一类连续可分离背包问题的直接算法

对于一类带有单个线性约束以及盒约束的一般连续可分离二次背包问题给出了一种直接的算法,根据模型特有的结构,通过调节线性约束的拉格朗日乘子λ 的取值范围,以及在算法求解过程中通过判断目标函数一次项中的变量是否在盒约束范围内,来逐步确定所有变量的最优值, 并通过该算法得到的实验结果与其他算法的比较,说明了这种算法的可行性和有效性．     

一种基于LVI求解二次规划问题的数值算法

给出并研究了一种数值算法(简称94LVI算法),用于求解带等式和双端约束的二次规划问题. 这类带约束的二次规划问题首先被转换为线性变分不等式问题,该问题等价于分段线性投影等式.接着使用94LVI算法求解上述分段线性投影等式,从而得到QP问题的最优解. 进一步给出了94LVI算法的全局收敛性证明. 94LVI算法与经典有效集算法的对比实验结果证实了给出的94LVI算法在求解二次规划问题上的高效性与优越性.     

一类带约束min-max-min问题的区间算法

构造了求解一类带不等式约束的min-max-min问题的区间算法,其中目标函数和约束函数都是一阶连续可微函数,证明了方法的收敛性,给出了数值算例.该方法可以同时求出问题的最优值和全部全局最优解,是有效和可靠的.     

流体流动中波状游动平板最优运动的数值方法

提出了一种求解波状游动平板最优运动方式的优化方法．最优化问题表述为固定推力的条件下,使得输入功率最小．由于存在不可见模态,使得该问题具有奇性,用通常的Lagrange乘子法计算得到的可能不是最优解,而是一个鞍点值．为了消除这一奇性,增加了一个关于幅值的不等式约束,并利用逐步二次规划的优化方法求解该问题．将该方法运用到二维和三维的波动板的几个例子上,获得了最优解．     

二次整数背包问题的新算法

给出了一种求解一般二次整数背包问题(quadratic integer knapsack problem,QIKP)的新算法.该方法把占优的概念与分支定界思想结合,旨在寻求全局最优解.对QIKP给出了占优的定义,通过变量系数之间的关系,很容易找到占优组和极小占优组,从而删除可行域中那些非最优点.新的占优定义对凹的二次函数尤其有效.在理论证明的基础上,设计相应的算法,并进行了数值计算.结果显示,在随机产生的例子中,该算法是有效的,并且与传统的分支定界算法相比,得到了更好的最优解,最优值有了较大的提升.     

一类0-1二次规划最优解的新算法

从矩阵的基础知识出发,给出了当目标函数矩阵是严格对角占优阵时,快速地获得0-1二次规划最优解的一个新算法;该方法具有很强的实用性,是此类问题的一个高效求解算法.     

基于粒子群算法的非线性二层规划问题的求解算法

粒子群算法（Particle Swarm Optimization，PSO）是一种新兴的优化技术，其思想来源于人工生命和演化计算理论。PSO通过粒子追随自己找到的最好解和整个群的最好解来完成优化。该算法简单易实现，可调参数少，已得到了广泛研究和应用。本文根据该算法能够有效的求出非凸数学规划全局最优解的特点，对非线性二层规划的上下层问题求解，并根据二层规划的特点，给出了求解非线性二层规划问题全局最优解的有效算法。数值计算结果表明该算法有效。     

带有不等式约束的非线性规划问题的一个精确增广Lagrange函数

对求解带有不等式约束的非线性非凸规划问题的一个精确增广Lagrange函数进行了研究．在适当的假设下,给出了原约束问题的局部极小点与增广Lagrange函数,在原问题变量空间上的无约束局部极小点之间的对应关系．进一步地,在对全局解的一定假设下,还提供了原约束问题的全局最优解与增广Lagrange函数,在原问题变量空间的一个紧子集上的全局最优解之间的一些对应关系．因此,从理论上讲,采用该文给出的增广Lagrange函数作为辅助函数的乘子法,可以求得不等式约束非线性规划问题的最优解和对应的Lagrange乘子．     

一类不可微规划的多项式型算法

本文考虑了由教育最优投资问题导出的一类不可微规划，讨论了可行解是最优解的充要条件，在对乘子作某些假设下，利用Kuhu－Tucker定理给出了求解的一种多项式算法．     

求解凸二次规划问题的势下降内点算法

1　引　言二次规划问题的求解是数学规划和工业应用等领域的一个重要课题 ,同时也是解一般非线性规划问题的序列二次规划算法的关键 .求解二次规划问题的早期技术是利用线性规划问题的单纯形方法求解二次规划问题的 KKT最优性必要条件[1 ] .这类算法比较直观 ,但在处理不等式约束时 ,松弛变量的引进很容易导致求解过程的明显减慢 .有效集策略是求解二次规划问题的另一类主要技术 .这类方法一般都是稳定的 ,但随着问题中大量不等式约束的出现 ,其收敛速度将越来越低[2 ] .简约空间技术将所求问题的 Hessian阵投影到自由变量所在的子空间中 …     

Extending Dantzig's bound to the bounded multiple-class binary Knapsack problem

 The bounded multiple-class binary knapsack problem is a variant of the knapsack problem where the items are partitioned into classes and the item weights in each class are a multiple of a class weight. Thus, each item has an associated multiplicity. The constraints consists of an upper bound on the total item weight that can be selected and upper bounds on the total multiplicity of items that can be selected in each class. The objective is to maximize the sum of the profits associated with the selected items. This problem arises as a sub-problem in a column generation approach to the cutting stock problem. A special case of this model, where item profits are restricted to be multiples of a class profit, corresponds to the problem obtained by transforming an integer knapsack problem into a 0-1 form. However, the transformation proposed here does not involve a duplication of solutions as the standard transformation typically does. The paper shows that the LP-relaxation of this model can be solved by a greedy algorithm in linear time, a result that extends those of Dantzig (1957) and Balas and Zemel (1980) for the 0-1 knapsack problem. Hence, one can derive exact algorithms for the multi-class binary knapsack problem by adapting existing algorithms for the 0-1 knapsack problem. Computational results are reported that compare solving a bounded integer knapsack problem by transforming it into a standard binary knapsack problem versus using the multiple-class model as a 0-1 form. Received: May 1998 / Accepted: February 2002-09-04 Published online: December 9, 2002 Key Words. Knapsack problem – integer programming – linear programming relaxation     

The constrained compartmentalized knapsack problem: mathematical models and solution methods

The constrained compartmentalized knapsack problem can be seen as an extension of the constrained knapsack problem. However, the items are grouped into different classes so that the overall knapsack has to be divided into compartments, and each compartment is loaded with items from the same class. Moreover, building a compartment incurs a fixed cost and a fixed loss of the capacity in the original knapsack, and the compartments are lower and upper bounded. The objective is to maximize the total value of the items loaded in the overall knapsack minus the cost of the compartments. This problem has been formulated as an integer non-linear program, and in this paper, we reformulate the non-linear model as an integer linear master problem with a large number of variables. Some heuristics based on the solution of the restricted master problem are investigated. A new and more compact integer linear model is also presented, which can be solved by a branch-and-bound commercial solver that found most of the optimal solutions for the constrained compartmentalized knapsack problem. On the other hand, heuristics provide good solutions with low computational effort.     

Exact and heuristic solution approaches for the mixed integer setup knapsack problem

We consider a class of knapsack problems that include setup costs for families of items. An individual item can be loaded into the knapsack only if a setup cost is incurred for the family to which it belongs. A mixed integer programming formulation for the problem is provided along with exact and heuristic solution methods. The exact algorithm uses cross decomposition. The proposed heuristic gives fast and tight bounds. In addition, a Benders decomposition algorithm is presented to solve the continuous relaxation of the problem. This method for solving the continuous relaxation can be used to improve the performance of a branch and bound algorithm for solving the integer problem. Computational performance of the algorithms are reported and compared to CPLEX.     

Knapsack problems with setups

Knapsack problems with setups find their application in many concrete industrial and financial problems. Moreover, they also arise as subproblems in a Dantzig–Wolfe decomposition approach to more complex combinatorial optimization problems, where they need to be solved repeatedly and therefore efficiently. Here, we consider the multiple-class integer knapsack problem with setups. Items are partitioned into classes whose use implies a setup cost and associated capacity consumption. Item weights are assumed to be a multiple of their class weight. The total weight of selected items and setups is bounded. The objective is to maximize the difference between the profits of selected items and the fixed costs incurred for setting-up classes. A special case is the bounded integer knapsack problem with setups where each class holds a single item and its continuous version where a fraction of an item can be selected while incurring a full setup. The paper shows the extent to which classical results for the knapsack problem can be generalized to these variants with setups. In particular, an extension of the branch-and-bound algorithm of Horowitz and Sahni is developed for problems with positive setup costs. Our direct approach is compared experimentally with the approach proposed in the literature consisting in converting the problem into a multiple choice knapsack with pseudo-polynomial size.     

Unbounded knapsack problem with controllable rates: the case of a random demand for items

This paper focuses on a dynamic, continuous-time control generalization of the unbounded knapsack problem. This generalization implies that putting items in a knapsack takes time and has a due date. Specifically, the problem is characterized by a limited production horizon and a number of item types. Given an unbounded number of copies of each type of item, the items can be put into a knapsack at a controllable production rate subject to the available capacity. The demand for items is not known until the end of the production horizon. The objective is to collect items of each type in order to minimize shortage and surplus costs with respect to the demand. We prove that this continuous-time problem can be reduced to a number of discrete-time problems. As a result, solvable cases are found and a polynomial-time algorithm is suggested to approximate the optimal solution with any desired precision.     

A generalized knapsack problem with variable coefficients

The ordinary knapsack problem is to find the optimal combination of items to be packed in a knapsack under a single constraint on the total allowable resources, where all coefficients in the objective function and in the constraint are constant.In this paper, a generalized knapsack problem with coefficients depending on variable parameters is proposed and discussed. Developing an effective branch and bound algorithm for this problem, the concept of relaxation and the efficiency function introduced here will play important roles. Furthermore, a relation between the algorithm and the dynamic programming approach is discussed, and subsequently it will be shown that the ordinary 0–1 knapsack problem, the linear programming knapsack problem and the single constrained linear programming problem with upper-bounded variables are special cases of the interested problem. Finally, practical applications of the problem and its computational experiences will be shown.     

Linear programming for the 0–1 quadratic knapsack problem

In this paper we consider the quadratic knapsack problem which consists in maximizing a positive quadratic pseudo-Boolean function subject to a linear capacity constraint. We propose a new method for computing an upper bound. This method is based on the solution of a continuous linear program constructed by adding to a classical linearization of the problem some constraints rebundant in 0–1 variables but nonredundant in continuous variables. The obtained upper bound is better than the bounds given by other known methods. We also propose an algorithm for computing a good feasible solution. This algorithm is an elaboration of the heuristic methods proposed by Chaillou, Hansen and Mahieu and by Gallo, Hammer and Simeone. The relative error between this feasible solution and the optimum solution is generally less than 1%. We show how these upper and lower bounds can be efficiently used to determine the values of some variables at the optimum. Finally we propose a branch-and-bound algorithm for solving the quadratic knapsack problem and report extensive computational tests.     

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.     

A column generation heuristic for the two-dimensional two-staged guillotine cutting stock problem with multiple stock size

We consider a two-dimensional cutting stock problem where stock of different sizes is available, and a set of rectangular items has to be obtained through two-staged guillotine cuts. We propose a heuristic algorithm, based on column generation, which requires as its subproblem the solution of a two-dimensional knapsack problem with two-staged guillotines cuts. A further contribution of the paper consists in the definition of a mixed integer linear programming model for the solution of this knapsack problem, as well as a heuristic procedure based on dynamic programming. Computational experiments show the effectiveness of the proposed approach, which obtains very small optimality gaps and outperforms the heuristic algorithm proposed by Cintra et al. [3].     

A reduction dynamic programming algorithm for the bi-objective integer knapsack problem

This paper presents a backward state reduction dynamic programming algorithm for generating the exact Pareto frontier for the bi-objective integer knapsack problem. The algorithm is developed addressing a reduced problem built after applying variable fixing techniques based on the core concept. First, an approximate core is obtained by eliminating dominated items. Second, the items included in the approximate core are subject to the reduction of the upper bounds by applying a set of weighted-sum functions associated with the efficient extreme solutions of the linear relaxation of the multi-objective integer knapsack problem. Third, the items are classified according to the values of their upper bounds; items with zero upper bounds can be eliminated. Finally, the remaining items are used to form a mixed network with different upper bounds. The numerical results obtained from different types of bi-objective instances show the effectiveness of the mixed network and associated dynamic programming algorithm.