A mathematical programming system for preference and compatibility maximized menu planning and scheduling

In this paper, the analytical representation of food preference is used in a separable non-linear program to yield the serving frequencies of menu items for a finite time horizon. The frequencies obtained in this way insure cost and nutritional control. Subsequently, the scheduling problem dealing with item assignments to meals and days is formulated as an integer program consisting of several transportation problems linked by weekly nutritional constraints. This problem is solved using a branch and bound algorithm which employs Lagrangian relaxation to obtain bounds and to decide on branching strategy.     

Integer programming duality: Price functions and sensitivity analysis

Recently a duality theory for integer programming has been developed. Here we examine some of the economic implications of this theory, in particular the necessity of using price functions in place of prices, and the possibility of carrying out sensitivity analysis of optimal solutions. In addition we consider the form of price functions that are generated by known algorithms for integer programming.This research was supported in part by a Senior Visiting Research Fellowship from the Science Research Council at the London School of Economics while the author was on leave from CORE, Université Catholique de Louvain, at Louvain-la-Neuve.     

Fixed-parameter algorithms for vertex cover P3

Let $P_{\ell }$ denote a path in a graph $G=(V,E)$ with $\ell$ vertices. A vertex cover $P_{\ell }$ set $C$ in $G$ is a vertex subset such that every $P_{\ell }$ in $G$ has at least a vertex in $C$. The Vertex Cover$P_{\ell }$ problem is to find a vertex cover $P_{\ell }$ set of minimum cardinality in a given graph. This problem is NP-hard for any integer $\ell ⩾2$. The parameterized version of Vertex Cover$P_{\ell }$ problem called $k$-Vertex Cover$P_{\ell }$ asks whether there exists a vertex cover $P_{\ell }$ set of size at most $k$ in the input graph. In this paper, we give two fixed parameter algorithms to solve the $k$-Vertex Cover$P_3$ problem. The first algorithm runs in time $O^1\left(1.7964^k\right)$ in polynomial space and the second algorithm runs in time $O^1\left(1.7485^k\right)$ in exponential space. Both algorithms are faster than previous known fixed-parameter algorithms.     

Two-Dimensional Multiple-Valued Dirichlet Minimizing Functions

We give some remarks on two-dimensional multiple-valued Dirichlet minimizing functions, including frequency, classification of branch points and their connections. As an application, we prove that blowing-up functions of a two-dimensional multiple-valued Dirichlet minimizing function are unique. This article is concluded with a boundary regularity theorem for two-dimensional multiple-valued Dirichlet minimizing functions.     

Shortest path problem with forbidden paths: The elementary version

This paper addresses the elementary shortest path problem with forbidden paths. The main aim is to find the shortest paths from a single origin node to every other node of a directed graph, such that the solution does not contain any path belonging to a given set (i.e., the forbidden set). It is imposed that no cycle can be included in the solution. The problem at hand is formulated as a particular instance of the shortest path problem with resource constraints and two different solution approaches are defined and implemented. One is a Branch & Bound based algorithm, the other is a dynamic programming approach. Different versions of the proposed solution strategies are developed and tested on a large set of test problems.     

The single vehicle routing problem with toll-by-weight scheme: A branch-and-bound approach

Expressways in China make use of the toll-by-weight scheme, in which expressway tolls are collected based on the weight and traveling distance of the vehicle. Most vehicle routing models assume that the cost of traversing each edge is equivalent to edge length or some constant; as a result, such models cannot be practically applied to the Chinese expressway transportation system. This study addresses a new single vehicle routing problem that takes the vehicle’s (laden and unladen) weight into account. To solve this problem exactly, we provide a branch-and-bound algorithm with a provably valid lower bound measure, along with five dominance checkers for additional pruning. We analyze our algorithm using instances generated from standard TSP test cases, as well as two new sets of test cases based on real expressway information from the Gansu and Jiangxi provinces in China. The algorithm can be applied to any toll scheme in which the toll per unit distance monotonically increases with weight, even if the toll function is non-linear.     

一类新的非线性比式和问题的分枝定界算法(英文)

对一类新的非线性比式和问题(SNR)提出分枝定界算法,该问题的研究还很少.首先,通过两层线性化技术,构造一个松弛线性规划,求解该线性规划问题,得到问题(SNR)最优值的下界.其次,介绍新的下界更新技术,证明所给算法的收敛性.数值试验显示了算法的可行性和有效性     

DC programming techniques for solving a class of nonlinear bilevel programs

We propose a method for finding a global solution of a class of nonlinear bilevel programs, in which the objective function in the first level is a DC function, and the second level consists of finding a Karush-Kuhn-Tucker point of a quadratic programming problem. This method is a combination of the local algorithm DCA in DC programming with a branch and bound scheme well known in discrete and global optimization. Computational results on a class of quadratic bilevel programs are reported.     

布尔环上的分支Gr(o)bner基算法

众所周知Gr\"obner基在很多领域都有着十分重要的应用.近些年来Gr\"obner基算法有了很大的改进,其中最著名的是Faug\`ere提出的F4和F5算法. 这两个算法具有很高的效率但通常需要消耗大量的内存.鉴于此,将给出一个布尔环上基于zdd数据结构的分支Gr\"obner基算法,该算法不仅可以大大降低对内存的消耗,还能有效的控制矩阵规模,从而提高算法的整体效率.详细阐述并证明了算法的基本理论,介绍该分支算法的数据结构及分支策略.最后通过实验数据可以发现,在很多例子中此算法都要优于Magma中的F4算法.     

带有分支输流管道控制方程的参数变换、

输流管道的非线性振动问题研究为当前学术界的热点课题。本文在研究带有分支管路输流管道并推导出其控制微分方程的基础上,进一步分析了与控制微分方程相关的各项参数之间的关系,为了更加方便地采用数学方法进行研究．对该方程应用无量纲化的方法进行了参数变换,从而将该物理方程的研究转换为计算数学问题,为后续Galerkin离散化及数值计算的进行提供了可能。