A Cutting Plane Algorithm for Linear Reverse Convex Programs

In this paper, global optimization of linear programs with an additional reverse convex constraint is considered. This type of problem arises in many applications such as engineering design, communications networks, and many management decision support systems with budget constraints and economies-of-scale. The main difficulty with this type of problem is the presence of the complicated reverse convex constraint, which destroys the convexity and possibly the connectivity of the feasible region, putting the problem in a class of difficult and mathematically intractable problems. We present a cutting plane method within the scope of a branch-and-bound scheme that efficiently partitions the polytope associated with the linear constraints and systematically fathoms these portions through the use of the bounds. An upper bound and a lower bound for the optimal value is found and improved at each iteration. The algorithm terminates when all the generated subdivisions have been fathomed.     

求解标准二次优化问题的割平面算法

标准的二次优化问题是NP-hard问题,把该问题转化为半不定的线性规划问题,且提出了一个线性规划的割平面算法来求解这个半不定的线性规划问题,并给出了该算法的收敛性证明.     

一种求解线性二层规划的割平面方法

以下层问题的K-T最优性条件代替下层问题,将线性二层规划转化为相应的单层规划问题,通过分析单层规划可行解集合的结构特征,设计了一种求解线性二层规划全局最优解的割平面算法.数值结果表明所设计的割平面算法是可行、有效的.     

A Center Cutting Plane Algorithm for a Likelihood Estimate Problem

We describe a method for solving the maximum likelihood estimate problem of a mixing distribution, based on an interior cutting plane algorithm with cuts through analytic centers. From increasingly refined discretized statistical problem models we construct a sequence of inner non-linear problems and solve them approximately applying a primal-dual algorithm to the dual formulation. Refining the statistical problem is equivalent to adding cuts to the inner problems.     

A Long-Step, Cutting Plane Algorithm for Linear and Convex Programming

A cutting plane method for linear programming is described. This method is an extension of Atkinson and Vaidya's algorithm, and uses the central trajectory. The logarithmic barrier function is used explicitly, motivated partly by the successful implementation of such algorithms. This makes it possible to maintain primal and dual iterates, thus allowing termination at will, instead of having to solve to completion. This algorithm has the same complexity (O(nL ²) iterations) as Atkinson and Vaidya's algorithm, but improves upon it in that it is a long-step version, while theirs is a short-step one in some sense. For this reason, this algorithm is computationally much more promising as well. This algorithm can be of use in solving combinatorial optimization problems with large numbers of constraints, such as the Traveling Salesman Problem.     

A Finite Branch-and-Bound Algorithm for Linear Multiplicative Programming

On the basis of Soland's rectangular branch-and-bound, we develop an algorithm for minimizing a product of p (2) affine functions over a polytope. To tighten the lower bound on the value of each subproblem, we install a second-stage bounding procedure, which requires O(p) additional time in each iteration but remarkably reduces the number of branching operations. Computational results indicate that the algorithm is practical if p is less than 15, both in finding an exact optimal solution and an approximate solution.     

A Lagrangian Based Branch-and-Bound Algorithm for Production-transportation Problems

We propose a branch-and-bound algorithm of Falk–Soland's type for solving the minimum cost production-transportation problem with concave production costs. To accelerate the convergence of the algorithm, we reinforce the bounding operation using a Lagrangian relaxation, which is a concave minimization but yields a tighter bound than the usual linear programming relaxation in O(mn log n) additional time. Computational results indicate that the algorithm can solve fairly large scale problems.     

A Branch-and-Bound Algorithm for Concave Network Flow Problems

In this paper a Branch-and-Bound (BB) algorithm is developed to obtain an optimal solution to the single source uncapacitated minimum cost Network Flow Problem (NFP) with general concave costs. Concave NFPs are NP-Hard, even for the simplest version therefore, there is a scarcity of exact methods to address them in their full generality. The BB algorithm presented here can be used to solve optimally single source uncapacitated minimum cost NFPs with any kind of concave arc costs. The bounding is based on the computation of lower bounds derived from state space relaxations of a dynamic programming formulation. The relaxations, which are the subject of the paper (Fontes et al., 2005b) and also briefly discussed here, involve the use of non-injective mapping functions, which guarantee a reduction on the cardinality of the state space. Branching is performed by either fixing an arc as part of the final solution or by removing it from the final solution. Computational results are reported and compared to available alternative methods for addressing the same type of problems. It could be concluded that our BB algorithm has better performance and the results have also shown evidence that it has a sub-exponential time growth.     

A Revision of the Trapezoidal Branch-and-Bound Algorithm for Linear Sum-of-Ratios Problems

In this paper, we point out a theoretical flaw in Kuno [(2002)Journal of Global Optimization 22, 155–174] which deals with the linear sum-of-ratios problem, and show that the proposed branch-and-bound algorithm works correctly despite the flaw. We also note a relationship between a single ratio and the overestimator used in the bounding operation, and develop a procedure for tightening the upper bound on the optimal value. The procedure is not expensive, but the revised algorithms incorporating it improve significantly in efficiency. This is confirmed by numerical comparisons between the original and revised algorithms. The author was partially supported by the Grand-in-Aid for Scientific Research (C)(2) 15560048 from the Japan Society for the Promotion of Science.     

Branch-and-Bound Outer Approximation Algorithm for Sum-of-Ratios Fractional Programs

In this article, a branch and-bound outer approximation algorithm is presented for globally solving a sum-of-ratios fractional programming problem. To solve this problem, the algorithm instead solves an equivalent problem that involves minimizing an indefinite quadratic function over a nonempty, compact convex set. This problem is globally solved by a branch-and-bound outer approximation approach that can create several closed-form linear inequality cuts per iteration. In contrast to pure outer approximation techniques, the algorithm does not require computing the new vertices that are created as these cuts are added. Computationally, the main work of the algorithm involves solving a sequence of convex programming problems whose feasible regions are identical to one another except for certain linear constraints. As a result, to solve these problems, an optimal solution to one problem can potentially be used to good effect as a starting solution for the next problem.     

A Cutting Plane Method for Solving Quasimonotone Variational Inequalities

We present an iterative algorithm for solving variational inequalities under the weakest monotonicity condition proposed so far. The method relies on a new cutting plane and on analytic centers.     

Cone Adaptation Strategies for a Finite and Exact Cutting Plane Algorithm for Concave Minimization

In this paper we are concerned with pure cutting plane algorithms for concave minimization. One of the most common types of cutting planes for performing the cutting operation in such algorithm is the concavity cut. However, it is still unknown whether the finite convergence of a cutting plane algorithm can be enforced by the concavity cut itself or not. Furthermore, computational experiments have shown that concavity cuts tend to become shallower with increasing iteration. To overcome these problems we recently proposed a procedure, called cone adaptation, which deepens concavity cuts in such a way that the resulting cuts have at least a certain depth with 0, where is independent of the respective iteration, which enforces the finite convergence of the cutting plane algorithm. However, a crucial element of our proof that these cuts have a depth of at least was that we had to confine ourselves to -global optimal solutions, where is a prescribed strictly positive constant. In this paper we examine possible ways to ensure the finite convergence of a pure cutting plane algorithm for the case where = 0.     

A Branch-and-Bound Algorithm to Solve a Multi-level Network Optimization Problem

Multi-level network optimization problems arise in many contexts such as telecommunication, transportation, and electric power systems. A model for multi-level network design is formulated as a mixed-integer program. The approach is innovative because it integrates in the same model aspects of discrete facility location, topological network design, and dimensioning. We propose a branch-and-bound algorithm based on Lagrangian relaxation to solve the model. Computational results for randomly generated problems are presented showing the quality of our approach. We also present and discuss a real world problem of designing a two-level local access urban telecommunication network and solving it with the proposed methodology.     

Solving Convex MINLP Optimization Problems Using a Sequential Cutting Plane Algorithm

In this article we look at a new algorithm for solving convex mixed integer nonlinear programming problems. The algorithm uses an integrated approach, where a branch and bound strategy is mixed with solving nonlinear programming problems at each node of the tree. The nonlinear programming problems, at each node, are not solved to optimality, rather one iteration step is taken at each node and then branching is applied. A Sequential Cutting Plane (SCP) algorithm is used for solving the nonlinear programming problems by solving a sequence of linear programming problems. The proposed algorithm generates explicit lower bounds for the nodes in the branch and bound tree, which is a significant improvement over previous algorithms based on QP techniques. Initial numerical results indicate that the described algorithm is a competitive alternative to other existing algorithms for these types of problems.     

A “Branch-and-Bound” Algorithm for the Exact Solution of the Three-Machine Scheduling Problem

By combining Roy's graph-theoretical interpretation of the problem of job scheduling on machines and some general results of his theory with the “branch-and-bound” method recently applied by Little et al. to the travelling salesman problem an algorithm has been obtained for the exact solution of the scheduling problem in the case of three machines. The working of the algorithm has been illustrated by numerical examples and possibilities of further investigations have been indicated.     

Decomposition Branch-and-Bound Based Algorithm for Linear Programs with Additional Multiplicative Constraints

This article presents an algorithm for globally solving a linear program (P) that contains several additional multiterm multiplicative constraints. To our knowledge, this is the first algorithm proposed to date for globally solving Problem (P). The algorithm decomposes the problem to obtain a master problem of low rank. To solve the master problem, the algorithm uses a branch-and-bound scheme where Lagrange duality theory is used to obtain the lower bounds. As a result, the lower-bounding subproblems in the algorithm are ordinary linear programs. Convergence of the algorithm is shown and a solved sample problem is given.     

A Branch-and-Bound Algorithm to Minimize the Makespan in a Flowshop with Blocking

This work addresses the minimization of the makespan criterion for the flowshop problem with blocking. In this environment there are no buffers between successive machines, and therefore intermediate queues of jobs waiting in the system for their next operations are not allowed. We propose a lower bound which exploits the occurrence of blocking. A branch-and-bound algorithm that uses this lower bound is described and its efficiency is evaluated on several problems. Results of computational experiments are reported.     

关于问题Pm|intree;pj=1;rj|Cmax的分支定界算法

本文针对一个尚未解决的问题Pm|intree;pj=1;rj|Cmax进行了研究,借助于决策论中的递阶层次结构的概念提出一个全新的分支定界算法,并用这一算法得到了问题Pm|intree;pj=1;rj|Cmax的最优排序．