强化半定规划松驰模型 .理论证明该模型给出了原有问题的一个更好的下界 ,数值实验也说明了这一点 .     

求解最大割问题的半定规划松驰的序列线性规划方法

本文基于最大割问题的半定规划松弛,利用矩阵分解的方法给出了与半定规划松弛等价的非线性规划模型,提出一种序列线性规划方法求解该模型.并在适当的条件下,证明了算法的全局收敛性.数值实验表明:序列线性规划方法在时间上要优于半定规划的内点算法.所以序列线性规划方法能更有效地求解大规模的最大割问题的半定规划松弛.     

一种求解半定规划的邻近外梯度算法

本文提出了一种求解半定规划的邻近外梯度算法.通过转化半定规划的最优性条件为变分不等式,在变分不等式满足单调性和Lipschitz连续的前提下,构造包含原投影区域的半空间,产生邻近点序列来逼近变分不等式的解,简化了投影的求解过程.将该算法应用到教育测评问题中,数值实验结果表明,该方法是解大规模半定规划问题的一种可行方法.     

非线性半定规划若干算法介绍

介绍近几年国际上求解非线性半定规划的若干有效新算法, 包括增广Lagrangian函数法、序列半定规划法、序列线性方程组法以及交替方向乘子法. 最后, 对非线性半定规划的算法研究前景进行了探讨.     

约束全局整数规划问题的填充函数法

本文给出了一类新的求解箱约束全局整数规划问题的填充函数，并讨论了其填充性质．基于提出的填充函数，设计了一个求解带等式约束、不等式约束、及箱约束的全局整数规划问题的算法．初步的数值试验结果表明提出的算法是可行的。     

多项式分裂可行问题

本文研究多项式分裂可行问题,即由多项式不等式定义的分裂可行问题,包括凸与非凸、可行与不可行的问题;给出多项式分裂可行问题解集的半定松弛表示;研究其半定松弛化问题的性质;并基于这些性质建立求解多项式分裂可行问题的半定松弛算法.本文在较为一般的条件下证明了,如果分裂可行问题有解,则可通过本文建立的算法求得一个解点;如果问题无解,则该算法能够判别问题不可行.最后通过数值实验对算法进行验证.     

半定互补问题的全局收敛算法

基于某一效益函数,本文给出了求解半定互补问题的下降算法,并在适当的条件下证得其全局收敛性.     

带核箱覆盖问题的在线算法

经典的箱覆盖问题是组合优化中一个著名的问题,并且得到了广泛的研究．本文主要讨论带核元的箱覆盖问题的复杂性和在线条件下的算法．指出了带核的箱覆盖问题是强ＮＰ－ｈａｒｄ的．给出了在不同的在线条件下可行算法渐近比的上界,指出仅在条件三下才存在渐近比好于０的在线算法,并给出了在此条件下一个渐近比为１／２的最好的在线算法。     

Ant colony optimization and local search for bin packing and cutting stock problems

The Bin Packing Problem and the Cutting Stock Problem are two related classes of NP-hard combinatorial optimization problems. Exact solution methods can only be used for very small instances, so for real-world problems, we have to rely on heuristic methods. In recent years, researchers have started to apply evolutionary approaches to these problems, including Genetic Algorithms and Evolutionary Programming. In the work presented here, we used an ant colony optimization (ACO) approach to solve both Bin Packing and Cutting Stock Problems. We present a pure ACO approach, as well as an ACO approach augmented with a simple but very effective local search algorithm. It is shown that the pure ACO approach can compete with existing evolutionary methods, whereas the hybrid approach can outperform the best-known hybrid evolutionary solution methods for certain problem classes. The hybrid ACO approach is also shown to require different parameter values from the pure ACO approach and to give a more robust performance across different problems with a single set of parameter values. The local search algorithm is also run with random restarts and shown to perform significantly worse than when combined with ACO.     

A technique for speeding up the solution of the Lagrangean dual

We propose techniques for the solution of the LP relaxation and the Lagrangean dual in combinatorial optimization and nonlinear programming problems. Our techniques find the optimal solution value and the optimal dual multipliers of the LP relaxation and the Lagrangean dual in polynomial time using as a subroutine either the Ellipsoid algorithm or the recent algorithm of Vaidya. Moreover, in problems of a certain structure our techniques find not only the optimal solution value, but the solution as well. Our techniques lead to significant improvements in the theoretical running time compared with previously known methods (interior point methods, Ellipsoid algorithm, Vaidya's algorithm). We use our method to the solution of the LP relaxation and the Langrangean dual of several classical combinatorial problems, like the traveling salesman problem, the vehicle routing problem, the Steiner tree problem, thek-connected problem, multicommodity flows, network design problems, network flow problems with side constraints, facility location problems,K-polymatroid intersection, multiple item capacitated lot sizing problem, and stochastic programming. In all these problems our techniques significantly improve the theoretical running time and yield the fastest way to solve them.     

Probabilistische analyse von heuristiken der kombinatorischen optimierung – ein überbllck

To various problems of combinatorial optimization we consider the question how the value of the optimal solution resp. the values of some approximative solutions are predetermined with high probability to a given distribution. We present results to probabilistic analysis of heuristics. We consider the problems Traveling Salesman, Minimum Perfect Matching. Minimum Spanning Tree, Linear Optimization, Bin Packing, Multi-processor-Scheduling, Subset Sum and some problems to random graphs.     

A probabilistic greedy search algorithm for combinatorial optimisation with application to the set covering problem

We present a probabilistic greedy search method for combinatorial optimisation problems. This approach is implemented and evaluated for the Set Covering Problem (SCP) and shown to yield a simple, robust, and quite fast heuristic. Tests performed on a large set of benchmark instances with up to 1000 rows and 10?000 columns show that the algorithm consistently yields near-optimal solutions.     

Packing chained items in aligned bins with applications to container transshipment and project scheduling

Bin packing problems are at the core of many well-known combinatorial optimization problems and several practical applications alike. In this work we introduce a novel variant of an abstract bin packing problem which is subject to a chaining constraint among items. The problem stems from an application of container handling in rail freight terminals, but is also of relevance in other fields, such as project scheduling. The paper provides a structural analysis which establishes computational complexity of several problem versions and develops (pseudo-)polynomial algorithms for specific subproblems. We further propose and evaluate simple and fast heuristics for optimization versions of the problem.     

On relaxations of the max k-cut problem formulations

A tight continuous relaxation is a crucial factor in solving mixed integer formulations of many NP-hard combinatorial optimization problems. The (weighted) max k-cut problem is a fundamental combinatorial optimization problem with multiple notorious mixed integer optimization formulations. In this paper, we explore four existing mixed integer optimization formulations of the max k-cut problem. Specifically, we show that the continuous relaxation of a binary quadratic optimization formulation of the problem is: (i) stronger than the continuous relaxation of two mixed integer linear optimization formulations and (ii) at least as strong as the continuous relaxation of a mixed integer semidefinite optimization formulation. We also conduct a set of experiments on multiple sets of instances of the max k-cut problem using state-of-the-art solvers that empirically confirm the theoretical results in item (i). Furthermore, these numerical results illustrate the advances in the efficiency of global non-convex quadratic optimization solvers and more general mixed integer nonlinear optimization solvers. As a result, these solvers provide a promising option to solve combinatorial optimization problems. Our codes and data are available on GitHub.     

Two-phase method and Lagrangian relaxation to solve the Bi-Objective Set Covering Problem

This paper deals with the Bi-Objective Set Covering Problem, which is a generalization of the well-known Set Covering Problem. The proposed approach is a two-phase heuristic method which has the particularity to be a constructive method using the primal-dual Lagrangian relaxation to solve single objective Set Covering problems. The results show that this algorithm finds several potentially supported and unsupported solutions. A comparison with an exact method (up to a medium size), shows that many Pareto-optimal solutions are retrieved and that the other solutions are well spread and close to the optimal ones. Moreover, the method developed compares favorably with the Pareto Memetic Algorithm proposed by Jaszkiewicz.     

On the strength of recursive McCormick relaxations for binary polynomial optimization

Recursive McCormick relaxations are among the most popular convexification techniques for binary polynomial optimization. It is well-understood that both the quality and the size of these relaxations depend on the recursive sequence and finding an optimal sequence amounts to solving a difficult combinatorial optimization problem. We prove that any recursive McCormick relaxation is implied by the extended flower relaxation, a linear programming relaxation, which for binary polynomial optimization problems with fixed degree can be solved in strongly polynomial time.     

二次分配问题的大洪水算法求解

大洪水算法是一种求解组合优化问题的独特方法,该方法通过模拟洪水上涨的过程来达到求解一些组合优化难题的目的.本文运用该方法求解二次分配问题(QAP),设计了相应的算法程序,并对QAPLIB(二次分配基准问题库)中的算例进行了实验测试,结果表明,大洪水算法可以快速有效地求得二次分配问题的优化解,是求解二次分配问题的一个新的较好方案.