Greedy primal-dual algorithm for dynamic resource allocation in complex networks

In Stolyar (Queueing Systems 50 (2005) 401–457) a dynamic control strategy, called greedy primal-dual (GPD) algorithm, was introduced for the problem of maximizing queueing network utility subject to stability of the queues, and was proved to be (asymptotically) optimal. (The network utility is a concave function of the average rates at which the network generates several “commodities.”) Underlying the control problem of Stolyar (Queueing Systems 50 (2005) 401–457) is a convex optimization problem subject to a set of linear constraints. In this paper we introduce a generalized GPD algorithm, which applies to the network control problem with additional convex (possibly non-linear) constraints on the average commodity rates. The underlying optimization problem in this case is a convex problem subject to convex constraints. We prove asymptotic optimality of the generalized GPD algorithm. We illustrate key features and applications of the algorithm on simple examples. AMS Subject Classifications: 90B15 · 90C25 · 60K25 · 68M12     

A lexicographic minimax algorithm for multiperiod resource allocation

Resource allocation problems are typically formulated as mathematical programs with some special structure that facilitates the development of efficient algorithms. We consider a multiperiod problem in which excess resources in one period can be used in subsequent periods. The objective minimizes lexicographically the nonincreasingly sorted vector of weighted deviations of cumulative activity levels from cumulative demands. To this end, we first develop a new minimax algorithm that minimizes the largest weighted deviation among all cumulative activity levels. The minimax algorithm handles resource constraints, ordering constraints, and lower and upper bounds. At each iteration, it fixes certain variables at their lower bounds, and sets groups of other variables equal to each other as long as no lower bounds are violated. The algorithm takes advantage of the problem's special structure; e.g., each term in the objective is a linear decreasing function of only one variable. This algorithm solves large problems very fast, orders of magnitude faster than well known linear programming packages. (The latter are, of course, not designed to solve such minimax problems efficiently.) The lexicographic procedure repeatedly employs the minimax algorithm described above to solve problems, each of the same format but with smaller dimension.     

An improved branch and bound algorithm for single machine scheduling with deadlines to minimize total weighted completion time

We consider a scheduling problem in which n jobs with distinct deadlines are to be scheduled on a single machine. The objective is to find a feasible job sequence that minimizes the total weighted completion time. We present an efficient branch-and-bound algorithm that fully exploits the principle of optimality. Favorable numerical results are also reported on an extensive set of problem instances of 20-120 jobs.     

A branch and bound algorithm for globally solving a class of nonconvex programming problems

A branch and bound algorithm is proposed for globally solving a class of nonconvex programming problems (NP). For minimizing the problem, linear lower bounding functions (LLBFs) of objective function and constraint functions are constructed, then a relaxation linear programming is obtained which is solved by the simplex method and which provides the lower bound of the optimal value. The proposed algorithm is convergent to the global minimum through the successive refinement of linear relaxation of the feasible region and the solutions of a series of linear programming problems. And finally the numerical experiment is reported to show the feasibility and effectiveness of the proposed algorithm.     

A lower bound on the size of an absorbing set in an arc-coloured tournament

Bousquet, Lochet and Thomassé recently gave an elegant proof that for any integer $n$, there is a least integer $f\left(n\right)$ such that any tournament whose arcs are coloured with $n$ colours contains a subset of vertices $S$ of size $f\left(n\right)$ with the property that any vertex not in $S$ admits a monochromatic path to some vertex of $S$. In this note we provide a lower bound on the value $f\left(n\right)$.     

A tight bound for the boolean quadratic optimization problem and its use in a branch and bound algorithm 1

Known duality statements are used to find tight bounds for the branch and bound process in solving Boolean quadratic optimization problems. To solve the corresponding continuous partial problem, a NEWTON-like procedure is indicated. Superlinear convergence, however, is only obtained in partial cases.     

A simplicial branch and duality bound algorithm for the sum of convex–convex ratios problem

This article presents a simplicial branch and duality bound algorithm for globally solving the sum of convex–convex ratios problem with nonconvex feasible region. To our knowledge, little progress has been made for globally solving this problem so far. The algorithm uses a branch and bound scheme where the Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems during the algorithm search are all ordinary linear programs that can be solved very efficiently. It has been proved that the algorithm possesses global convergence. Finally, the numerical experiments are given to show the feasibility of the proposed algorithm.     

A branch and bound algorithm for bound constrained optimization problems without derivatives

In this paper, we give a new branch and bound algorithm for the global optimization problem with bound constraints. The algorithm is based on the use of inclusion functions. The bounds calculated for the global minimum value are proved to be correct, all rounding errors are rigorously estimated. Our scheme attempts to exclude most uninteresting parts of the search domain and concentrates on its promising subsets. This is done as fast as possible (by involving local descent methods), and uses little information as possible (no derivatives are required). Numerical results for many well-known problems as well as some comparisons with other methods are given.     

A new branching rule for the branch and bound algorithm for solving nonlinear integer programming problems

We examine a branch and bound algorithm for solving nonlinear (convex) integer programming problems. In this note we generalize previous results for the quadratic case. The variables are branched in such a way that the number of branch and bound nodes checked in the process is small. Numerical results confirm the efficiency.     

A new branch and bound algorithm for cell formation problem

Cell formation (CF) is the first and the most important problem in designing cellular manufacturing systems. Due to its non-polynomial nature, various heuristic and metaheuristic algorithms have been proposed to solve CF problem. Despite the popularity of heuristic algorithms, few studies have attempted to develop exact algorithms, such as branch and bound (B&B) algorithms, for this problem. We develop three types of branch and bound algorithms to deal with the cell formation problem. The first algorithm uses a binary branching scheme based on the definitions provided for the decision variables. Unlike the first algorithm, which relies on the mathematical model, the second one is designed based on the structure of the cell formation problem. The last algorithm has a similar structure to the second one, except that it has the ability to eliminate duplicated nodes in branching trees. The proposed branch and bound algorithms and a hybrid genetic algorithm are compared through some numerical examples. The results demonstrate the effectiveness of the modified problem-oriented branch and bound algorithm in solving relatively large size cell formation problems.     

A branch and bound algorithm for the maximum clique problem

We present a branch and bound algorithm for the maximum clique problem in arbitrary graphs. The main part of the algorithm consists in the determination of upper bounds by graph colorings. Using a modification of a known graph coloring method called DSATUR we simultaneously derive lower and upper bounds for the clique number.

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Zusammenfassung Wir stellen einen Branch and Bound Algorithmus für das Maximum Clique Problem in einem beliebigen Graphen vor. Das Hauptaugenmerk richtet sich dabei auf die Bestimmung oberer Schranken mit Hilfe von Färbungen von Graphen. Es wird eine Modifikation einer bekannten Färbungsmethode, genannt DSATUR, verwendet, mit der sich gleichzeitig obere und untere Schranken für die Cliquezahl erstellen lassen.

     

A branch and bound algorithm for the matrix bandwidth minimization

In this article, we first review previous exact approaches as well as theoretical contributions for the problem of reducing the bandwidth of a matrix. This problem consists of finding a permutation of the rows and columns of a given matrix which keeps the non-zero elements in a band that is as close as possible to the main diagonal. This NP-complete problem can also be formulated as a labeling of vertices on a graph, where edges are the non-zero elements of the corresponding symmetrical matrix. We propose a new branch and bound algorithm and new expressions for known lower bounds for this problem. Empirical results with a collection of previously reported instances indicate that the proposed algorithm compares favourably to previous methods.     

A branch and bound algorithm for the maximum diversity problem

This article begins with a review of previously proposed integer formulations for the maximum diversity problem (MDP). This problem consists of selecting a subset of elements from a larger set in such a way that the sum of the distances between the chosen elements is maximized. We propose a branch and bound algorithm and develop several upper bounds on the objective function values of partial solutions to the MDP. Empirical results with a collection of previously reported instances indicate that the proposed algorithm is able to solve all the medium-sized instances (with 50 elements) as well as some large-sized instances (with 100 elements). We compare our method with the best previous linear integer formulation solved with the well-known software Cplex. The comparison favors the proposed procedure.     

A discrete time optimal control model with uncertainty for dynamic machine allocation problem and its application to manufacturing and construction industries

This paper proposed a discrete time optimal control model in which machine failure time is modeled assuming a Weibull distribution and machine productivity is regarded as a fuzzy variable for dealing with a dynamic machine allocation problem (DMAP) in manufacturing and construction industries. The aim is to maximize total production or construction throughput when uncertainties such as machine breakdowns are taken into account. A failure probability-work time equation is presented to describe the relationship between machine failure probability and mean time to work. To transform the uncertain optimal control model into a deterministic one, the expected value model (EVM) was introduced for forming an equivalent crisp model. The fuzzy variables in the model are also defuzzified by using an expected value operator with an optimistic–pessimistic index. Then a number of lemmas and theorems are presented and proved to formulate the theoretical algorithm so that the crisp model of the DMAP can be solved. Three actual construction and production projects are used as practical application examples. The theoretical algorithm results for the three project examples are compared with a particle swarm optimization approach and a genetic algorithm method, which demonstrates the practicality and efficiency of our optimization method.     

A branch and bound algorithm for scheduling trains in a railway network

The paper studies a train scheduling problem faced by railway infrastructure managers during real-time traffic control. When train operations are perturbed, a new conflict-free timetable of feasible arrival and departure times needs to be re-computed, such that the deviation from the original one is minimized. The problem can be viewed as a huge job shop scheduling problem with no-store constraints. We make use of a careful estimation of time separation among trains, and model the scheduling problem with an alternative graph formulation. We develop a branch and bound algorithm which includes implication rules enabling to speed up the computation. An experimental study, based on a bottleneck area of the Dutch rail network, shows that a truncated version of the algorithm provides proven optimal or near optimal solutions within short time limits.     

A branch and bound algorithm for solving the multiple-choice knapsack problem

The multiple-choice knapsack problem is a binary knapsack problem with the addition of disjoint multiple-choice constraints. We describe a branch and bound algorithm based on embedding Glover and Klingman's method for the associated linear program within a depth-first search procedure. A heuristic is used to find a starting dual feasible solution to the associated linear program and a ‘pegging’ test is employed to reduce the size of the problem for the enumeration phase. Computational experience and comparisons with the code of Nauss and an algorithm of Armstrong et al. for the same problem are reported.     

A best-first branch and bound algorithm for unconstrained two-dimensional cutting problems

This paper is concerned with the problem of unconstrained two-dimensional cutting of small rectangular pieces, each of which has its own profit and size, from a large rectangular plate so as to maximize the profit-sum of the pieces produced. Hifi and Zissimopoulos's recursive algorithm using G and Kang's upper bound is presently the most efficient exact algorithm for the problem. We propose a best-first branch and bound algorithm based upon the bottom-up approach that is more efficient than their recursive algorithm. The proposed algorithm uses efficient upper bound and branching strategies that can reduce the number of nodes that must be searched significantly. We demonstrate the efficiency of the proposed algorithm through computational experiments.     

A two-stage parallel branch and bound algorithm for mixed integer programs

Mixed integer programming (MIP) models are extensively usedto aid strategic and tactical decision making in many businesssectors. Solving MIP models is a computationally intensive processand there is a need to develop solution approaches that enablelarger models to be solved within acceptable timeframes. Inthis paper, we describe the implementation of a two-stage parallelbranch and bound (PB & B) algorithm for MIP. In stage 1of the algorithm, a multiple heuristic search is implementedin which a number of alternative search trees are investigatedusing a forest search in the hope of finding a good solutionquickly. In stage 2, the search is reorganized so that the branchesof a chosen tree are investigated in parallel. A new heuristicis introduced, based on a best projection criterion, which evaluatesalternative B & B trees in order to choose one for investigationin stage 2 of the algorithm. The heuristic also serves as away of implementing a quality load balancing scheme for stage2 of the algorithm. The results of experimental investigationsare reported for a range of models taken from the MIPLIB libraryof benchmark problems.     

A branch and bound algorithm for class based storage location assignment

Class-based storage implementation decisions have significant impact on the required storage space and the material handling cost in a warehouse. In this paper, a nonlinear integer programming model is proposed to capture the above. Effects of storage area reduction on order picking and storage space cost are incorporated. A branch and bound algorithm is developed to solve the model. Computational experience with randomly generated data sets and an industrial case shows that branch and bound algorithm is computationally more efficient than a baseline dynamic programming algorithm. It is further observed that the class based policy results in lower total cost of order picking and storage space than the dedicated policy.     

A simplicial branch and bound duality-bounds algorithm for the linear sum-of-ratios problem

In this article, we present and validate a simplicial branch and bound duality-bounds algorithm for globally solving the linear sum-of-ratios fractional program. The algorithm computes the lower bounds called for during the branch and bound search by solving ordinary linear programming problems. These problems are derived by using Lagrangian duality theory. The algorithm applies to a wide class of linear sum-of-ratios fractional programs. Two sample problems are solved, and the potential practical and computational advantages of the algorithm are indicated.