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.     

A hybrid method for multidimensional scaling using city-block distances

The problem of multidimensional scaling with city-block distances in the embedding space is reduced to a two level optimization problem consisting of a combinatorial problem at the upper level and a quadratic programming problem at the lower level. A hybrid method is proposed combining randomized search for the upper level problem with a standard quadratic programming algorithm for the lower level problem. Several algorithms for the combinatorial problem have been tested and an evolutionary global search algorithm has been proved most suitable. An experimental code of the proposed hybrid multidimensional scaling algorithm is developed and tested using several test problems of two- and three-dimensional scaling.     

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.     

A Branch and Bound Algorithm for Separable Concave Programming

In this paper, we propose a new branch and bound algorithm for the solution of large scale separable concave programming problems. The largest distance bisection (LDB) technique is proposed to divide rectangle into sub-rectangles when one problem is branched into two subproblems. It is proved that the LDB method is a normal rectangle subdivision(NRS). Numerical tests on problems with dimensions from 100 to 10000 show that the proposed branch and bound algorithm is efficient for solving large scale separable concave programming problems, and convergence rate is faster than ω-subdivision method.     

Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds

Speed and memory requirements of branch and bound algorithms depend on the selection strategy of which candidate node to process next. The goal of this paper is to experimentally investigate this influence to the performance of sequential and parallel branch and bound algorithms. The experiments have been performed solving a number of multidimensional test problems for global optimization. Branch and bound algorithm using simplicial partitions and combination of Lipschitz bounds has been investigated. Similar results may be expected for other branch and bound algorithms.     

求线性比试和问题的确定性全局优化算法

为求线性比试和问题的全局最优解,本文给出了一个分支定界算法.通过一个等价问题和一个新的线性化松弛技巧,初始的非凸规划问题归结为一系列线性规划问题的求解.借助于这一系列线性规划问题的解,算法可收敛于初始非凸规划问题的最优解.算法的计算量主要是一些线性规划问题的求解.数值算例表明算法是切实可行的.     

MEASURING OPTIMAL CONNECTIONS IN LARGE NETWORKS: A NEW ALGORITHM AND ITS APPLICATIONS

Researchers have identified several problems in measuring the strongest path connecting pairs of actors in valued graphs. To address these problems, it has been proposed that average path value be used to indicate optimal connections between dyads. However, a lack of proper computer algorithm and its implementation has hindered a wide-range application of the proposed solution. In this paper we develop a computer algorithm and fully implement it with four JAVA programs, which are available on request. These programs produce an optimal connection matrix, which is subsequently inputted into UCINET for further multidimensional scaling and clustering analysis. We demonstrate this procedure with a data matrix containing 38 organizations in information technology. We discuss the methodological implications of the application of our algorithm to future social network studies.     

Uncertain constrained optimization by interval-oriented algorithm

This paper deals with an interval-oriented approach to solve general interval constrained optimization problems. Generally, this type of problems has infinitely many compromise solutions. The aim of this approach is to obtain one of such solutions with higher accuracy and lower computational cost. The proposed algorithm is nothing but a different kind of branch and bound algorithm with multi-section division criterion of the search region (or box). In the proposed technique, the prescribed/accepted region is divided into several distinct subregions and in each feasible subregion the interval objective function value is computed. Then the subregion containing the best objective value is found by applying a specific interval ranking rule defined with respect to the pessimistic decision makers’ point of view. The process is continued until the interval width for each variable in the accepted subregion is negligible. Finally, the algorithm converges to a compromise solution in interval form. To illustrate the method and also to test the efficiency as well as the effectiveness of the proposed algorithm, we have solved some numerical examples.     

A Branch and Bound Method for the Multiconstraint Zero-One Knapsack Problem

This paper presents an efficient solution algorithm for the multiconstraint zero-one knapsack problem through a branch and bound search process. The algorithm has been coded in FORTRAN; and a group of thirty 5-constraint knapsack problems with 30-90 variables were run on IBM 360/75 using two other codes as well, in order to compare the computational efficiency of the proposed method with that of the original Balas and an improved Balas additive algorithms. The computational results show that the proposed method is markedly faster with regard to the total as well as the individual solution times for these test problems, and such superiority becomes more evident as the number of variables and the difficulty of the problems increase. The results also indicated that the original Balas method is extremely inefficient for the type of problems being considered here. The total solution time for the thirty problems is 13 min for the proposed method, 109 min for the improved Balas algorithm, and over 380 min for the original Balas algorithm. Extension of the solution algorithm to the generalized knapsack problem is also suggested.     

Global optimization for sum of geometric fractional functions

This paper presents an efficient branch and bound algorithm for globally solving sum of geometric fractional functions under geometric constraints, which arise in various practical problems. By using an equivalent transformation and a new linear relaxation technique, a linear relaxation programming problem of the equivalent problem is obtained. The proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems. Numerical results are reported to show the feasibility of our algorithm.     

解带有二次约束非凸二次规划问题的一个分枝缩减方法

在这篇论文里,有机地把外逼近方法与分枝定界技术结合起来,提出了解带有二次约束非凸二次规划问题的一个分枝缩减方法;给出了原问题的一个新的线性规划松弛,以便确定它在超矩形上全局最优值的一个下界;利用超矩形的一个深度二级剖分方法,以及超矩形的缩减和删除技术,提高算法的收敛速度;证明了在知道原问题可行点的条件下,该算法在有限步里就可以获得原问题的一个全局最优化解,并且用一个例子说明了该算法是有效的.     

带非凸二次约束的二次比式和问题的全局优化算法(英文)

对带非凸二次约束的二次比式和问题(P)给出分枝定界算法,首先将问题(P)转化为其等价问题(Q),然后利用线性化技术,建立了(Q)松弛线性规划问题(RLP),通过对(RLP)可行域的细分及求解一系列线性规划问题,不断更新(Q)的上下界,从理论上证明了算法的收敛性,数值实验表明了算法的可行性和有效性.     

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 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 trust region affine scaling method for bound constrained optimization

We study a new trust region affine scaling method for general bound constrained optimization problems. At each iteration, we compute two trial steps. We compute one along some direction obtained by solving an appropriate quadratic model in an ellipsoidal region. This region is defined by an affine scaling technique. It depends on both the distances of current iterate to boundaries and the trust region radius. For convergence and avoiding iterations trapped around nonstationary points, an auxiliary step is defined along some newly defined approximate projected gradient. By choosing the one which achieves more reduction of the quadratic model from the two above steps as the trial step to generate next iterate, we prove that the iterates generated by the new algorithm are not bounded away from stationary points. And also assuming that the second-order sufficient condition holds at some nondegenerate stationary point, we prove the Q-linear convergence of the objective function values. Preliminary numerical experience for problems with bound constraints from the CUTEr collection is also reported.     

On the Biobjective Adjacent Only Quadratic Spanning Tree Problem

The adjacent only quadratic minimum spanning tree problem is an NP-hard version of the minimum spanning tree where the costs of interaction effects between every pair of adjacent edges are included in the objective function. This paper addresses the biobjective version of this problem. A Pareto local search algorithm is proposed. The algorithm is applied to a set of 108 benchmark instances. The results are compared to the optimal Pareto front generated by a branch and bound algorithm, which is a multiobjective adaptation of a well known algorithm for the mono-objective case.     

A parallel branch and bound algorithm for solving large asymmetric traveling salesman problems

A parallel branch and bound algorithm that solves the asymmetric traveling salesman problem to optimality is described. The algorithm uses an assignment problem based lower bounding technique, subtour elimination branching rules, and a subtour patching algorithm as an upper bounding procedure. The algorithm is organized around a data flow framework for parallel branch and bound. The algorithm begins by converting the cost matrix to a sparser version in such a fashion as to retain the optimality of the final solution. Computational results are presented for three different classes of problem instances: (1) matrix elements drawn from a uniform distribution of integers for instances of size 250 to 10 000 cities, (2) instances of size 250 to 1000 cities that concentrate small elements in the upper left portion of the cost matrix, and (3) instances of size 300 to 3000 cities that are designed to confound neighborhood search heuristics.     

Graphical Sensitivity Analysis for Multidimensional Scaling

This article introduces graphical sensitivity analysis for multidimensional scaling. This new technique is designed to combat two problems associated with multidimensional scaling analyses: The possibility of local minima and the uncertainty regarding sensitivity of the solution to changes in the parameters. Graphical sensitivity analysis is currently available in ViSta-MDS, a test bed for graphical model examination. By graphically manipulating points in the solution space, analysts may examine the sensitivity of the solution to changes in the model parameters. Furthermore, the analyst may search for alternative solutions that represent local minima. An example of graphical sensitivity analysis using ViSta-MDS is described.     

一种求解二次约束二次规划问题的自适应全局优化算法

为了更好地解决二次约束二次规划问题(QCQP), 本文基于分支定界算法框架提出了自适应线性松弛技术, 在理论上证明了这种新的定界技术对于解决(QCQP)是可观的。文中分支操作采用条件二分法便于对矩形进行有效剖分; 通过缩减技术删除不包含全局最优解的部分区域, 以加快算法的收敛速度。最后, 通过数值结果表明提出的算法是有效可行的。     

A branch and bound method for the solution of multiparametric mixed integer linear programming problems

In this paper, we present a novel algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems that exhibit uncertain objective function coefficients and uncertain entries in the right-hand side constraint vector. The algorithmic procedure employs a branch and bound strategy that involves the solution of a multiparametric linear programming sub-problem at leaf nodes and appropriate comparison procedures to update the tree. McCormick relaxation procedures are employed to overcome the presence of bilinear terms in the model. The algorithm generates an envelope of parametric profiles, containing the optimal solution of the mp-MILP problem. The parameter space is partitioned into polyhedral convex critical regions. Two examples are presented to illustrate the steps of the proposed algorithm.