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 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 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 branch and bound algorithm for the robust shortest path problem with interval data

Many real problems can be modelled as robust shortest path problems on interval digraphs, where intervals represent uncertainty about real costs and a robust path is not too far from the shortest path for each possible configuration of the arc costs.A branch and bound algorithm for this problem is presented.     

An adaptive Chebyshev iterative method\newline for nonsymmetric linear systems based on modified moments

Summary. Large, sparse nonsymmetric systems of linear equations with a matrix whose eigenvalues lie in the right half plane may be solved by an iterative method based on Chebyshev polynomials for an interval in the complex plane. Knowledge of the convex hull of the spectrum of the matrix is required in order to choose parameters upon which the iteration depends. Adaptive Chebyshev algorithms, in which these parameters are determined by using eigenvalue estimates computed by the power method or modifications thereof, have been described by Manteuffel [18]. This paper presents an adaptive Chebyshev iterative method, in which eigenvalue estimates are computed from modified moments determined during the iterations. The computation of eigenvalue estimates from modified moments requires less computer storage than when eigenvalue estimates are computed by a power method and yields faster convergence for many problems. Received May 13, 1992/Revised version received May 13, 1993     

Branch and bound algorithm for multidimensional scaling with city-block metric

A two level global optimization algorithm for multidimensional scaling (MDS) with city-block metric is proposed. The piecewise quadratic structure of the objective function is employed. At the upper level a combinatorial global optimization problem is solved by means of branch and bound method, where an objective function is defined as the minimum of a quadratic programming problem. The later is solved at the lower level by a standard quadratic programming algorithm. The proposed algorithm has been applied for auxiliary and practical problems whose global optimization counterpart was of dimensionality up to 24.     

Transportation problem with nonlinear side constraints a branch and bound approach

In a container terminal management, we are often confronted with the following problem: how to assign a reasonable depositing position for an arriving container, so that the efficiency of searching for and loading of a container later can be increased. In this paper, the problem is modeled as a transportation problem with nonlinear side constraints (TPNSC). The reason of nonlinear side constraints arising is that some kinds of containers cannot be stacked in the same row (the space of storage yard is properly divided into several rows). A branch and bound algorithm is designed to solve this problem. The algorithm is based on the idea of using disjunctive arcs (branches) for resolving conflicts that are created whenever some conflicting kinds of containers are deposited in the same row. During the branch and bound, the candidate problems are transformed into classical transportation problems, so that the efficient transportation algorithm can be applied, at the same time the reoptimization technique is employed during the branch and bound. Further, we design a heuristic to obtain a feasible initial solution for TPNSC in order to prune some candidates as early and/or as much as possible. We report computational results on randomly generated problems.     

A Projection Method for the Integer Quadratic Knapsack Problem

In this paper we present a new branch and bound algorithm for solving a class of integer quadratic knapsack problems. A previously published algorithm solves the continuous variable subproblems in the branch and bound tree by performing a binary search over the breakpoints of a piecewise linear equation resulting from the Kuhn-Tucker conditions. Here, we first present modifications to a projection method for solving the continuous subproblems. Then we implement the modified projection method in a branch and bound framework and report computational results indicating that the new branch and bound algorithm is superior to the earlier method.     

Numerical approximation of young measuresin non-convex variational problems

Summary. In non-convex optimisation problems, in particular in non-convex variational problems, there usually does not exist any classical solution but only generalised solutions which involve Young measures. In this paper, first a suitable relaxation and approximation theory is developed together with optimality conditions, and then an adaptive scheme is proposed for the efficient numerical treatment. The Young measures solving the approximate problems are usually composed only from a few atoms. This is the main argument our effective active-set type algorithm is based on. The support of those atoms is estimated from the Weierstrass maximum principle which involves a Hamiltonian whose good guess is obtained by a multilevel technique. Numerical experiments are performed in a one-dimensional variational problem and support efficiency of the algorithm. Received November 26, 1997 / Published online September 24, 1999     

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 simulation tool for the performance evaluation of parallel branch and bound algorithms

Parallel computation offers a challenging opportunity to speed up the time consuming enumerative procedures that are necessary to solve hard combinatorial problems. Theoretical analysis of such a parallel branch and bound algorithm is very hard and empirical analysis is not straightforward because the performance of a parallel algorithm cannot be evaluated simply by executing the algorithm on a few parallel systems. Among the difficulties encountered are the noise produced by other users on the system, the limited variation in parallelism (the number of processors in the system is strictly bounded) and the waste of resources involved: most of the time, the outcomes of all computations are already known and the only issue of interest is when these outcomes are produced.We will describe a way to simulate the execution of parallel branch and bound algorithms on arbitrary parallel systems in such a way that the memory and cpu requirements are very reasonable. The use of simulation has only minor consequences for the formulation of the algorithm.     

Modified subspace limited memory BFGS algorithm for large-scale bound constrained optimization

In this paper, a subspace limited memory BFGS algorithm for solving large-scale bound constrained optimization problems is developed. It is modifications of the subspace limited memory quasi-Newton method proposed by Ni and Yuan [Q. Ni, Y.X. Yuan, A subspace limited memory quasi-Newton algorithm for large-scale nonlinear bound constrained optimization, Math. Comput. 66 (1997) 1509–1520]. An important property of our proposed method is that more limited memory BFGS update is used. Under appropriate conditions, the global convergence of the method is established. The implementations of the method on CUTE test problems are presented, which indicate the modifications are beneficial to the performance of the algorithm.     

A branch and bound algorithm for the robust spanning tree problem with interval data

The robust spanning tree problem is a variation, motivated by telecommunications applications, of the classic minimum spanning tree problem. In the robust spanning tree problem edge costs lie in an interval instead of having a fixed value.Interval numbers model uncertainty about the exact cost values. A robust spanning tree is a spanning tree whose total cost minimizes the maximum deviation from the optimal spanning tree over all realizations of the edge costs. This robustness concept is formalized in mathematical terms and is used to drive optimization.In this paper a branch and bound algorithm for the robust spanning tree problem is proposed. The method embeds the extension of some results previously presented in the literature and some new elements, such as a new lower bound and some new reduction rules, all based on the exploitation of some peculiarities of the branching strategy adopted.Computational results obtained by the algorithm are presented. The technique we propose is up to 210 faster than methods recently appeared in the literature.     

Project scheduling with resource constraints: A branch and bound approach

This paper describes a branch and bound algorithm for project scheduling with resource constraints. The algorithmis based on the idea of using disjunctive arcs for resolving conflicts that are created whenever sets of activities have to be scheduled whose total resource requirements exceed the resource availabilities in some periods. Four lower bounds are examined. The first is a simple lower bound based on longest path computations. The second and third bounds are derived from a relaxed integer programming formulation of the problem. The second bound is based on the Linear Programming relaxation with the addition of cutting planes, and the third bound is based on a Lagrangean relaxation of the formulation. This last relaxation involves a problem which is a generalization of the longest path computation and for which an efficient, though not polynomial, algorithm is given. The fourth bound is based on the disjunctive arcs used to model the problem as a graph.We report computational results on the performance of each bound on randomly generated problems involving up to 25 activities and 3 resources.     

An extended branch and bound algorithm for linear bilevel programming

For linear bilevel programming, the branch and bound algorithm is the most successful algorithm to deal with the complementary constraints arising from Kuhn–Tucker conditions. However, one principle challenge is that it could not well handle a linear bilevel programming problem when the constraint functions at the upper-level are of arbitrary linear form. This paper proposes an extended branch and bound algorithm to solve this problem. The results have demonstrated that the extended branch and bound algorithm can solve a wider class of linear bilevel problems can than current capabilities permit.     

Algorithms for bound constrained quadratic programming problems

Summary We present an algorithm which combines standard active set strategies with the gradient projection method for the solution of quadratic programming problems subject to bounds. We show, in particular, that if the quadratic is bounded below on the feasible set then termination occurs at a stationary point in a finite number of iterations. Moreover, if all stationary points are nondegenerate, termination occurs at a local minimizer. A numerical comparison of the algorithm based on the gradient projection algorithm with a standard active set strategy shows that on mildly degenerate problems the gradient projection algorithm requires considerable less iterations and time than the active set strategy. On nondegenerate problems the number of iterations typically decreases by at least a factor of 10. For strongly degenerate problems, the performance of the gradient projection algorithm deteriorates, but it still performs better than the active set method.Work supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38     

QMR: a quasi-minimal residual method for non-Hermitian linear systems

Summary The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. In this paper, we present a novel BCG-like approach, the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported.This work was supported in part by DARPA via Cooperative Agreement NCC 2-387 between NASA and the Universities Space Research Association (USRA).     

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 determining locations of long-term care facilities

This paper focuses on the problem of determining locations for long-term care facilities with the objective of balancing the numbers of patients assigned to the facilities. We present a branch and bound algorithm by developing dominance properties, a lower bounding scheme and a heuristic algorithm for obtaining an upper bound for the problem. For evaluation of the suggested branch and bound algorithm, computational experiments are performed on a number of test problems. Results of the experiments show that the suggested algorithm gives optimal solutions of problems of practical sizes in a reasonable amount of computation time.     

Application of a block modified Chebyshev algorithm to the iterative solution of symmetric linear systems with multiple right hand side vectors

Summary. An adaptive Richardson iteration method is described for the solution of large sparse symmetric positive definite linear systems of equations with multiple right-hand side vectors. This scheme ``learns' about the linear system to be solved by computing inner products of residual matrices during the iterations. These inner products are interpreted as block modified moments. A block version of the modified Chebyshev algorithm is presented which yields a block tridiagonal matrix from the block modified moments and the recursion coefficients of the residual polynomials. The eigenvalues of this block tridiagonal matrix define an interval, which determines the choice of relaxation parameters for Richardson iteration. Only minor modifications are necessary in order to obtain a scheme for the solution of symmetric indefinite linear systems with multiple right-hand side vectors. We outline the changes required. Received April 22, 1993