基于粒子群算法的非线性二层规划问题的求解算法

粒子群算法（Particle Swarm Optimization，PSO）是一种新兴的优化技术，其思想来源于人工生命和演化计算理论。PSO通过粒子追随自己找到的最好解和整个群的最好解来完成优化。该算法简单易实现，可调参数少，已得到了广泛研究和应用。本文根据该算法能够有效的求出非凸数学规划全局最优解的特点，对非线性二层规划的上下层问题求解，并根据二层规划的特点，给出了求解非线性二层规划问题全局最优解的有效算法。数值计算结果表明该算法有效。     

最小平均长度回路问题的一个有效算法

本文提出了求解最小平均长度回路问题的一个有效算法。从本质上来说,这一算法是原始——对偶算法。该算法通过反复求解一系列织合特点更强的子问题,从而有效地求得最优解。我们还证明了算法的复杂性为 O(n~4),并且通过计算大量随机产生的实例并与已有算比较,证明该算法是很有效的。     

校园最佳游览路线问题的数学模型分析

将某高校的校园示意图转化为赋权连通图，求得该连通图的邻接矩阵，利用Floyd算法及图论软件包构造一个最短路径矩阵，得到一个赋权完全图，将求校园最佳游览路线问题归结为图论中的最佳推销员回路问题，建立混合整数线性规划模型，并利用优化软件求得最优解．从而解决了校园开放日游览计划中提出的关于校园最佳游览路线和校园游览车最优配置问题．     

一种求解救灾物资运输问题的改进解法

本文通过对B运输问题建立数学模型,提出了一种求解B运输问题的改进解法。改进解法首先通过最小元素法求出初始解,然后进行变量闭回路法调整,直到求出最优解,并给出了一个计算实例证明了解法的有效性。文章还对改进解法和另外两种现有的算法进行了综合的分析,由于改进解法计算过程中采用的变量闭回路法省略了求检验数的环节,使得新算法比两种现有的算法更简便。     

复合二项对偶模型中的周期性分红问题

该文主要在有界红利率的条件下讨论复合二项对偶模型的周期性分红问题.通过对值函数进行变换,得到了最优红利策略的一些性质,并且证明了最优值函数是一个HJB方程的唯一解.从而得到了最优策略和最优值函数的一个简单计算方法.根据最优红利策略的一些性质,该文还得到了最优值函数的可无限逼近的上界和下界.最后提供一些数值计算实例来说明该算法.     

空中加油问题的递推模型与调度策略

首先对空中加油问题进行了分析,提取了相关性质,在此基础上建立了问题的递推模型.根据该模型,提出了一种启发式搜索算法.该算法计算复杂度低,适用性好.对应于辅机是否可以多次起飞,该算法分为两子算法.对这两种不同情况下的具体问题,设计了相关的优化函数.所有算法都在计算机中运行,并得到了相应结果.值得指出的是,提出的启发式搜索算法十分高效.对于问题1和问题2,该算法所得解是约束条件下的最优调度策略.对于问题3,问题4,问题5,该算法所得解逼近最优调度策略.     

离散变量结构优化设计的组合算法*

本文首先给出了离散变量优化设计局部最优解的定义,然后提出了一种综合的组合算法.该算法采用分级优化的方法,第一级优化首先采用计算效率很高且经过随机抽样性能实验表明性能较高的启发式算法─—相对差商法,求解离散变量结构优化设计问题近似最优解 X ;第二级采用组合算法,在 X 的离散邻集内建立离散变量结构优化设计问题的(-1,0.1)规划模型,再进一步将其化为(0,1)规划模型,应用定界组合算法或相对差商法求解该(0,1)规划模型,求得局部最优解.解决了采用启发式算法无法判断近似最优解是否为局部最优解这一长期未得到解决的问题,提高了计算精度,同时,由于相对差商法的高效率与高精度,以上综合的组合算法的计算效率也还是较高的.     

资源公平分配的一种贪婪算法

对资源公平分配模型提出了一种简单的贪婪算法，在一定条件下可得到全局最优解且在相当多的情况下所得解都为最优解。该方法效率极高，编程简单，计算量很小，从大量模拟情况来看相当有效。     

有界变量线性规划的基线算法

本文对有界变量线性规划的算法进行了研究，得到了一种解此问题的新算法。文中根据基线算法的算法原理，通过对BL表的旋转，在各变量满足界约束的条件下，使目标函数值不断增大，直至得到有界硬上界，从而得到问题的最优解。文中给出了有界变量线性规划基线算法的计算步骤，并给出了一个例子。与单纯形法相比，采用基线算法解有界变量线性规划操作更简单。迭代次数少，解题速度更快。     

基于部分基变量的LP问题矩阵算法

基于部分基变量提出了LP问题的矩阵算法. 该算法以最优基矩阵的一个充分必要条件为基础,首先将一个初始矩阵转化为右端项和检验数均满足要求的矩阵,再转为检验数满足要求的基矩阵,最后转化为最优基矩阵.该算法具有使用范围广、计算规模小、计算过程简化、计算机易于实现的优势.矩阵算法的核心运算是求逆矩阵的运算,提出了矩阵算法的求逆问题,讨论并给出了求逆快速算法,该算法充分利用了矩阵算法迭代过程中提供的原来的逆矩阵的信息经过简单的变换得到新的逆矩阵,该算法比直接求逆法计算效率更高.     

Finding low cost TSP and 2-matching solutions using certain half-integer subtour vertices

Consider the traveling salesman problem (TSP) defined on the complete graph, where the edge costs satisfy the triangle inequality. Let TOUR denote the optimal solution value for the TSP. Two well-known relaxations of the TSP are the subtour elimination problem and the 2-matching problem. If we let SUBT and 2M represent the optimal solution values for these two relaxations, then it has been conjectured that TOUR/SUBT ≤4/3, and that 2M/SUBT ≤10/9.In this paper, we exploit the structure of certain 1/2-integer solutions for the subtour elimination problem in order to obtain low cost TSP and 2-matching solutions. In particular, we show that for cost functions for which the optimal subtour elimination solution found falls into our classes, the above two conjectures are true. Our proofs are constructive and could be implemented in polynomial time, and thus, for such cost functions, provide a 4/3 (or better) approximation algorithm for the TSP.     

A polynomial-time solution to Papadimitriou and Steiglitz's ‘traps’

Papadimitriou and Steiglitz constructed ‘traps’ for the symmetric travelling salesman problem (TSP) with n = 8k cities. The constructed problem instances have exponentially many suboptimal solutions with arbitrarily large weight, which differ from the unique optimal solution in exactly 3k edges, and hence local search algorithms are ineffective to solve this problem. However, we show that this class of ‘catastrophic’ examples can be solved by linear programming relaxation appended with k subtour elimination constraints. It follows that this class of problem instances of TSP can be optimized in polynomial time.     

Min-weight double-tree shortcutting for Metric TSP: Bounding the approximation ratio

The Metric Traveling Salesman Problem (TSP) is a classical NP-hard optimization problem. The double-tree shortcutting method for Metric TSP yields an exponentially-sized space of TSP tours, each of which approximates the optimal solution within at most a factor of 2. We consider the problem of finding among these tours the one that gives the closest approximation, i.e. the minimum-weight double-tree shortcutting. Previously, we gave an efficient algorithm for this problem, and carried out its experimental analysis. In this paper, we address the related question of the worst-case approximation ratio for the minimum-weight double-tree shortcutting method. In particular, we give lower bounds on the approximation ratio in some specific metric spaces: the ratio of 2 in the discrete shortest path metric, 1.622 in the planar Euclidean metric, and 1.666 in the planar Minkowski metric. The first of these lower bounds is tight; we conjecture that the other two bounds are also tight, and in particular that the minimum-weight double-tree method provides a 1.622-approximation for planar Euclidean TSP.     

Revisiting dynamic programming for precedence-constrained traveling salesman problem and its time-dependent generalization

The precedence constrained traveling salesman problem (TSP-PC), or the sequential ordering problem (SOP), consists of finding an optimal TSP tour that will also satisfy the namesake precedence constraints, typically specified as a partial order or a directed acyclic graph. Its dynamic programming (DP) solution was proposed as early as 1979, however, by late 1990s, it mostly fell out of use in plain TSP-PC. Revisiting this method, we are able to close one of the long-standing TSPLIB SOP problem instances, ry48p.3.sop, and provide improved bounds on its time complexity. Harnessing the “omnivorous” nature of DP, we prove the validity of DP optimality principle for TSP-PC with both (i) abstract cost aggregation function, which may be the arithmetic + operation as in “ordinary” TSP or $\max$ as in Bottleneck TSP, or any other left-associative nondecreasing in the first argument operation and (ii) travel cost functions depending on the set of pending tasks (“sequence dependence”). Using the latter generalization, we close several TD-SOP (time-dependent TSP-PC) instances based on TSPLIB SOP as proposed by Kinable et al., including rbg253a.sop. Through the restricted DP heuristic, which was originally formulated for time-dependent TSP by Malandraki and Dial, we improve the state-of-the-art upper bounds for all yet unsolved TSPLIB-based TD-SOP instances, including those with more than 100 cities. We also improve worst-case complexity estimates for DP in TSP-PC.     

Experimental analysis of heuristics for the bottleneck traveling salesman problem

In this paper we develop efficient heuristic algorithms to solve the bottleneck traveling salesman problem (BTSP). Results of extensive computational experiments are reported. Our heuristics produced optimal solutions for all the test problems considered from TSPLIB, JM-instances, National TSP instances, and VLSI TSP instances in very reasonable running time. We also conducted experiments with specially constructed ‘hard’ instances of the BTSP that produced optimal solutions for all but seven problems. Some fast construction heuristics are also discussed. Our algorithms could easily be modified to solve related problems such as the maximum scatter TSP and testing hamiltonicity of a graph.     

Maximizing the profit per unit of time for the TSP with convex resource-dependent travelling times

This paper introduces a new problem that is an extension of the travelling salesman problem (TSP) in which the travelling times are resource dependent and the objective is to maximize the profit per unit of time. We present an optimal solution approach comprised of three main steps: (1) calculating the optimal amount of total resource required (regardless of the selected tour); (2) constructing the tour; and (3) assigning the optimal resource to each connection between vertices using the equivalent load method. This solution approach finds the optimal solution with the same computational complexity for solving the classic TSP.     

The travelling salesman problem: new solvable cases and linkages with the development of approximation algorithms

We identify new solvable cases of the travelling salesman problem (TSP) by an indirect analysis that has useful consequences. First, we develop new procedures for the TSP that require only linear time to execute and yield TSP tours that are better than an exponential number of alternative tours. We then identify special subgraphs, easily generated, so that our method yields these outcomes for every instance of these subgraphs. Finally, when the associated costs satisfy prescribed conditions, we show the solutions produced by these algorithms are optimal and thus we have new solvable cases of the TSP. Besides possible practical applications to problems that may exhibit these cost conditions, our algorithms may also be applied as subroutines within more complex metaheuristics. Our methods extend in a natural way to bottleneck TSP formulations, and their underlying results raise new theoretical questions about the analysis of heuristics for hard combinatorial problems.     

On estimating the distribution of optimal traveling salesman tour lengths using heuristics

The traveling salesman problem is an important combinatorial optimization problem due to its significance in academic research and its real world applications. The problem has been extensively studied and much is known about its polyhedral structure and algorithms for exact and heuristic solutions. While most work is concentrated on solving the deterministic version of the problem, there also has been some research on the stochastic TSP. Research on the stochastic TSP has concentrated on asymptotic properties and estimation of the TSP-constant. Not much is, however, known about the probability distribution of the optimal tour length. In this paper, we present some empirical results based on Monte Carlo simulations for the symmetric Euclidean and Rectilinear TSPs. We derive regression equations for predicting the first four moments of the distribution of estimated TSP tour lengths using heuristics. We then show that a Beta distribution gives excellent fits for small to moderate sized TSP problems. We derive regression equations for predicting the parameters of the Beta distribution. Finally we predict the TSP constant using two alternative approaches.     

Algorithms for the universal and a priori TSP

We present two simple results for generalizations of the traveling salesman problem (TSP): for the universal TSP, we show that one can compute a tour that is universally optimal whenever the input is a tree metric. A (randomized) O(logn)-approximation algorithm for the a priori TSP follows as a corollary.     

On the asymptotic behavior of subtour-patching heuristics in solving the TSP on permuted Monge matrices

We examine the performance of different subtour-patching heuristics for solving the strongly NP\mathcal{NP}-hard traveling salesman problem (TSP) on permuted Monge matrices. We prove that a well-known heuristic is asymptotically optimal for the TSP on product matrices and k-root cost matrices. We also show that the heuristic is provably asymptotically optimal for general permuted Monge matrices under some mild conditions. Our theoretical results are strongly supported by the findings of a large-scale experimental study on randomly generated numerical examples, which show that the heuristic is not only asymptotically optimal, but also finds optimal TSP tours with high probability that increases with the problem size. Thus the heuristic represents a practical tool to solve large instances of the problem.