LP问题的λ算法

本文给出了求LP问题最优解的λ算法,并指出了此法旋转运算的次数.此算法不需要基本可行解或对偶基本可行解.     

LP问题的λ算法

本给出了求LP问题的最优解的λ算法，并指出了此法旋转运算的次数，此算法不需要基本可行解或对偶基本可行解。     

基于种群分类排序的约束优化遗传算法

针对约束优化问题,提出了一类将种群中的个体分类排序的思想.算法的特点在于:先将种群中的解分为可行解和不可行解两类,然后分别按照不同的标准排序.由于很多约束优化问题的最优解位于可行域的边界上或附近,所以排序时并不认为可行解一定优于不可行解.基于此分类排队思想,特别设计了只允许同等级个体进行交叉的新的交叉算子,称之为同等级交叉算子,以及基于一维搜索的变异算子.算法同时采用了保证固定比例不可行解的自适应策略.4个标准测试函数的数值仿真结果验证了算法的有效性.     

车间作业调度的可行化判定及不可行解的纠正

生产调度过程中出现不可行解是调度研究经常遇到的问题之一.提出了对JSP调度方案进行可行化判定和纠正不可行解的可行算子,算子包括了基于有向图拓扑排序原理对车间作业调度方案进行可行判定的方法和将不可行解纠正为可行解的算法.证明了该纠正算法总能成功,并对算子的功能进行了拓展使之还可应用于不完备调度.最后讨论了可行算子的特点、时间效率和应用前景.     

一种原始——对偶单纯形算法的枢轴准则选择

Curet曾提出了一种有趣的原始一对偶技术,在优化对偶问题的同时单调减少原始不可行约束的数量,当原始可行性产生时也就产生了原问题的最优解.然而该算法需要一个初始对偶可行解来启动,目标行的选择也是灵活、不确定的.根据Curet的原始一对偶算法原理,提出了两种目标行选择准则,并通过数值试验进行比较和选择.对不存在初始对偶可行解的情形,通过适当改变目标函数的系数来构造一个对偶可行解,以求得一个原始可行解,再应用原始单纯形算法求得原问题的最优解.数值试验对这种算法的计算性能进行验证,通过与经典两阶段单纯形算法比较,结果表明,提出的算法在大部分问题上具有更高的计算效率.     

求解约束优化问题的一种新的遗传算法

提出一种新的求解约束优化问题的遗传算法,算法通过重新定义可行解与不可行解的适应度函数分别对它们进行选择,有效避免了惩罚函数法引入参数所带来的困难,重新设计的交叉算子使得算法对解空间的寻优范围扩大了.数值实验结果表明算法具有较好的鲁棒性,且对最优解位于约束边界上的一类问题具有很大优势.     

基于改进遗传算法的集合覆盖问题

集合覆盖问题是组合优化中的典型问题,在日常生活中有着广泛的应用.提出了一种改进遗传算法来解决集合覆盖问题.算法对标准遗传算法的改进主要表现在:1)结合启发式算法和随机生成,设计了新的产生初始种群的方法;2)引入修补操作处理不可行解使其转换成可行解;3)对重复个体进行处理再利用;4)对多点交叉进行推广,提出了新的交叉算子;5)针对可行解和不可行解,采取两种自适应多位变异操作.数值实验结果表明该算法对于解决规模较大的集合覆盖问题是有效的.     

约束尺度和算子自适应变化的差分进化算法

由于可行域不连续和函数形式复杂使得许多算法难以有效求解约束优化问题,提出了一种约束尺度和算子自适应变化的差分进化算法.通过统计新个体中可行解和不可行解的数量以自适应调整惩罚系数,使个体能够分布在多个不连续的可行域中,从而找到最优解所在区域.同时,算法还采用了两种不同的差分算子,分别用于局部区域的快速寻优和整个可行域的全局探索.在两种算子的选择上,则根据新个体的存活情况和约束违反情况来自适应调整其选择的概率.最后通过3组标准约束优化问题在10维和30维变量下的测试结果显示:所提算法的性能整体优于对比算法,其平均最优解在10维时至少提升了4.75%.     

具有模糊关系不等式约束的单项几何规划的算法

本文研究具有取大取小模糊关系不等式约束的单项几何规划的解法.首先证明它的最优解由最大可行解与一个极小可行解组成,然后提出简化问题的几个规则,最后根据简化规划与分支定界法提出一个不需要求解全部可行极小解的算法.数值实例表明提出的算法是可行的.     

哈密顿矩阵的逆特征值问题

该文探讨了哈密顿矩阵的逆特征值问题, 得到了有解的充要条件、通解的表达式以及最小范数解.并给出了最佳逼近解的求法. 给出了相应的算法, 数值实例说明算法是可行的.     

A feasible direction method for linear programming

We discuss a finite method of a feasible direction for linear programming problems. The method begins with a feasible basic vector for the problem, constructs a profitable direction to move using the updated column vectors of the nonbasic variables eligible to enter this basic vector. It then moves in this direction as far as possible, while retaining feasibility. This move in general takes it though the relative interior of a face of th set of a feasible solutions. The final point, $\text{x}$, obtained at the end of this move will not in general be a basic solution. Using $\text{x}$ the method then constructs a basic feasible solution at which the objective value is better than, or the same as that at $\text{x}$. The whole process repeats with the new basic feasible solution. We show that this method can be implemented using basis inverses. Initial computer runs of this method in comparison with the usual edge following primary simplex algorithms are very encouraging.     

The double pivot simplex method

The simplex method, created by George Dantzig, optimally solves a linear program by pivoting. Dantzig’s pivots move from a basic feasible solution to a different basic feasible solution by exchanging exactly one basic variable with a nonbasic variable. This paper introduces the double pivot simplex method, which can transition between basic feasible solutions using two variables instead of one. Double pivots are performed by identifying the optimal basis in a two variable linear program using a new method called the slope algorithm. The slope algorithm is fast and allows an iteration of DPSM to have the same theoretical running time as an iteration of the simplex method. Computational experiments demonstrate that DPSM decreases the average number of pivots by approximately 41% on a small set of benchmark instances.     

Some fundamental issues of basic line search algorithm for linear programming problems

In this article we present the fundamental idea, concepts and theorems of a basic line search algorithm for solving linear programming problems which can be regarded as an extension of the simplex method. However, unlike the iteration of the simplex method from a basic point to an improved adjacent basic point via pivot operation, the basic line search algorithm, also by pivot operation, moves from a basic line which contains two basic feasible points to an improved basic line which also contains two basic feasible points whose objective values are no worse than that of the two basic feasible points on the previous basic line. The basic line search algorithm may skip some adjacent vertices so that it converges to an optimal solution faster than the simplex method. For example, for a 2-dimensional problem, the basic line search algorithm can find an optimal solution with only one iteration.     

直接求线性规划可行基的一种方法

本文给出直接求线性规划问题基可行解的一种简易方法,该方法既避免了引入人工变量,减少存储,一般又能较快地得到一个较好的基可行解.     

A class of methods for linear programming

A class of methods is presented for solving standard linear programming problems. Like the simplex method, these methods move from one feasible solution to another at each iteration, improving the objective function as they go. Each such feasible solution is also associated with a basis. However, this feasible solution need not be an extreme point and the basic solution corresponding to the associated basis need not be feasible. Nevertheless, an optimal solution, if one exists, is found in a finite number of iterations (under nondegeneracy). An important example of a method in the class is the reduced gradient method with a slight modification regarding selection of the entering variable.     

Some new results in linear programs with trapezoidal fuzzy numbers: Finite convergence of the Ganesan and Veeramani’s method and a fuzzy revised simplex method

In a recent paper, Ganesan and Veermani [K. Ganesan, P. Veeramani, Fuzzy linear programs with trapezoidal fuzzy numbers, Ann. Oper. Res. 143 (2006) 305–315] considered a kind of linear programming involving symmetric trapezoidal fuzzy numbers without converting them to the crisp linear programming problems and then proved fuzzy analogues of some important theorems of linear programming that lead to a new method for solving fuzzy linear programming (FLP) problems. In this paper, we obtain some another new results for FLP problems. In fact, we show that if an FLP problem has a fuzzy feasible solution, it also has a fuzzy basic feasible solution and if an FLP problem has an optimal fuzzy solution, it has an optimal fuzzy basic solution too. We also prove that in the absence of degeneracy, the method proposed by Ganesan and Veermani stops in a finite number of iterations. Then, we propose a revised kind of their method that is more efficient and robust in practice. Finally, we give a new method to obtain an initial fuzzy basic feasible solution for solving FLP problems.     

不需加人工变量的两阶段法

A method is provided to achieve an initial basic feasible solution of a linear programming in this paper. This method dose not need introducing any artificial variable, but needs only solving an auxiliary linear programming. Compared with the traditional two-phase method, it has advantages of saving the memories and reducing the computational efforts.     

New criteria for the simplex algorithm

The standard linear programming problem with a finite optimum value is considered. We derive new criteria which guarantee that(i) a non-basic variable of a basic feasible solution will remain a non-basic variable of an optimal basic solution; (ii) a basic variable of a basic feasible solution will remain a basic variable of an optimal basic solution.     

Comment on a paper by M.C. Cheng

In a recent paper M.C. Cheng proposed new criteria for the simplex algorithm which guarantee that (i) a nonbasic variable of a basic feasible solution will remain nonbasic in an optimal basic solution, (ii) a basic variable of a basic solution will remain basic in an optimal basic solution. This comment gives (i) a slight generalization of the first result and (ii) a counterexample to the second proposition.     

On relaxation methods for systems of linear inequalities

In their classical papers Agmon and Motzkin and Schoenberg introduced a relaxation method to find a feasible solution for a system of linear inequalities. So far the method was believed to require infinitely many iterations on some problem instances since it could (depending on the dimension of the set of feasible soltions) converge asymptotically to a feasible solution, if one exists. Hence it could not be used to determine infeasibility.Using two lemma's basic to Khachian's polynomially bounded algorithm we can show that the relaxation method is finite in all cases and thus can handle infeasible systems as well. In spite of more refined stopping criteria the worst case behaviour of the relaxation method is not polynomially bounded as examplified by a class of problems constructed here.