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Secondly, a finite element method is used to approximate the constrained minimization problem, both existence of minimizer for cost functional in finite demensional space and the convergence of minimizer are proven.

然后利用有限元方法对上述定解问题进行离散，并证明了有限维空间上泛函的极小元存在以及有限维空间上的极小元收敛到无限维空间上的极小元。

Based upon all above-mentioned, a mathematical representation for multi-material structures is proposed by means of vector level sets, and the general structure topology optimization can be expressed by a constrained functional minimization problem of a set of level set functions.

同时，还提出了在有效约束切空间上构造的非线性映射技术，以改进向量水平集的演化方向，可提高计算效率2到3倍。

The layer assignment, also called constrained via minimization, is to determine which layers can be used for routing the wire segments such that the number of vias can be minimized.

布局层指定是决定布局中各线段的布局层位置使得布局所产生的穿孔数能尽量少。

As for 3-layer constrained via minimization problem (3CVM), a heuristic approach on the basis of segment-crossing graph model is presented. Two types of algorithm, global sense and local sense are combined into our algorithm.

对三层布线的通孔最少化问题（3CVM），提出了一个启发式算法，以线段—相交图为模型，分总体优化和局部优化两步对三层详细布线进行通孔优化处理。

Thirdly, by applying Frenchet derivative analysis, the iteration formula is derived to solve the constrained functional minimization problem, essentially the formula is a level set equation, in which the motion velocity of the level set is the shape sensitivity of material interfaces.

为了有效处理违反约束，这里还提出了在违反约束水平集的负法凸锥内构造水平集的法向修正速度的返回映射算法，它可以方便地处理多材料、多约束和含有非设计域的结构设计问题，使初值给定容易。

Using level-set method, mathematical representation for contimuum structures is proposed by means of the vector of level-set, and the general structure topology optimization can be expressed by a constrained functional minimization problem of a set of level set functions.

其次利用水平集方法将一般拓扑优化问题描述为一组水平集函数的约束泛函极小化问题，应用敏度分析，给出了此泛函极小化数值迭代求解公式，即水平集演化方程。

By establishing MINLP model, the computation of phase equilibrium is transformed into a constrained two-level minimization problem. On the inner level, equilibrium composition is computed; On the outer level, the optimum phase combination configuration is searched for. The whole solving process is achieved by Genetic Algorithm.

通过建立MINLP模型，将相平衡计算问题转化为有约束的双层最优化问题，内层用来计算给定相组合结构下的平衡组成，外层则是搜索最优的相组合结构，整个求解过程通过遗传算法来实现。

In this paper, we focused on the Thermal Power Generator unit commitment optimization. A mathematical model is presented and consuming minimization of generator is chosen for its objective function, with the power balance, generation capacity limits, maximum allowable start and stop times limits and time limits taken into consideration. The unit commitment optimization is constrained minimization problem.

本文主要对火电机组优化组合问题进行了研究，提出了机组优化组合的数学模型，该模型以发电总成本最小为目标函数，同时考虑了功率平衡约束、机组输出功率上、下限约束，机组最大允许启停次数和时间约束等约束条件。

Our approach consists of the following steps: providing a sketch based mapping between the sources and the target; deforming the target based on the deformations of the corresponding sources by means of differential mean value coordinates; smoothing the deformed target with constrained minimization method to form a key frame of the target animation; and interpolating between key frames created from to produce a complete 3D animation of the target.

该框架由以下几个步骤组成：使用草图在源网格和目标网格间建立映射；使用非闭合网格均值坐标，根据源动画的运动风格来变形目标网格；对变形后的目标网格求解最小化约束函数以获取光滑的目标网格，从而生成目标动画的关键帧；对由产生的结果进行插值，产生完整的目标对象的三维动画。

By applying Frenchet derivative analysis, the iteration formula is derived to solve the constrained functional minimization problem, essentially the formula is a level set equation, in which the motion velocity of the level set is the shape sensitivity of material interfaces.

这样就把材料界面表示为一组高维函数的水平集，使拓扑优化问题转化为材料界面追踪问题，这时材料界面的形状敏度就(来源：9aABb5C论文网www.abclunwen.com)是水平集的运动速度。

The statistical analysis of this large set of mutations has led us to propose a diagnostic strategy that should help with the molecular work-up of optic neuropathies.

这一大系列基因突变的统计学分析引导我们提出一个诊断策略，可以帮助我们对视神经病变进行分子学的诊断检查。

The results show that the pipe grouting changes the mechanical parameters of surrounding rocks to achieve the security of engineering objectives by reinforcing surrounding rocks and bolcking the water.

研究表明：小导管注浆一方面改变了围岩的力学参数，另一方面利用浆液的固结特性可以实现堵水和加固围岩的工程目标，使隧道在开挖过程中的安全得到保障。