可行约束

Under the assumption that the constrained set is bounded and nonempty , we discussed the geometric properties of the feasible set for MFBLP model and Optimality conditions of MFBLP .

在约束集为非空有界的前提下讨论了可行集的几何性质，得到了可行集为约束集的若干个面的并，可行集为弱拟凸集等一些较好的几何性质。

The crucial point is to reformulate online optimization problem corresponding to nominal model with an appropriate additional robust and feasible constraint.

其关键在于为针对标称系统设计的在线优化问题附加适当的额外的鲁棒可行约束。

Secondly, make amendments to the initial feasible direction in order to adapt it to all the inequations of the optimization model.

具体做法是：首先对非线性优化模型中所有等式约束条件实施拟牛顿迭代法，求得一个可行方向，然后对这个可行方向进行修正，使之满足线性和非线性的不等式约束条件，最后收敛于问题的最优解。

ABSTRACT The infeasibility problem in voltage/reactive power optimization is analyzed in this paper, and this problem is handled with fuzzy constraints to obtain a solution as close as possible to the original space.

分析了电力系统电压无功实时优化控制中出现的不可行问题，通过引入模糊约束，从一个新的角度对不可行问题进行了处理，最终获得尽量靠近原不等式约束空间的解。

To solve a heavily constrained optimization problem includes linear constraint conditions and non-linear constraint conditions, in this paper, a new comprehensive solution mechanism, based on specialized genetic operators, exact penalty function and initialization with multiple linear feasible solutions, is proposed.

针对一类同时具有线性约束条件和非线性约束条件的重约束优化问题，本文成功地综合采用了专门化遗传算子、精确惩罚函数及多个线性可行解初始化的求解方法，取得了很好的效果。

Through calculating the fundus of the null space of matrix A in the constraint,there exists at least one feasible descent direction at every nonstationary point and every iterate generated by the methods is feasible so long as the initial iterate is feasible.

通过计算约束条件中矩阵A的零空间的基底，使得在每一个非稳定点处至少存在一个可行的下降方向，并且在初始点是可行点的条件下，该算法产生的每一个迭代点都是可行点。

Based on the method of multi-objective fuzzy optimization and the overhead traveling crane design, a mathematical model of fuzzy optimization for main box beam section is established. A fuzzy goal anthology is constructed. The fuzziness of multi-objective function and constraints is defined. The transformation of the fuzzy constraints set to normal constraints set in real number domain is realized by applying optimal level cut set method. The optimal solution and optimal constraint value of sub-objective function in the feasible field are found using the genetic algorithms.

应用多目标模糊优化方法，以工程设计中桥式起重机箱形主梁设计为实例，建立了桥式起重机箱形主梁多目标模糊优化设计的数学模型，构造了模糊目标集，确定了多目标函数的模糊性和约束的模糊性，并应用最优水平截集法实现了模糊约束到实数论域上的集合转换，从而用遗传算法寻求到子目标函数在可行域空间的最优解和约束最优值。

The projection gradient method will be a possible way to solve the problem that we just get. It has been shown that the projections of the every directions, of which is the boundary point in linear restraint problems, are the possible decent directions, and the projection of negative grads direction is a decent direction. In 1960, Rosen proposed the basic idea of projection gradient methods, and then lots of researchers have been tried to find the convergence of this method. But most of them get the convergence with the condition to amend the convergence itself.

在约束最优化问题的算法中怎样寻找有效的下降方向是构造算法的重要内容，在寻找下降方向方面可行方向法中的投影梯度法有效的解决了下降方向的寻找问题，利用线性约束问题边界点的任意方向在边界上的投影都是可行方向，而负梯度方向的投影就是一个下降方向。60年代初Rosen提出投影梯度法的基本思想，自从Rosen提出该方法以后，对它的收敛性问题不少人进行了研究，但一般都是对算法作出某些修正后才能证明其收敛的，直到最近对Rosen算法本身的收敛性的证明才予以解决。

Embedding the gradient direction search and new kinds of evaluation functions into the Genetic Algorithm, a special hybrid Genetic Algorithm with mutation along the weighted gradient direction to solve non-linear programming problems is developed. Then in Chapt 4, a type of nonlinear programming problems with fuzzy objective and fuzzy resources constraints is discussed, in which the objective is ill-defined and the resources possess two different types of fuzzy resources, i. e., tolerance and imprecision. By way of description and formulation of fuzzy objective and fuzzy resources, the problem with fuzzy objective and two different types of fuzzy resources is modelled into a fuzzy nonlinear programming model FO/RNLP and interpreted as three equivalent alternative crisp model to reflect the variety of the environment and the subjective and preference of the decision maker .

本文首先针对遗传算法在求解非线性规划问题中存在的问题，首次提出非可行点／染色体的主导准可行方向，准可行方向，及非可行点属于可行域的可行度等新概念，借助于模糊思想，提出了描述和度量非可行染色体的新方法；通过嵌入非可行染色体的信息于评价函数中，提出了三种新的评价非可行染色体的评价函数；把传统的启发式方法-梯度方向搜索和新的评价函数嵌入到遗传算法中，提出了一种沿加权梯度方向变异的新型混合式遗传算法；第四章针对生产实际中非清晰定义的模糊目标和两种类型的模糊资源，建立了一类连续型模糊目标／资源约束非线性规划问题的模型FO／RNLP。

There exits several MDO algorithms. But they are in our opinion either nonefficient or complicated. So we devised an algorithm called Subspace Approximation Optimization . In the SAO algorithm, the whole system is decomposed into one system-level optimization and several disciplinary optimizations so that a large and complicated problem can be divided into several easy-solving sub-problems. The coupling relationships and the coordination among disciplines are presented by equality constraints and these equality constraints are assigned to relevant disciplines. The optimums of design variables in system level optimization are transferred to discipline level optimization. The optimums of design variables in discipline level correspond to the point that is the nearest to the optimums of design variables in system level. If the optimums of design variables in system level are out of feasible region of discipline 1eve1, linear constraints can be built in the system level optimization using the design variable optimums obtained by the discipline level optimization. The system level optimization would improve the design of the whole system with these linear constraints.

目前，国内外已经发展出了多种飞机多学科设计优化算法，本文的重点是针对协同优化算法的不足，提出了子空间近似优化算法(Subspace ApproximatingOptimization，SAO)，SAO算法中，整个系统的优化问题被分解成一个系统级优化和若干学科级优化，而各个学科之间的耦合与权衡关系则被当作等式约束，这些等式约束将被分配到各个学科级优化中去，系统级优化的任务是寻找整个系统的最优解，而学科级优化的目标函数是以系统级优化分配下来的设计点为圆心的超球半径的平方，因此，如果系统级优化分配下来的设计点在学科级优化可行域内，则学科级优化目标函数为0，反之，则学科级优化的最优点是系统级优化当前设计点距离可行域最近的点。

The absorption and distribution of chromium were studied in ryeusing nutrient culture technique and pot experiment.

采用不同浓度K2CrO4(0，0.4，0.8和1.2 mmol/L)的Hoagland营养液处理黑麦幼苗，测定铬在黑麦体内的亚细胞分布、铬化学形态及不同部位的积累。

By analyzing theory foundation of mathematical morphology in the digital image processing, researching morphology arithmetic of the binary Image, discussing two basic forms for the least structure element: dilation and erosion.

通过分析数学形态学在图像中的理论基础，研究二值图像的形态分析算法，探讨最小结构元素的两种基本形态：膨胀和腐蚀；分析了数学形态学复杂算法的基本原理，把数学形态学的部分并行处理理念引入到家实际应用中。