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 new dynamic programming procedure for three-staged cutting patterns

Three-staged patterns are often used to solve the 2D cutting stock problem of rectangular items. They can be divided into items in three stages: Vertical cuts divide the plate into segments; then horizontal cuts divide the segments into strips, and finally vertical cuts divide the strips into items. An algorithm for unconstrained three-staged patterns is presented, where a set of rectangular item types are packed into the plate so as to maximize the pattern value, and there is no constraint on the frequencies of each item type. It can be used jointly with the linear programming approach to solve the cutting stock problem. The algorithm solves three large knapsack problems to obtain the optimal pattern: One for the item layout on the widest strip, one for the strip layout on the longest segment, and the third for the segment layout on the plate. The computational results indicate that the algorithm is efficient.     

广义几何规划的压缩共轭梯度路径非单调信赖域算法(英文)

In this paper,on the basis of making full use of the characteristics of unconstrained generalized geometric programming（GGP）,we establish a nonmonotonic trust region algorithm via the conjugate path for solving unconstrained GGP problem.A new type of condensation problem is presented,then a particular conjugate path is constructed for the problem,along which we get the approximate solution of the problem by nonmonotonic trust region algorithm,and further prove that the algorithm has global convergence and quadratic convergence properties.     

Conjugate gradient techniques for the optimal control evolution dam problem

This paper considers the numerical simulation of optimal control evolution dam problem by using conjugate gradient method.The paper considers the free boundary value problem related to time dependent fluid flow in a homogeneous earth rectangular dam.The dam is taken to be sufficiently long that the flow is considered to be two dimensional.On the left and right walls of the dam there is a reservoir of fluid at a level dependent on time.This problem can be transformed into a variational inequality on a fixed domain.The numerical techniques we use are based on a linear finite element method to approximate the state equations and a conjugate gradient algorithm to solve the discrete optimal control problem.This algorithm is based on Armijo's rule in the unconstrained optimization theory.The convergence of the discrete optimal solutions to the continuous optimal solutions,and the convergence of the conjugate gradient algorithm are proved.A numerical example is given to determine the location of the minimum surface     

ABS算法在求解MPEC问题中的应用

应用ABS—隐式LU算法，简化MPEC问题的约束条件，将简化后的MPEC问题转化为目标函数带有罚函数子项的非线性无约束优化问题，给出收敛性定理，证明当罚因子足够大时，此非线性无约束问题的极小点就是简化后的MPEC问题的极小点，将此极小点代入本中给出的一个转换公式可得原MPEC问题的极小点，末给出一算例。     

On the convergence of asynchronous parallel algorithm for large-scale linearly constrained minimization problem

As a synchronization parallel framework, the parallel variable transformation (PVT) algorithm is effective to solve unconstrained optimization problems. In this paper, based on the idea that a constrained optimization problem is equivalent to a differentiable unconstrained optimization problem by introducing the Fischer Function, we propose an asynchronous PVT algorithm for solving large-scale linearly constrained convex minimization problems. This new algorithm can terminate when some processor satisfies terminal condition without waiting for other processors. Meanwhile, it can enhances practical efficiency for large-scale optimization problem. Global convergence of the new algorithm is established under suitable assumptions. And in particular, the linear rate of convergence does not depend on the number of processors.     

Minimum norm solution to the positive semidefinite linear complementarity problem

In this article, we present an algorithm to compute the minimum norm solution of the positive semidefinite linear complementarity problem. We show that its solution can be obtained using the alternative theorems and a convenient characterization of the solution set of a convex quadratic programming problem. This problem reduces to an unconstrained minimization problem with once differentiable convex objective function. We propose an extension of Newton's method for solving the unconstrained optimization problem. Computational results show that convergence to high accuracy often occurs in just a few iterations.     

A 2.75-approximation algorithm for the unconstrained traveling tournament problem

A 2.75-approximation algorithm is proposed for the unconstrained traveling tournament problem, which is a variant of the traveling tournament problem. For the unconstrained traveling tournament problem, this is the first proposal of an approximation algorithm with a constant approximation ratio. In addition, the proposed algorithm yields a solution that meets both the no-repeater and mirrored constraints. Computational experiments show that the algorithm generates solutions of good quality.     

Minimum Norm Solution to the Absolute Value Equation in the Convex Case

In this paper, we give an algorithm to compute the minimum norm solution to the absolute value equation (AVE) in a special case. We show that this solution can be obtained from theorems of the alternative and a useful characterization of solution sets of convex quadratic programs. By using an exterior penalty method, this problem can be reduced to an unconstrained minimization problem with once differentiable convex objective function. Also, we propose a quasi-Newton method for solving unconstrained optimization problem. Computational results show that convergence to high accuracy often occurs in just a few iterations.     

半监督度量学习内蕴最速下降算法的收敛性分析

主要研究对称正定矩阵群上的内蕴最速下降算法的收敛性问题.首先针对一个可转化为对称正定矩阵群上无约束优化问题的半监督度量学习模型,提出对称正定矩阵群上一种自适应变步长的内蕴最速下降算法.然后利用李群上的光滑函数在任意一点处带积分余项的泰勒展开式,证明所提算法在对称正定矩阵群上是线性收敛的.最后通过在分类问题中的数值实验说明算法的有效性.     

Simple block patterns for the two-dimensional cutting problem

This paper presents an algorithm for the unconstrained two-dimensional cutting problem of rectangular pieces. It proposes the simple block (SB) pattern consisting of simple blocks. The SB pattern is defined recursively. Each cut on the stock plate produces just one simple block. A horizontal cut produces a horizontal block with width equal to that of the leftmost piece in the block. A vertical cut produces a vertical block with length equal to that of the bottommost piece in the block. The algorithm generates the optimal SB pattern recursively, and selects optimally the first piece in each block. It uses upper bound to prune some unpromising branches during the searching process. The computational results indicate that the algorithm is highly efficient in improving material utilization, and the computation time is reasonable.     

An algorithm based on the generalized D-gap function for equilibrium problems

The equilibrium problem (EP) can be reformulated as an unconstrained minimization problem through the generalized D-gap function. In this paper, we propose an algorithm for minimizing the problem and analyze some convergence properties of the proposed algorithm. Under some reasonable conditions, we show that the iteration sequence generated by the algorithm is globally convergent and converges to a solution to the EP and the generalized D-gap function provides a global error bound for the algorithm.     

A worst case analysis of a dynamic programming-based heuristic algorithm for 2D unconstrained guillotine cutting

In this paper, a dynamic programming-based recursive method is proposed for solving an unconstrained 2D rectangular cutting problem. The algorithm is an incomplete method, in which some intricate cutting patterns may not be obtained. The worst case performance of the algorithm is evaluated and some theoretical analyses for the algorithm are performed. Compared to traditional dynamic programming, this algorithm gives a high percentage of optimal solutions (94.84%, 86.67% and 77.83% for small, medium and large sized unweighted instances, 99.67%, 99.50% and 97.00% for small, medium and large sized weighted instances) but features a far lower computational complexity. Computational results are also presented for some known benchmarks.     

Algorithms for Unconstrained Two-Dimensional Guillotine Cutting

In this paper we consider the unconstrained, two-dimensional, guillotine cutting problem. This is the problem that occurs in the cutting of a number of rectangular pieces from a single large rectangle, so as to maximize the value of the pieces cut, where any cuts that are made are restricted to be guillotine cuts.We consider both the staged version of the problem (where the cutting is performed in a number of distinct stages) and the general (non-staged) version of the problem.A number of algorithms, both heuristic and optimal, based upon dynamic programming are presented. Computational results are given for large problems.     

一类无约束复合非光滑优化的信赖域算法

本文提出了一个易实施的处理一类无约束复合非光滑优化的信赖域算法，并在一定条件下证明了该算法所产生的迭代序列的任何聚点都是原问题的稳定点．     

一个解非线性0-1整数规划问题基于罚函数的混合粒子群优化算法

利用罚函数思想把非线性0-1整数规划问题转化为无约束最优化问题,然后把粒子群优化和罚函数方法结合构造出一个基于罚函数的混合粒子群优化算法,数值结果表明所提出的算法是有效的.     

基于支持向量机的多类分类问题的一种新算法

在K-SVCR算法结构的基础上构造了新的模型.模型的特点是它的一阶最优化条件可以转化为一个线性互补问题,通过Lagrangian隐含数,可以将其进一步转化成一个强凸的无约束优化问题.利用共轭梯度技术对其进行求解,在有限步内得到分类超平面.最后在标准数据集进行了初步试验.试验结果显示了提出的算法在分类的精度和速度上都有明显提高.     

Singular values of a real rectangular tensor

Real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we systematically study properties of singular values of a real rectangular tensor, and give an algorithm to find the largest singular value of a nonnegative rectangular tensor. Numerical results show that the algorithm is efficient.     

A recursive algorithm for constrained two-dimensional cutting problems

This paper presents a recursive algorithm for constrained two-dimensional guillotine cutting problems of rectangular items. The algorithm divides a stock plate into a sequence of small rectangular blocks. For the current block considered, it selects an item, puts it at the left-bottom corner of the block, and determines the direction of the dividing cut that divides the unoccupied region of the block into two smaller blocks for further consideration. The dividing cut is either along the upper edge or along the right edge of the selected item. The upper bound obtained from the unconstrained solution is used to shorten the searching space. The computational results on benchmark problems indicate that the algorithm can improve the solutions, and is faster than other algorithms.     

基于非单调线搜索的对角二阶拟牛顿法

在二阶拟牛顿方程的基础上,结合Zhang H.C.提出的非单调线搜索构造了一种求解大规模无约束优化问题的对角二阶拟牛顿算法.算法在每次迭代中利用对角矩阵逼近Hessian矩阵的逆,使计算搜索方向的存储量和工作量明显减少,为大型无约束优化问题的求解提供了新的思路.在通常的假设条件下,证明了算法的全局收敛性和超线性收敛性.数值实验表明算法是有效可行的.