A new effective branch-and-bound algorithm to the high order MIMO detection problem

This paper develops a branch-and-bound method based on a new convex reformulation to solve the high order MIMO detection problem. First, we transform the original problem into a \(\{-1,1\}\) constrained quadratic programming problem with the smallest size. The size of the reformulated problem is smaller than those problems derived by some traditional transformation methods. Then, we propose a new convex reformulation which gets the maximized average objective value as the lower bound estimator in the branch-and-bound scheme. This estimator balances very well between effectiveness and computational cost. Thus, the branch-and-bound algorithm achieves a high total efficiency. Several simulations are used to compare the performances of our method and other benchmark methods. The results show that this proposed algorithm is very competitive for high accuracy and relatively good efficiency.     

A Combined D.C. Optimization—Ellipsoidal Branch-and-Bound Algorithm for Solving Nonconvex Quadratic Programming Problems

In this paper we propose a new branch-and-bound algorithm by using an ellipsoidal partition for minimizing an indefinite quadratic function over a bounded polyhedral convex set which is not necessarily given explicitly by a system of linear inequalities and/or equalities. It is required that for this set there exists an efficient algorithm to verify whether a point is feasible, and to find a violated constraint if this point is not feasible. The algorithm is based upon the fact that the problem of minimizing an indefinite quadratic form over an ellipsoid can be efficiently solved by some available (polynomial and nonpolynomial time) algorithms. In particular, the d.c. (difference of convex functions) algorithm (DCA) with restarting procedure recently introduced by Pham Dinh Tao and L.T. Hoai An is applied to globally solving this problem. DCA is also used for locally solving the nonconvex quadratic program. It is restarted with current best feasible points in the branch-and-bound scheme, and improved them in its turn. The combined DCA-ellipsoidal branch-and-bound algorithm then enhances the convergence: it reduces considerably the upper bound and thereby a lot of ellipsoids can be eliminated from further consideration. Several numerical experiments are given.     

On global integer extrema of real-valued box-constrained multivariate quadratic functions

Determining global integer extrema of an real-valued box-constrained multivariate quadratic functions is a very difficult task. In this paper, we present an analytic method, which is based on a combinatorial optimization approach in order to calculate global integer extrema of a real-valued box-constrained multivariate quadratic function, whereby this problem will be proven to be as NP-hard via solving it by a Travelling Salesman instance. Instead, we solve it using eigenvalue theory, which allows us to calculate the eigenvalues of an arbitrary symmetric matrix using Newton’s method, which converges quadratically and in addition yields a Jordan normal form with \(1 \times 1\)-blocks, from which a special representation of the multivariate quadratic function based on affine linear functions can be derived. Finally, global integer minimizers can be calculated dynamically and efficiently most often in a small amount of time using the Fourier–Motzkin- and a Branch and Bound like Dijkstra-algorithm. As an application, we consider a box-constrained bivariate and multivariate quadratic function with ten arguments.     

Protein Threading by Linear Programming: Theoretical Analysis and Computational Results

In a previous paper (Xu, Li, Kim, and Xu, Journal of Bioinformatics and Computational Biology, vol. 1, no. 1, pp. 95–117, 2003), we have used an integer programming approach to implement a protein threading program RAPTOR for protein 3D structure prediction, based on the threading model treating pairwise contacts rigorously and allowing variable gaps. We have solved the integer program by the canonical branch-and-bound method. In this paper we present a branch-and-cut method, a careful theoretical analysis of our formulation and why our approach is so effective. The result of cutting plane analysis is that two types of well-known cuts for this problem are already implied in the constraint set, which provides us some intuition that our formulation would be very effective. Experimental results show that for about 99 percent of real threading instances, the linear relaxations of their integer programs solve to integral optimal solutions directly. For the rest one percent of real instances, the integral solutions can be obtained with only several branch nodes. Experimental results also show that no special template or sequence features result in more possibilities of fractional solutions. This indicates that extra effort to seek for cutting planes to strengthen the existing formulation is unnecessary.     

Improving an exact approach for solving separable integer quadratic knapsack problems

We consider the specially structured (pure) integer Quadratic Multi-Knapsack Problem (QMKP) tackled in the paper “Exact solution methods to solve large scale integer quadratic knapsack problems” by D. Quadri, E. Soutif and P. Tolla (2009), recently appeared on this journal, where the problem is solved by transforming it into an equivalent 0–1 linearized Multi-Knapsack Problem (MKP). We show that, by taking advantage of the structure of the transformed (MKP), it is possible to derive an effective variable fixing procedure leading to an improved branch-and-bound approach. This procedure reduces dramatically the resulting linear problem size inducing an impressive improvement in the performances of the related branch and bound approach when compared to the results of the approach proposed by D. Quadri, E. Soutif and P. Tolla.     

SIRALINA: efficient two-steps heuristic for storage optimisation in single period task scheduling

In this paper, we study the general problem of one-dimensional periodic task scheduling under storage requirement, irrespective of machine constraints. We have already presented in (Touati and Eisenbeis, Parallel Process. Lett. 14(2):287–313, 2004) a theoretical framework that allows an optimal optimisation of periodic storage requirement in a cyclic schedule. Since our optimisation problem is NP-hard (Touati, PhD thesis, 2002), solving an exact integer linear programming formulation is too expensive in practice. In this article, we propose an efficient two-steps heuristic using model’s properties that allows fast computation times while providing highly satisfactory results. This method includes the solution of an integer linear program with a totally unimodular constraints matrix in first step, then the solution of a linear assignment problem. Our heuristic is implemented for an industrial compiler for embedded VLIW processors.     

On approximation of the fixed charge transportation problem

In this paper we present a new approximation for computing lower bound for the fixed charge transportation problem (FCTP). The lower bounds thus generated delivered 87% optimal solutions for 56 randomly generated small, up to 6×10 in size, problems in an experimental design. For somewhat larger, 10×10 and 10×15 size problems, the lower bounds delivered an average error of 5%, approximately, using a fraction of CPU times as compared to CPLEX to solve these problems. The proposed lower bound may be used as a superior initial solution with any other existing branch-and-bound method or tabu search heuristic procedure to enhance convergence to the optimal solution for large size problems which cannot be solved by CPLEX due to time constraints.     

An improved MIP heuristic for the intermodal hub location problem

Optimization methods have been commonly developed for the intermodal hub location problem because it has a broad range of practical applications. These methods include exact methods (limited on solving large-size problems) and heuristics (no guarantee on solution quality). In order to avoid their weakness but to leverage their strength, we develop an improved MIP heuristic combining branch-and-bound, Lagrangian relaxation, and linear programming relaxation. In the heuristic, we generate a population of initial feasible solutions using the branch-and-bound and Lagrangian relaxation methods and create a linear-relaxed solution using the linear programming relaxation method. We combine these feasible and linear-relaxed solutions to fix a portion of hub location variables so as to create a number of restricted hub location subproblems. We then combine the branch-and-bound method to solve these restricted subproblems for iteratively improving solution quality. We discuss in detail the application of the method to the intermodal hub location problem. The discussion is followed by extensive statistical analysis and computational tests, where the analysis shows statistical significance of solutions for guiding the heuristic search and comparisons with other methods indicate that the proposed approach is computationally tractable and is able to obtain competitive results.     

基于持续运营机会约束的竞争设施点选址研究——一种有效的实数编码遗传求解算法

竞争设施点选址是空间经济、区域发展、组合优化和系统工程的重要课题之一。本文以市场份额最大化为目标,研究了基于持续运营机会约束的竞争设施点选址问题,并给出了一种有效的实数编码遗传求解算法。在求解模型方面,首先假定运营成本是竞争设施点规模大小的函数,并对设施点持续运营概率进行机会约束,借鉴引力模型建立竞争设施点选址-设计问题的非线性混合整数规划模型。其次,考虑到选址变量和规模变量的数值类型,以及编码变换问题,设计了一种实数编码遗传求解算法。通过数值实验表明,对不同规模问题的实际计算结果,该算法可以在较短时间内获得最优解,可行解和精确解之间误差小于0.5%,相关比较分析也讨论了该算法的优越性和实用性,为竞争设施点选址问题的研究提供了不同的视角和实用求解算法。     

A Lagrangian relaxation approach for the multiple sequence alignment problem

We present a branch-and-bound (bb) algorithm for the multiple sequence alignment problem (MSA), one of the most important problems in computational biology. The upper bound at each bb node is based on a Lagrangian relaxation of an integer linear programming formulation for MSA. Dualizing certain inequalities, the Lagrangian subproblem becomes a pairwise alignment problem, which can be solved efficiently by a dynamic programming approach. Due to a reformulation w.r.t. additionally introduced variables prior to relaxation we improve the convergence rate dramatically while at the same time being able to solve the Lagrangian problem efficiently. Our experiments show that our implementation, although preliminary, outperforms all exact algorithms for the multiple sequence alignment problem. Furthermore, the quality of the alignments is among the best computed so far.     

Efficient Benders decomposition for distance-based critical node detection problem

This paper addresses the critical node detection problem which seeks a subset of nodes for removal in order to maximize the disconnectivity of the residual graph with respect to a specific distance-based measure, namely the Wiener index. Such a measure is defined based on the all-pair shortest path distances in the residual graph so that the longer the total length of shortest paths, the greater the value of the disconnectivity measure. In the literature, a mixed integer linear programming model and an exact iterative-based method have been presented for this problem; however, both approaches become very time-consuming on graphs having large diameter and non-unit edge lengths. To overcome this shortcoming, in this paper, we present a new formulation for the problem and solve it by Benders decomposition algorithm. We improve the performance of Benders algorithm by several techniques (including analytical calculation of dual variables, generation of good-quality initial optimality cuts, considering master's optimality cuts as lazy constraints, etc.) to reduce the total running time. The extensive computational experiments on instances, taken from the literature or generated randomly, confirm the effectiveness of the new approaches.     

Constructive procedures to solve 2-dimensional bin packing problems with irregular pieces and guillotine cuts

This paper presents an approach for solving a new real problem in cutting and packing. At its core is an innovative mixed integer programme model that places irregular pieces and defines guillotine cuts. The two-dimensional irregular shape bin packing problem with guillotine constraints arises in the glass cutting industry, for example, the cutting of glass for conservatories. Almost all cutting and packing problems that include guillotine cuts deal with rectangles only, where all cuts are orthogonal to the edges of the stock sheet and a maximum of two angles of rotation are permitted. The literature tackling packing problems with irregular shapes largely focuses on strip packing i.e. minimizing the length of a single fixed width stock sheet, and does not consider guillotine cuts. Hence, this problem combines the challenges of tackling the complexity of packing irregular pieces with free rotation, guaranteeing guillotine cuts that are not always orthogonal to the edges of the stock sheet, and allocating pieces to bins. To our knowledge only one other recent paper tackles this problem. We present a hybrid algorithm that is a constructive heuristic that determines the relative position of pieces in the bin and guillotine constraints via a mixed integer programme model. We investigate two approaches for allocating guillotine cuts at the same time as determining the placement of the piece, and a two phase approach that delays the allocation of cuts to provide flexibility in space usage. Finally we describe an improvement procedure that is applied to each bin before it is closed. This approach improves on the results of the only other publication on this problem, and gives competitive results for the classic rectangle bin packing problem with guillotine constraints.     

On the <Emphasis Type="Italic">m</Emphasis>-clique free interval subgraphs polytope: polyhedral analysis and applications

In this paper we study the m-clique free interval subgraphs. We investigate the facial structure of the polytope defined as the convex hull of the incidence vectors associated with these subgraphs. We also present some facet-defining inequalities to strengthen the associated linear relaxation. As an application, the generalized open-shop problem with disjunctive constraints (GOSDC) is considered. Indeed, by a projection on a set of variables, the m-clique free interval subgraphs represent the solution of an integer linear program solving the GOSDC presented in this paper. Moreover, we propose exact and heuristic separation algorithms, which are exploited into a Branch-and-cut algorithm for solving the GOSDC. Finally, we present and discuss some computational results.     

中小呼叫中心月度排班优化模型与算法

国内中小呼叫中心制定坐席人员月度排班表时,通常考虑劳动法规合同约束和体现企业自身用工管理诉求。构建坐席人员月度排班优化问题的二次整数规划模型。鉴于问题模型难解性,依据调研企业需求和模型逻辑结构分析,把问题分解成三个子问题。通过构建整数规划模型和提出启发式算法来求出子问题解,从而生成排班问题优化解。问题实例计算表明,模型算法能够有效控制人力成本和兼顾员工同班次管理目标。与周排班方法比较,该方法能够充分体现月度排班人力灵活性来实现人力优化配置。     

A genetic algorithm-based decomposition approach to solve an integrated equipment-workforce-service planning problem

We develop a new genetic algorithm to solve an integrated Equipment-Workforce-Service Planning problem, which features extremely large scales and complex constraints. Compared with the canonical genetic algorithm, the new algorithm is innovative in four respects: (1) The new algorithm addresses epistasis of genes by decomposing the problem variables into evolutionary variables, which evolve with the genetic operators, and the optimization variables, which are derived by solving corresponding optimization problems. (2) The new algorithm introduces the concept of Capacity Threshold and calculates the Set of Efficient and Valid Equipment Assignments to preclude unpromising solution spaces, which allows the algorithm to search much narrowed but promising solution spaces in a more efficient way. (3) The new algorithm modifies the traditional genetic crossover and mutation operators to incorporate the gene dependency in the evolutionary procedure. (4) The new algorithm proposes a new genetic operator, self-evolution, to simulate the growth procedure of an individual in nature and use it for guided improvements of individuals. The new genetic algorithm design is proven very effective and robust in various numerical tests, compared to the integer programming algorithm and the canonical genetic algorithm. When the integer programming algorithm is unable to solve the large-scale problem instances or cannot provide good solutions in acceptable times, and the canonical genetic algorithm is incapable of handling the complex constraints of these instances, the new genetic algorithm obtains the optimal or close-to-optimal solutions within seconds for instances as large as 84 million integer variables and 82 thousand constraints.     

Exact methods in optimum disassembly sequence search for problems subject to sequence dependent costs

Disassembling complex products is formally approached via network representation and subsequent mathematical modeling, aimed at selecting a good or optimum sequence of disassembly operations. This is done via heuristics, metaheuristics or mathematical programming. In contrast with heuristics and metaheuristics, which select a near-optimum solution, mathematical programming guarantees the selection of the global optimum. This problem is relatively simple if the disassembly costs can be assumed sequence independent. In practice, however, sequence dependent disassembly costs are frequently encountered, which causes NP-completeness of the problem. Although methods, e.g., based on the two-commodity network flow approach, are available to solve this constrained asymmetric Traveling Salesperson problem rigorously, this requires the introduction of integer variables. In this paper, a modification of the two-commodity network flow approach is proposed, which reduces the number of integer variables. This is applied to product structures that can be represented by a disassembly precedence graph. It is demonstrated that use of integer variables is completely avoided by iteratively solving a binary integer linear programming problem. This appears to be more efficient than solving the corresponding integer linear programming problem. It is demonstrated, on the basis of some cases, that this method might provide the exact solution of problems with increased complexity compared to those discussed so far in the literature. This appears particularly useful for evaluating heuristic and metaheuristic approaches.     

Using branch-and-price approach to solve the directed network design problem with relays

We present node-arc and arc-path formulations, and develop a branch-and-price approach for the directed network design problem with relays (DNDR). The DNDR problem can be used to model many network design problems in transportation, service, and telecommunication system, where relay points are necessary. The DNDR problem consists of introducing a subset of arcs and locating relays on a subset of nodes such that in the resulting network, the total cost (arc cost plus relay cost) is minimized, and there exists a directed path linking the origin and destination of each commodity, in which the distances between the origin and the first relay, any two consecutive relays, and the last relay and the destination do not exceed a predefined distance limit. With the node-arc formulation, we can directly solve small DNDR instances using mixed integer programming solver. With the arc-path formulation, we design a branch-and-price approach, which is a variant of branch-and-bound with bounds provided by solving linear programs using column generation at each node of the branch-and-bound tree. We design two methods to efficiently price out columns and present computational results on a set of 290 generated instances. Results demonstrate that our proposed branch-and-price approach is a computationally efficient procedure for solving the DNDR problem.     

考虑定价和需求关系的供应链网络优化研究

本文通过引入价格-需求函数描述产品价格变化对消费者需求的影响，构建了一个由制造商、仓库和消费者组成的三级供应链网络。在此基础上，以最大化供应链总利润为目标、以定价和需求为决策变量，建立了混合整数非线性规划（MINLP）模型。基于该模型目标函数非线性的复杂性，采用外部近似法将目标函数近似线性化，即切割成有限条切线，使其可以求解。最后通过不同规模的算例分析验证了模型和算法的可行性与有效性，从而指导供应链上的企业权衡其成本和收益，提高顾客的满意度。     

具有最小交易量限制的多阶段均值-半方差投资组合优化

考虑交易成本,借款约束和阈值约束,文章提出了具有最小交易量限制的多阶段均值-半方差投资组合模型。该模型是具有路径依赖性的混合整数动态优化问题,还是NP完全问题。文章提出了前向动态规划方法求解。最后,通过一个算例比较不同风险约束下的最优投资策略,从而验证模型和算法的有效性。     

An access network design problem with end-to-end QoS constraints

In this paper, we present an access network design problem with end-to-end quality of service (QoS) requirement. The problem can be conceptualized as a two-level hierarchical location-allocation problem on the tree topology with nonlinear side constraints. The objective function of the nonlinear mixed integer programming model minimizes the total cost of switch and fiber cable, while satisfying demand within the prescribed level of QoS. By exploiting the inherent structure of the nonlinear QoS constraints, we develop linearization techniques for finding an optimal solution. Also, we devise an effective exact optimal algorithm within the context of disjunctive constraint generation. We present promising computational results that demonstrate the effectiveness of the proposed solution procedure.