A Branch and Contract Algorithm for Problems with Concave Univariate, Bilinear and Linear Fractional Terms

A new deterministic branch and bound algorithm is presented in this paper for the global optimization of continuous problems that involve concave univariate, bilinear and linear fractional terms. The proposed algorithm, the branch and contract algorithm, relies on the use of a bounds-contraction subproblem that aims at reducing the size of the search region by eliminating portions of the domain in which the objective function takes only values above a known upper bound. The solution of contraction subproblems at selected branch and bound nodes is performed within a finite contraction operation that helps reducing the total number of nodes in the branch and bound solution tree. The use of the proposed algorithm is illustrated with several numerical examples.     

Discounted MEAN bound for the optimal searcher path problem with non-uniform travel times

We consider an extension of the optimal searcher path problem (OSP), where a searcher moving through a discretised environment may now need to spend a non-uniform amount of time travelling from one region to another before being able to search it for the presence of a moving target. In constraining not only where but when the search of each cell can take place, the problem more appropriately models the search of environments which cannot be easily partitioned into equally sized cells. An existing OSP bounding method in literature, the MEAN bound, is generalised to provide bounds for solving the new problem in a branch and bound framework. The main contribution of this paper is an enhancement, discounted MEAN (DMEAN), which greatly tightens the bound for the new and existing problems alike with almost no additional computation. We test the new algorithm against existing OSP bounding methods and show it leads to faster solution times for moving target search 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.     

A branch and bound algorithm with constraint partitioning for integer goal programming problems

Using constraint partitioning and variable elimination, the authors have recently developed an efficient algorithm for solving linear goal programming problems. However, many goal programs require some or all of the decision variables to be integer valued. This paper shows how the new partitioning algorithm can be extended with a modified branch and bound strategy to solve both pure and mixed type integer goal programming problems. A potential problem in combining the partitioning algorithm and the branch and bound search scheme is presented and resolved.     

可靠性网络中费用最小化问题的一种新的分枝定界算法

本文对可靠性网络中串-并系统的费用最小化问题提出一种新的分枝定界算法.我们根据这类网络的特殊结构和性质,建立了新的最优性必要条件,在分枝搜索过程中增加新的剪枝准则,从而加速了算法的收敛速度.有效的数值试验表明,该算法可求解大规模可靠性网络的费用最小化问题.     

Global optimization method for maximizing the sum of difference of convex functions ratios over nonconvex region

A global optimization algorithm is presented for maximizing the sum of difference of convex functions ratios problem over nonconvex feasible region. This algorithm is based on branch and bound framework. To obtain a difference of convex programming, the considered problem is first reformulated by introducing new variables as few as possible. By using subgradient and convex envelope, the fundamental problem of estimating lower bound in the branch and bound algorithm is transformed into a relaxed linear programming problem which can be solved efficiently. Furthermore, the size of the relaxed linear programming problem does not change during the algorithm search. Lastly, the convergence of the algorithm is analyzed and the numerical results are reported.     

Using concave envelopes to globally solve the nonlinear sum of ratios problem

This article presents a branch and bound algorithm for globally solving the nonlinear sum of ratios problem (P). The algorithm works by globally solving a sum of ratios problem that is equivalent to problem (P). In the algorithm, upper bounds are computed by maximizing concave envelopes of a sum of ratios function over intersections of the feasible region of the equivalent problem with rectangular sets. The rectangular sets are systematically subdivided as the branch and bound search proceeds. Two versions of the algorithm, with convergence results, are presented. Computational advantages of these algorithms are indicated, and some computational results are given that were obtained by globally solving some sample problems with one of these algorithms.     

Nonlinear integer programming for optimal allocation in stratified sampling

A stratified random sampling plan is one in which the elements of the population are first divided into nonoverlapping groups, and then a simple random sample is selected from each group. In this paper, we focus on determining the optimal sample size of each group. We show that various versions of this problem can be transformed into a particular nonlinear program with a convex objective function, a single linear constraint, and bounded variables. Two branch and bound algorithms are presented for solving the problem. The first algorithm solves the transformed subproblems in the branch and bound tree using a variable pegging procedure. The second algorithm solves the subproblems by performing a search to identify the optimal Lagrange multiplier of the single constraint. We also present linearization and dynamic programming methods that can be used for solving the stratified sampling problem. Computational testing indicates that the pegging branch and bound algorithm is fastest for some classes of problems, and the linearization method is fastest for other classes of problems.     

Optimal allocation and processing time decisions on non-identical parallel CNC machines: ϵ-constraint approach

When the processing times of jobs are controllable, selected processing times affect both the manufacturing cost and the scheduling performance. A well known example for such a case that this paper specifically deals with is the turning operation on a CNC machine. Manufacturing cost of a turning operation is a nonlinear convex function of its processing time. In this paper, we deal with making optimal machine-job assignments and processing time decisions so as to minimize total manufacturing cost while the makespan being upper bounded by a known value, denoted as ?-constraint approach for a bicriteria problem. We then give optimality properties for the resulting single criterion problem. We provide alternative methods to compute cost lower bounds for partial schedules, which are used in developing an exact (branch and bound) algorithm. For the cases where the exact algorithm is not efficient in terms of computation time, we present a recovering beam search algorithm equipped with an improvement search procedure. In order to find improving search directions, the improvement search algorithm uses the proposed cost bounding properties. Computational results show that our lower bounding methods in branch and bound algorithm achieve a significant reduction in the search tree size that we need to traverse. Also, our recovering beam search and improvement search heuristics achieve solutions within 1% of the optimum on the average while they spent much less computational effort than the exact algorithm.     

Computational experience with a group theoretic integer programming algorithm

This paper gives specific computational details and experience with a group theoretic integer programming algorithm. Included among the subroutines are a matrix reduction scheme for obtaining group representations, network algorithms for solving group optimization problems, and a branch and bound search for finding optimal integer programming solutions. The innovative subroutines are shown to be efficient to compute and effective in finding good integer programming solutions and providing strong lower bounds for the branch and bound search.This research was supported in part by the U.S. Army Research Office (Durham) under contract no. DAHC04-70-C-0058. This paper is not an official National Bureau of Economic Research publication.     

Using conical partition to globally maximizing the nonlinear sum of ratios

This article presents a global optimization algorithm for globally maximizing the sum of concave–convex ratios problem with a convex feasible region. The algorithm uses a branch and bound scheme where a concave envelope of the objective function is constructed to obtain an upper bound of the optimal value by using conical partition. As a result, the upper-bound subproblems during the algorithm search are all ordinary convex programs with less variables and constraints and do not grow in size from iterations to iterations in the computation procedure, and furthermore a new bounding tightening strategy is proposed such that the upper-bound convex relaxation subproblems are closer to the original nonconvex problem to enhance solution procedure. At last, some numerical examples are given to vindicate our conclusions.     

Branch and bound crossed with GA to solve hybrid flowshops

This article deals with an optimal methods for solving a k-stage hybrid flowshop scheduling problem. This problem is known to be NP-hard. In 1991, Brah and Hunsucker proposed a branch and bound algorithm to solve this problem. However, for some medium size problems, the computation time is not acceptable. The aim of this article is to present an improvement of this algorithm. As a matter of fact, we prove that the value of their lower bound (LB) may decrease along a path of the search tree. First of all, we present an improvement of their LB. Then, we introduce several heuristics at the beginning of the search in order to compute an initial upper bound and genetic algorithm (GA) to improve during the search the value of the upper bound. More precisely, our GA takes into account the set of partial decisions made by the branch and bound and builds a series of populations of complete solutions with the aim of improving the upper bound (the best found criterion value corresponding to a complete solution). Experimentation show that the optimality of branch and bound is more often reached and the value of criterion is improved when our improvements are taken into account.     

A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions

We present a branch and bound algorithm for the global optimization of a twice differentiable nonconvex objective function with a Lipschitz continuous Hessian over a compact, convex set. The algorithm is based on applying cubic regularisation techniques to the objective function within an overlapping branch and bound algorithm for convex constrained global optimization. Unlike other branch and bound algorithms, lower bounds are obtained via nonconvex underestimators of the function. For a numerical example, we apply the proposed branch and bound algorithm to radial basis function approximations.     

A hybrid tabu search/branch & bound approach to solving the generalized assignment problem

A new approach for solving the generalized assignment problem (GAP) is proposed that combines the exact branch & bound approach with the heuristic strategy of tabu search (TS) to produce a hybrid algorithm for solving GAP. The algorithm described uses commercial software to solve sub-problems generated by the TS guiding strategy. The TS approach makes use of the concept of referent domain optimisation and introduces novel add/drop strategies. In addition, the linear programming relaxation of GAP that forms part of the branch & bound approach is itself helpful in suggesting which variables might take binary values. Computational results on benchmark test instances are presented and compared with results obtained by the standard branch & bound approach and also several other heuristic approaches from the literature. The results show the new algorithm performs competitively against the alternatives and is able to find some new best solutions for several benchmark instances.     

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.     

基于D.C.分解的一类箱型约束的非凸二次规划的新型分支定界算法

提出了一类求解带有箱约束的非凸二次规划的新型分支定界算法.首先,把原问题目标函数进行D.C.分解(分解为两个凸函数之差),利用次梯度方法,求出其线性下界逼近函数的一个最优值,也即原问题的一个下界.然后,利用全局椭球算法获得原问题的一个上界,并根据分支定界方法把原问题的求解转化为一系列子问题的求解.最后,理论上证明了算法的收敛性,数值算例表明算法是有效可行的.     

A branch and bound to minimize the number of late jobs on a single machine with release time constraints

This paper presents a branch and bound algorithm for the single machine scheduling problem 1|r_i|∑U_i where the objective function is to minimize the number of late jobs. Lower bounds based on a Lagrangian relaxation and no reductions to polynomially solvable cases are proposed. Efficient elimination rules together with strong dominance relations are also used to reduce the search space. A branch and bound exploiting these techniques solves to optimality instances with up to 200 jobs, improving drastically the size of problems that could be solved by exact methods up to now.     

A simple,all primal branch and bound approach to pure and mixed integer binary programs

Many branch and bound procedures for integer programming employ linear programming to obtain bound information. Nodes in the tree structure are defined by explicitly changing bounds on certain variables and/or adding one or more constraints to the parent LP; thus, primal feasibility is destroyed. The design and analysis of the resulting tree structure requires that basis information be stored for each node and that feasibility restoring pivots be used to obtain the node bound. In turn, this may require the introduction of artificial variables and/or dual simplex pivots.This paper describes a simple procedure for branch and bound that does not destroy primal feasibility. Moreover, the information required to be stored to define the node problems is minimal.     

An outcome space algorithm for optimization over the weakly efficient set of a multiple objective nonlinear programming problem

This article presents for the first time an algorithm specifically designed for globally minimizing a finite, convex function over the weakly efficient set of a multiple objective nonlinear programming problem (V1) that has both nonlinear objective functions and a convex, nonpolyhedral feasible region. The algorithm uses a branch and bound search in the outcome space of problem (V1), rather than in the decision space of the problem, to find a global optimal solution. Since the dimension of the outcome space is usually much smaller than the dimension of the decision space, often by one or more orders of magnitude, this approach can be expected to considerably shorten the search. In addition, the algorithm can be easily modified to obtain an approximate global optimal weakly efficient solution after a finite number of iterations. Furthermore, all of the subproblems that the algorithm must solve can be easily solved, since they are all convex programming problems. The key, and sometimes quite interesting, convergence properties of the algorithm are proven, and an example problem is solved.     

BBPH: Using progressive hedging within branch and bound to solve multi-stage stochastic mixed integer programs

Progressive hedging, though an effective heuristic for solving stochastic mixed integer programs (SMIPs), is not guaranteed to converge in this case. Here, we describe BBPH, a branch and bound algorithm that uses PH at each node in the search tree such that, given sufficient time, it will always converge to a globally optimal solution. In addition to providing a theoretically convergent “wrapper” for PH applied to SMIPs, computational results demonstrate that for some difficult problem instances branch and bound can find improved solutions after exploring only a few nodes.