A primal–dual algorithm for the economic lot-sizing problem with multi-mode replenishment

The classical economic lot-sizing problem assumes that a single supplier and a single transportation mode are used to replenish the inventory. This paper studies an extension of this problem where several suppliers and transportation modes are available. The decision-making process in this case involves identifying (i) the timing for an order; (ii) the choice of shipment modes; and (iii) the order size for each mode. The problem is defined as a network flow problem with multiple setups cost function and additional side constraints. This study provides an MIP formulation for the problem. We also provide an additional formulation of the problem by redefining its decision variables and show that the dual of the corresponding LP-relaxation has a special structure. We take advantage of the structure of the dual problem to develop a primal–dual algorithm that generates tight lower and upper bounds. Computational results demonstrate the effectiveness of the algorithm.     

Fast algorithm for solving the Hankel/Toeplitz Structured Total Least Squares problem

The Structured Total Least Squares (STLS) problem is a natural extension of the Total Least Squares (TLS) problem when constraints on the matrix structure need to be imposed. Similar to the ordinary TLS approach, the STLS approach can be used to determine the parameter vector of a linear model, given some noisy measurements. In many signal processing applications, the imposition of this matrix structure constraint is necessary for obtaining Maximum Likelihood (ML) estimates of the parameter vector. In this paper we consider the Toeplitz (Hankel) STLS problem (i.e., an STLS problem in which the Toeplitz (Hankel) structure needs to be preserved). A fast implementation of an algorithm for solving this frequently occurring STLS problem is proposed. The increased efficiency is obtained by exploiting the low displacement rank of the involved matrices and the sparsity of the associated generators. The fast implementation is compared to two other implementations of algorithms for solving the Toeplitz (Hankel) STLS problem. The comparison is carried out on a recently proposed speech compression scheme. The numerical results confirm the high efficiency of the newly proposed fast implementation: the straightforward implementations have a complexity of O((m+n)³) and O(m³) whereas the proposed implementation has a complexity of O(mn+n²). This revised version was published online in June 2006 with corrections to the Cover Date.     

Modelling,simulation and optimization of an elastic structure under moving loads

We consider an elastic structure that is subject to moving loads representing e.g. heavy trucks on a bridge or a trolley on a crane beam. A model for the quasi-static mechanical behaviour of the structure is derived, yielding a coupled problem involving partial differential equations (PDE) and ordinary differential equations (ODE). The problem is simulated numerically and validated by comparison with a standard formula used in engineering. We derive an optimal policy for passing over potentially fragile bridges. In general, our problem class leads to optimal control problems subject to coupled ODE and PDE. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Affine Gelfand-Dickey brackets and holomorphic vector bundles

We define the (second) Adler-Gelfand-Dickey Poisson structure on differential operators over an elliptic curve and classify symplectic leaves of this structure. This problem leads to the problem of classification of coadjoint orbits for double loop algebras, conjugacy classes in loop groups, and holomorphic vector bundles over the elliptic curve. We show that symplectic leaves have a finite but (unlike the traditional case of operators on the circle) arbitrarily large codimension, and compute it explicitly.     

The inverse solution of the atomic mixing equations by an operator-splitting method

The quantification problem of recovering the original material distribution from secondary ion mass spectrometry (SIMS) data is considered in this paper. It is an inverse problem, is ill-posed and hence it requires a special technique for its solution. The quantification problem is essentially an inverse diffusion or (classically) a backward heat conduction problem. In this paper an operator-splitting method (that is proposed in a previous paper by the first author for the solution of inverse diffusion problems) is developed for the solution of the problem of recovering the original structure from the SIMS data. A detailed development of the quantification method is given and it is applied to typical data to demonstrate its effectiveness.     

On the Computational Complexity of a Rigidity Problem

Consider a structure of flexible joints connected by rigid bars.These bars will constrain the possible motions of the jointsof this structure. By "pinning down" some of the joints so thatthey cannot move further constraints will be added. In thisway the entire structure can be made rigid. A problem consideredby Bolker & Crapo (1977) and others, is that of findingthe minimum number of joints that must be pinned in order tomake a given two- or three-dimensional structure rigid. We considerthe computational complexity of this problem. Lovasz (1980)gives a somewhat complicated but polynomial time procedure forthis problem in the two-dimensional case. In this paper we showthat in three or more dimensions the problem is NP-complete,and so is unlikely to have a polynomial time algorithm.     

A composite very large-scale neighborhood structure for the capacitated minimum spanning tree problem

The capacitated minimum spanning tree (CMST) problem is to find a minimum cost spanning tree in a network where nodes have specified demands, with an additional capacity constraints on the subtrees incident to a given source node s. The capacitated minimum spanning tree problem arises as an important subproblem in many telecommunication network design problems. In a recent paper, Ahuja et al. (Math. Program. 91 (2001) 71) proposed two very large-scale neighborhood search algorithms for the capacitated minimum spanning tree problem. Their first node-based neighborhood structure is obtained by performing multi-exchanges involving several trees where each tree contributes a single node. Their second tree-based neighborhood structure is obtained by performing multi-exchanges where each tree contributes a subtree. The computational investigations found that node-based multi-exchange neighborhood gives the best performance for the homogenous demand case (when all nodes have the same demand), and the tree-based multi-exchange neighborhood gives the best performance for the heterogeneous demand case (when nodes may have different demands). In this paper, we propose a composite neighborhood structure that subsumes both the node-based and tree-based neighborhoods, and outperforms both the previous neighborhood search algorithms for solving the capacitated minimum spanning tree problem on standard benchmark instances. We also develop improved dynamic programming based exact algorithms for searching the composite neighborhood.     

Protein structure optimization by side-chain positioning via beta-complex

A molecular structure determines a molecular function(s) and a correct understanding of molecular structure is important for biotechnology. The computational prediction of molecular structure is a frequent requirement for important biomolecular applications such as a homology modeling, a docking simulation, a protein design, etc. where the optimization of molecular structure is fundamental. One of the core problems in the optimization of protein structure is the optimization of side-chains called the side-chain positioning problem. The side-chain positioning problem, assuming the rigidity of backbone and a rotamer library, attempts to optimally assign a rotamer to each residue so that the potential energy of protein is minimized in its entirety. The optimal solution approach using (mixed) integer linear programming, with the dead-end elimination technique, suffers even for moderate-sized proteins because the side-chain positioning problem is NP-hard. On the other hand, popular heuristic approaches focusing on speed produce solutions of low quality. This paper presents an efficient algorithm, called the BetaSCP, for the side-chain positioning problem based on the beta-complex which is a derivative geometric construct of the Voronoi diagram. Placing a higher priority on solution quality, the BetaSCP algorithm produces a solution very close to the optima within a reasonable computation time. The effectiveness and efficiency of the BetaSCP are experimentally shown via a benchmark test against well-known algorithms using twenty test models selected from Protein Data Bank.     

The normal sub-Riemannian geodesic flow on E(2) generated by a left-invariant metric and a right-invariant distribution

We consider a sub-Riemannian problem on the three-dimensional solvable Lie group E(2) endowed with a left-invariant metric and a right-invariant distribution. The problem is based on construction of a Hamilton structure for the given metric by the Pontryagin maximum principle.     

Project Scheduling under Time Dependent Costs – A Branch and Bound Algorithm

In a given project network, execution of each activity in normal duration requires utilization of certain resources. If faster execution of an activity is desired then additional resources at extra cost would be required. Given a project network, the cost structure for each activity and a planning horizon, the project compression problem is concerned with the determination of optimal schedule (duration) of performing each activity while satisfying given restrictions and minimizing the total cost of project execution. This paper considers the project compression problem with time dependent cost structure for each activity. The planning horizon is divided into several regular time intervals over which the cost structure of an activity may vary. But the cost structure of the activities remains the same (constant) within a time interval. Key events of the project attract penalty for finishing earlier or later than the corresponding target times. The objective is to find an optimal project schedule minimizing the total project cost. We present a mathematical model for this problem, develop some heuristics and an exact branch and bound algorithm. Using simulated problems we provide an insight into the computational performances of heuristics and the branch and bound algorithm.     

On robust maximum flow with polyhedral uncertainty sets

We address the problem of determining a robust maximum flow value in a network with uncertain link capacities taken in a polyhedral uncertainty set. Besides a few polynomial cases, we focus on the case where the uncertainty set is taken to be the solution set of an associated (continuous) knapsack problem. This class of problems is shown to be polynomially solvable for planar graphs, but NP-hard for graphs without special structure. The latter result provides evidence of the fact that the problem investigated here has a structure fundamentally different from the robust network flow models proposed in various other published works.     

决策者具有(严格)凸性偏好结构下的一类交互式多目标决策方法

为了更好地解决决策者具有(严格)凸性偏好结构下的多目标决策问题,一般目标空间为有界凸域的情形常常可以转化为目标空间为有界闭凸区域的情形,首先分析了切割平面及该平面上偏好最优点与被切割平面分割成的为有界闭凸区域的目标空间或目标空间的子集的两个部分之间的关系;然后分析并指出了对于包含全局偏好最优目标方案点的为有界闭凸域的目标空间及其子集(准最优目标集),在确定了切割平面上的偏好最优点后,通过适当地选取供决策者与切割平面的偏好最优点进行比较判断的目标方案点,经过一次比较就可以确定一个新的范围更小的包含全局偏好最优目标方案点的目标空间的有界闭凸子区域(准最优目标集).为获取切割平面上的偏好最优点,提出了改进的坐标轮换法.在这些结论和方法的基础上,提出了决策者具有(严格)凸性偏好结构下的一类交互式多目标决策方法,要求决策者提供较易的偏好性息,决策效能较好.     

Incremental accelerated gradient methods for SVM classification: study of the constrained approach

We investigate constrained first order techniques for training support vector machines (SVM) for online classification tasks. The methods exploit the structure of the SVM training problem and combine ideas of incremental gradient technique, gradient acceleration and successive simple calculations of Lagrange multipliers. Both primal and dual formulations are studied and compared. Experiments show that the constrained incremental algorithms working in the dual space achieve the best trade-off between prediction accuracy and training time. We perform comparisons with an unconstrained large scale learning algorithm (Pegasos stochastic gradient) to emphasize that our choice can remain competitive for large scale learning due to the very special structure of the training problem.     

Mathematical Modeling of Near-Field Optics

Consider a time-harmonic electromagnetic plane wave incident on a periodic (grating) structure. An inhomogeneous (subwavelength) object is placed inside the periodic structure. The scattering problem is to study the electromagnetic field distributions. The problem arises in the study of near-field optics and has many physical and biological applications. This work is devoted to modeling and analysis of a near-field optics problem. An integral representation approach is presented to solve the problem. It is shown particularly that the perturbation due to the object decays exponentially along the periodic direction of the structure, provided that no surface waves occur. Based on the approach, a general solution method is discussed and the well-posedness of the model problems are established.     

Optimal three-level presort loading of commercial bulk mail in the postal service industry

This study focuses on the presort loading of commercial bulk mail. Here, presort is the process by which a mailer prepares mail such that it is sorted to at least the finest extent required by the postal service provider for a claimed (discounted) price. We formulated this presort loading problem (PLP) as a special case of transportation problem with minimum quantity commitment (MQC) constraints. In addition, we developed a polynomial time optimal solution algorithm for the PLP and performed computational experiments on randomly generated problem instances under various discount structures. Results of the computational experiments show that mailers can potentially reduce their costs by sending mail less frequently, using small-sized mail trays; however, the discount structure does not affect the main results. There is some evidence that smaller cost reductions on mailing fees occur as the variation in the discount rate increases; however, the effects of discount structure are nominal compared with the gains from changing the mailing frequency. Mailing fee savings are more heavily influenced by the discount structure when MQC constraints become tighter.     

无粘、可压、绝热流体的Euler方程初值问题的适定性

根据分层理论提供的基本方法,讨论Euler方程的初值问题的适定性,给出了方程的典型初边值问题适定性的判别条件,确定了Euler方程的局部(准确)解的解空间构造,对适定问题给出了解析解的计算公式．     

Mixed integer formulations for the probabilistic minimum energy broadcast problem in wireless networks

In this paper we study a new variant of the minimum energy broadcast (MEB) problem, namely the probabilistic MEB (PMEB). The objective of the classic MEB problem is to assign transmission powers to the nodes of a wireless network is such a way that the total energy dissipated on the network is minimized, while a connected broadcasting structure is guaranteed by the assigned transmission powers. In the new variant of the problem treated in this paper, node failure is taken into account, aiming at providing solutions with a chosen reliability level for the broadcasting structure. Three mixed integer linear programming formulations for the new problem are presented, together with efficient formulation-dependent methods for their solution. Computational results are proposed and discussed. One method emerges as the most promising one under realistic settings. It is able to handle problems with up to fifty nodes.     

Finding DEA-efficient hyperplanes using MOLP efficient faces

This paper suggests a method for finding efficient hyperplanes with variable returns to scale the technology in data envelopment analysis (DEA) by using the multiple objective linear programming (MOLP) structure. By presenting an MOLP problem for finding the gradient of efficient hyperplanes, We characterize the efficient faces. Thus, without finding the extreme efficient points of the MOLP problem and only by identifying the efficient faces of the MOLP problem, we characterize the efficient hyperplanes which make up the DEA efficient frontier. Finally, we provide an algorithm for finding the efficient supporting hyperplanes and efficient defining hyperplanes, which uses only one linear programming problem.     

Constructing efficient solutions structure of multiobjective linear programming

It is not a difficult task to find a weak Pareto or Pareto solution in a multiobjective linear programming (MOLP) problem. The difficulty lies in finding all these solutions and representing their structure. This paper develops an algorithm for solving this problem. We investigate the solutions and their relationships in the objective space. The algorithm determines finite number of weights, each of which corresponds to a weighted sum problems. By solving these problems, we further obtain all weak Pareto and Pareto solutions of the MOLP and their structure in the constraint space. The algorithm avoids the degeneration problem, which is a major hurdle of previous works, and presents an easy and clear solution structure.     

A hierarchical multi-label classification ant colony algorithm for protein function prediction

This paper proposes a novel ant colony optimisation (ACO) algorithm tailored for the hierarchical multi-label classification problem of protein function prediction. This problem is a very active research field, given the large increase in the number of uncharacterised proteins available for analysis and the importance of determining their functions in order to improve the current biological knowledge. Since it is known that a protein can perform more than one function and many protein functional-definition schemes are organised in a hierarchical structure, the classification problem in this case is an instance of a hierarchical multi-label problem. In this type of problem, each example may belong to multiple class labels and class labels are organised in a hierarchical structure—either a tree or a directed acyclic graph structure. It presents a more complex problem than conventional flat classification, given that the classification algorithm has to take into account hierarchical relationships between class labels and be able to predict multiple class labels for the same example. The proposed ACO algorithm discovers an ordered list of hierarchical multi-label classification rules. It is evaluated on sixteen challenging bioinformatics data sets involving hundreds or thousands of class labels to be predicted and compared against state-of-the-art decision tree induction algorithms for hierarchical multi-label classification.