次梯度

Chapter 4 establishes the theory of convex fuzzy mappings: The concepts, such as Jensen's inequality, positively homogeneous, infimal convolution, right scalar multiplication and convex hull are introduced. The corresponding theorems are demonstrated by using the parametric representations of fuzzy numbers. In anti-fuzzy number space, the conjugate mapping of convex fuzzy mapping is concerned, and convexities of conjugate set and conjugate mapping of convex fuzzy mapping are proved. The notions of subgradient, subdifferential, differential with respect to convex fuzzy mappings are investigated, which provides the basis of the theory of fuzzy extremum problems.

在第4章中，建立了有关凸模糊映射的理论：建立了关于凸模糊映射的Jensen不等式、模糊正齐次映射、凸模糊映射的下卷积、右数乘和凸包等概念，利用模糊数的参数化表示，给出了相应的定理；在反模糊数空间，对凸模糊映射的共轭也作了探讨，证明了凸模糊映射的共轭集合和共轭映射都是凸的；最后对凸模糊映射的次梯度、次微分和微分等概念进行了研究，为模糊极值理论打下了基础。

The sub-gradient algorithm is designed to solve lagrangian dual problem according to its characteristic.

根据问题的特性设计了次梯度算法求解拉格朗日对偶问题。

Then we prove a nonsmooth minimax theorem by usingsome relative results about multi-valued maximal monotone mappings. Finally, we construct a nonsmooth minimax network to search for thesaddle points of the saddle functional. We have also proved that thetrajectory of the network is asymptotically convergent to thesaddle point of the saddle functional by usying Lyapunov function.

我们先介绍有关Clarke的广义梯度，次梯度及鞍泛函的有关结论；然后利用多值极大单调映象的满射性结果证明了一个非光滑的minimax定理；最后建立了一个非光滑的minimax神经网络来求解泛函的鞍点，并在一定的条件下构造出LYAPUNOV函数来证明网络的轨道渐近收敛于鞍泛函的鞍点。

An accelerated acoustic echo cancellation algorithm is proposed in this paper based on Parallel Subgradient Projection technique.

摘要该文在并行次梯度投影技术(Parallel Subgradient Projection，PSP)的基础上，提出一种加速回波抵消算法。

A surrogate subgradient with penalty function method is also used to solve unit commitment.

针对以上两点，分别提出了带惩罚项的伪次梯度法和启发式方法解决经济调度问题的两种新方法。

As the objective function is not strictly concave in the path rate, the penalty function method is applied to transform the primal problem into a new formulation to obtain the optimal solution based on the subgradient method. And the multiPath congestion control algorithm is also proposed.

由于多径效用最大化问题中的目标函数对路径速率而言不是严格凹的，所以运用罚函数法将此最大化问题转化成新的等价形式，再运用次梯度法获得了原问题的最优解，由此提出了用于Ad hoc网络的多径路由优化拥塞控制算法MPCC。

The general algorithm steps based on subgradient optimization mathematics method are presented.

建立了这类问题的数学规划模型，并采用拉格朗日松弛算法对模型进行求解，给出了次梯度优化求解算法的一般步骤。

A hybrid method based on differential evolution and subgradient optimization is used to solve the problem. It is found that the proposed method can find solutions better than the de facto commercial software LINGO.

在采用一种混合微分演化及次梯度优化法来解决此问题之后，实验结果发现这种混合法比业界公认的商业软体LINGO能够找到更好的答案。

The Lipschitzian property of functions involved is researched. An approach of computing a subgradient of the objective functions of the problems is investigated. The basic idea and steps of the algorithm are discussed. Finally, the convergence analysis is given.

研究了模型构成函数的Lipschitzian性，给出了计算目标函数次梯度的方法，分析了算法思想、步骤，最后讨论了算法的收敛性。

A lagrangian relaxation heuristic algorithm based on subgradient optimization is applicated to solving the practical scale problems. The solution algorithms solving sub-problems are also listed.

提出了求解实际规模问题的基于次梯度优化的拉格朗日松弛启发式算法，并给出了各子问题的求解算法。

However, as the name（read－only memory）implies, CD disks cannot be written onorchanged in any way.

然而，正如其名字所指出的那样，CD盘不能写，也不能用任何方式改变其内容。

Galvanizes steel pallet is mainly export which suits standard packing of European Union, the North America. galvanizes steel pallet is suitable to heavy rack. Pallet surface can design plate type, corrugated and the gap form, satisfies the different requirements.

镀锌钢托盘多用于出口，替代木托盘，免薰蒸，符合欧盟、北美各国对出口货物包装材料的法令要求；喷涂钢托盘适用于重载上货架之用，托盘表面根据需要制作成平板状、波纹状及间隔形式，满足不同的使用要求。