Cutting plane algorithms for solving a stochastic edge-partition problem

We consider the edge-partition problem, which is a graph theoretic problem arising in the design of Synchronous Optical Networks. The deterministic edge-partition problem considers an undirected graph with weighted edges, and simultaneously assigns nodes and edges to subgraphs such that each edge appears in exactly one subgraph, and such that no edge is assigned to a subgraph unless both of its incident nodes are also assigned to that subgraph. Additionally, there are limitations on the number of nodes and on the sum of edge weights that can be assigned to each subgraph. In this paper, we consider a stochastic version of the edge-partition problem in which we assign nodes to subgraphs in a first stage, realize a set of edge weights from a finite set of alternatives, and then assign edges to subgraphs. We first prescribe a two-stage cutting plane approach with integer variables in both stages, and examine computational difficulties associated with the proposed cutting planes. As an alternative, we prescribe a hybrid integer programming/constraint programming algorithm capable of solving a suite of test instances within practical computational limits.     

Limit vector variational inequality problems via scalarization

This paper presents a global optimization approach for solving signomial geometric programming (SGP) problems. We employ an accelerated extended cutting plane (ECP) approach integrated with piecewise linear (PWL) approximations to solve the global optimization of SGP problems. In this approach, we separate the feasible regions determined by the constraints into convex and nonconvex ones in the logarithmic domain. In the nonconvex feasible regions, the corresponding constraint functions are converted into mixed integer linear constraints using PWL approximations, while the other constraints with convex feasible regions are handled by the ECP method. We also use pre-processed initial cuts and batched cuts to accelerate the proposed algorithm. Numerical results show that the proposed approach can solve the global optimization of SGP problems efficiently and effectively.     

Generation of assembly graphs by systematic analysis of assembly structures

In assembly line balancing problems, parallel execution of assembly operations is often advocated because of its enhanced flexibility and minimum lead-time. Although the theoretical maximum number of possible assembly sequences combinatorially explodes with the number of components in a product, graphical representations can depict these sequences in a surveyable way. The AND/OR graph representation is an appropriate basis for optimum sequence selection, which can be achieved via heuristic, metaheuristic, and exact methods. The exact method, based on binary linear programming, is described. To arrive at the appropriate model, a novel approach for AND/OR graph generation, based on subassembly detection, is presented. The method is demonstrated with simple cases and next extended to increasingly complex products. A modification of the optimization method is applied, which enables a search for sequences with maximum parallelism.     

A branch and cut algorithm for the hierarchical network design problem

The Hierarchical Network Design Problem consists of locating a minimum cost bi-level network on a graph. The higher level sub-network is a path visiting two or more nodes. The lower level sub-network is a forest connecting the remaining nodes to the path. We optimally solve the problem using an ad hoc branch and cut procedure. Relaxed versions of a base model are solved using an optimization package and, if binary variables have fractional values or if some of the relaxed constraints are violated in the solution, cutting planes are added. Once no more cuts can be added, branch and bound is used. The method for finding valid cutting planes is presented. Finally, we use different available test instances to compare the procedure with the best known published optimal procedure, with good results. In none of the instances we needed to apply branch and bound, but only the cutting planes.     

Optimizing Bernoulli Routing Policies for Balancing Loads on Call Centers and Minimizing Transmission Costs

We address the problem of assigning probabilities at discrete time instants for routing toll-free calls to a given set of call centers to minimize a weighted sum of transmission costs and load variability at the call centers during the next time interval.We model the problem as a tripartite graph and decompose the finding of an optimal probability assignment in the graph into the following problems: (i) estimating the true arrival rates at the nodes for the last time period; (ii) computing routing probabilities assuming that the estimates are correct. We use a simple approach for arrival rate estimation and solve the routing probability assignment by formulating it as a convex quadratic program and using the affine scaling algorithm to obtain an optimal solution.We further address a practical variant of the problem that involves changing routing probabilities associated with k nodes in the graph, where k is a prespecified number, to minimize the objective function. This involves deciding which k nodes to select for changing probabilities and determining the optimal value of the probabilities. We solve this problem using a heuristic that ranks all subsets of k nodes using gradient information around a given probability assignment.The routing model and the heuristic are evaluated for speed of computation of optimal probabilities and load balancing performance using a Monte Carlo simulation. Empirical results for load balancing are presented for a tripartite graph with 99 nodes and 17 call center gates.     

Time-optimal velocity planning by a bound-tightening technique

Range reduction techniques often considerably enhance the performance of algorithmic approaches for the solution of nonconvex problems. In this paper we propose a range reduction technique for a class of optimization problems with some special structured constraints. The procedure explores and updates the values associated to the nodes of a suitably defined graph. Convergence of the procedure and some efficiency issues, in particular related to the order into which the nodes of the graph are explored. The proposed technique is applied to solve problems arising from a relevant practical application, namely velocity planning along a given trajectory. The computational experiments show the efficiency of the procedure and its ability of returning solutions within times much lower than those of nonlinear solvers and compatible with real-time applications.     

Solving VRPTWs with Constraint Programming Based Column Generation

Constraint programming based column generation is a hybrid optimization framework recently proposed (Junker et al., 1999) that uses constraint programming to solve column generation subproblems. In the past, this framework has been used to solve scheduling problems where the associated graph is naturally acyclic and has done so very efficiently. This paper attempts to solve problems whose graph is cyclic by nature, such as routing problems, by solving the elementary shortest path problem with constraint programming. We also introduce new redundant constraints which can be useful in the general framework. The experimental results are comparable to those of the similar method in the literature (Desrochers, Desrosiers, and Solomon, 1992) but the proposed method yields a much more flexible approach.     

A Cut and Branch Approach for the Capacitated <Emphasis Type="Italic">p</Emphasis>-Median Problem Based on Fenchel Cutting Planes

The capacitated p-median problem (CPMP) consists of finding p nodes (the median nodes) minimizing the total distance to the other nodes of the graph, with the constraint that the total demand of the nodes assigned to each median does not exceed its given capacity. In this paper we propose a cutting plane algorithm, based on Fenchel cuts, which allows us to considerably reduce the integrality gap of hard CPMP instances. The formulation strengthened with Fenchel cuts is solved by a commercial MIP solver. Computational results show that this approach is effective in solving hard instances or considerably reducing their integrality gap.      

一种连续的谱聚类优化模型

聚类与图的划分问题在大数据分析中有着重要的应用.这类问题一般被描述为组合优化问题,因此较难快速求解.本文设计了一种新的连续优化模型,并提出了一种块坐标下降算法,数值实验显示我们的新方法在求解聚类与图的划分问题上很有潜力.我们还更进一步分析了我们的连续优化模型和组合优化模型的关系.     

边界元法中奇异积分计算的极坐标变换法

在边界元法中,奇异积分的处理是一个极为引人注目的问题.本文提出了一种在单元状态作极坐标变换的新的处理方法,它能显式地消除奇异积分的奇异性,使之成为常规积分,因而易于在边界元法中使用高次单元.计算实例表明,本文所提出的方法是有效的、方便的.     

Branch-and-cut for linear programs with overlapping SOS1 constraints

SOS1 constraints require that at most one of a given set of variables is nonzero. In this article, we investigate a branch-and-cut algorithm to solve linear programs with SOS1 constraints. We focus on the case in which the SOS1 constraints overlap. The corresponding conflict graph can algorithmically be exploited, for instance, for improved branching rules, preprocessing, primal heuristics, and cutting planes. In an extensive computational study, we evaluate the components of our implementation on instances for three different applications. We also demonstrate the effectiveness of this approach by comparing it to the solution of a mixed-integer programming formulation, if the variables appearing in SOS1 constraints ar bounded.     

3D Multiphase Piecewise Constant Level Set Method Based on Graph Cut Minimization

Segmentation of three-dimensional (3D) complicated structures is of great importance for many real applications. In this work we combine graph cut minimization method with a variant of the level set idea for 3D segmentation based on the Mumford-Shah model. Compared with the traditional approach for solving the Euler-Lagrange equation we do not need to solve any partial differential equations. Instead, the minimum cut on a special designed graph need to be computed. The method is tested on data with complicated structures. It is rather stable with respect to initial value and the algorithm is nearly parameter free. Experiments show that it can solve large problems much faster than traditional approaches.     

Computational study of large-scale <Emphasis Type="Italic">p</Emphasis>-Median problems

Given a directed graph G(V,A), the p-Median problem consists of determining p nodes (the median nodes) minimizing the total distance from the other nodes of the graph. We present a Branch-and-Cut-and-Price algorithm yielding provably good solutions for instances with |V|≤3795. Key ingredients of the algorithm are a delayed column-and-row generation technique, exploiting the special structure of the formulation, to solve the LP-relaxation, and cutting planes to strengthen the formulation and limit the size of the enumeration tree.     

Elliptic problems on networks with constrictions

We investigate the asymptotic behavior of minimizers to sequences of elliptic variational problems posed on thin three-dimensional domains. These domains arise as thin neighborhoods of artibrary graphs that contain severe constrictions near the graph nodes. We characterize an appropriate limit of minimizers as a function of one variable defined on the graph that necessarily minimizes a one-dimensional variational problem. The most salient feature of these limits of minimizers is the emergence of jump discontinuities across the graph nodes. While the approach can handle quite general elliptic problems, we pay particular attention to an application to generalized Josephson junctions within the Ginzburg-Landau theory of superconductivity. Mathematics Subject Classification (2000) 35Q60, 78M30, 78M35     

An algorithm of global optimization for solving layout problems

The two-dimensional layout problem is known to be NP-complete, and the current research work is basically in the heuristic way. In this paper, we mainly discuss the methods for solving layout problem about the artificial satellite module by virtue of graph theory and group theory. Also, an algorithm of global optimization is presented first time. The method given here can be extended to solve other type of layout problems.     

Optimal competence set expansion using deduction graphs

A competence set is a collection of skills used to solve a problem. Based on deduction graph concepts, this paper proposes a method of finding an optimal process so as to expand a decision maker's competence set to enable him to solve his problem confidently. Using the concept of minimum spanning tree, Yu and Zhang addressed the problem of the optimal expansion of competence sets. In contrast, the method proposed here enjoys the following advantages: it can deal with more general problems involving intermediate skills and compound skills; it can find the optimal solution by utilizing a 0–1 integer program; and it can be directly extended to treat multilevel competence set problems, and thus is more practically useful.This work was supported by the National Science Council, Taipei, Taiwan, Republic of China, Grant No. NSC-81-0301-H-009-501.     

L. V. Kantorovich and Cutting-Packing Problems: New Approaches to Combinatorial Problems of Linear Cutting and Rectangular Packing

In the middle of the XXth century, L. V. Kantorovich and V. A. Zalgaller suggested to solve problems of economical use of material at the cutting stage with the help of linear programming. This led to the continuous relaxation of the problem of rational cutting and, in fact, settled the problem in mass production. The paper briefly describes ways of realization of the method for the one-dimensional cutting. The problem is extended to the integer case, which is typical for any cutting problem. For two-dimensional cutting-packing problems, the block structure technology was developed. This technology reduces to solving a certain special planning problem of one-dimensional cutting that can be solved by linear programming using simple heuristics. We present some computing circuits and results of numerical experiments with wasteless packings. The comparison with other algorithms confirms the efficiency of the block method. Bibliography: 22 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 312, 2004, pp. 239–255.     

Solving the 2-rooted mini-max spanning forest problem by branch-and-bound

The 2-rooted mini-max spanning forest problem is to find a spanning forest with two given root nodes on an undirected graph, such that the maximum tree cost in this forest is minimized. We introduce a branch-and-bound algorithm based on selecting nodes. On test instances up to 50 nodes the algorithm gives much better computational results than a known algorithm that is based on selecting edges. Furthermore the new algorithm can easily solve problem instances up to 80 nodes.     

Cutting planes for semidefinite relaxations based on triangle-free subgraphs

We show how to separate a doubly nonnegative matrix, which is not completely positive and has a triangle-free graph, from the completely positive cone. This method can be used to compute cutting planes for semidefinite relaxations of combinatorial problems. We illustrate our approach by numerical tests on the stable set problem.     

A new fractional finite volume method for solving the fractional diffusion equation

The inherent heterogeneities of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement. One approach to model solute transport through such media is by fractional diffusion equations with a space–time dependent variable coefficient. In this paper, a two-sided space fractional diffusion model with a space–time dependent variable coefficient and a nonlinear source term subject to zero Dirichlet boundary conditions is considered.Some finite volume methods to solve a fractional differential equation with a constant dispersion coefficient have been proposed. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann–Liouville fractional derivatives at control volume faces in terms of function values at the nodes. However, these finite volume methods have not been extended to two-dimensional and three-dimensional problems in a natural manner. In this paper, a new weighted fractional finite volume method with a nonlocal operator (using nodal basis functions) for solving this two-sided space fractional diffusion equation is proposed. Some numerical results for the Crank–Nicholson fractional finite volume method are given to show the stability, consistency and convergence of our computational approach. This novel simulation technique provides excellent tools for practical problems even when a complex transition zone is involved. This technique can be extend to two-dimensional and three-dimensional problems with complex regions.