cutting plane

简明释义

剖切面

英英释义

例句

1.The cutting plane is essential for determining the intersection of geometric shapes.

切割平面对于确定几何形状的交点至关重要。

2.To optimize the manufacturing process, we need to analyze the cutting plane used in production.

为了优化制造过程，我们需要分析生产中使用的切割平面。

3.The engineer used a cutting plane to refine the shape of the metal part.

工程师使用了一个切割平面来精细化金属部件的形状。

4.In geometry, a cutting plane can help visualize three-dimensional objects.

在几何学中，切割平面可以帮助可视化三维物体。

5.The architect designed a cutting plane that intersects the building at a unique angle.

建筑师设计了一个以独特角度与建筑相交的切割平面。

作文

In the field of optimization and computational mathematics, the concept of a cutting plane plays a crucial role in solving various types of linear programming problems. A cutting plane is essentially a hyperplane that separates feasible solutions from infeasible ones, thus helping to refine the search space for optimal solutions. This technique is particularly useful in integer programming, where the goal is to find integer solutions to linear inequalities. By introducing a cutting plane, we can eliminate certain portions of the solution space that do not contain any feasible integer solutions, thereby streamlining the optimization process.The history of cutting planes dates back to the work of mathematicians such as Ralph Gomory in the 1950s, who developed the first cutting plane algorithms. These algorithms have since evolved and are now widely implemented in various optimization software. The fundamental idea is to iteratively add cutting planes to the problem formulation, which progressively narrows down the set of possible solutions until an optimal solution is found.One of the key advantages of using cutting planes is their ability to improve the efficiency of the optimization process. Traditional methods may require examining all potential solutions, which can be computationally expensive, especially for large-scale problems. By contrast, cutting planes allow us to focus only on the most promising regions of the solution space, significantly reducing the number of necessary calculations.Moreover, cutting planes can be derived from various sources, including valid inequalities and relaxations of the original problem. For example, in a typical integer programming scenario, one might start with a relaxation of the problem that allows for fractional solutions. Once these solutions are obtained, cutting planes can be generated to exclude those fractional solutions that do not correspond to integer values, thus guiding the search towards feasible integer solutions.Another important aspect of cutting planes is their application in real-world problems. Many industries, from logistics to finance, rely on optimization techniques to make informed decisions. For instance, in supply chain management, companies must determine the most efficient way to distribute products while minimizing costs. By employing cutting planes, they can quickly arrive at optimal distribution strategies that take into account various constraints, such as delivery times and vehicle capacities.In conclusion, the concept of a cutting plane is integral to modern optimization techniques. It allows for the efficient narrowing of solution spaces in complex mathematical problems, making it easier to find optimal solutions in various applications. As computational power continues to advance, the use of cutting planes is likely to become even more prevalent, driving innovation across multiple fields and enhancing our ability to solve intricate problems effectively. Understanding and mastering the concept of cutting planes is essential for anyone looking to delve deeper into optimization and computational mathematics, as it offers powerful tools for tackling a wide range of challenges.

在优化和计算数学领域，切割平面的概念在解决各种线性规划问题中发挥着至关重要的作用。切割平面本质上是一个超平面，它将可行解与不可行解分开，从而帮助精炼最优解的搜索空间。这种技术在整数规划中尤为有用，其目标是找到线性不等式的整数解。通过引入切割平面，我们可以消除不包含任何可行整数解的解空间某些部分，从而简化优化过程。切割平面的历史可以追溯到20世纪50年代拉尔夫·戈莫里等数学家的工作，他开发了第一个切割平面算法。这些算法自那时以来不断演变，现在广泛应用于各种优化软件中。其基本思想是迭代地向问题公式中添加切割平面，逐步缩小可能解的集合，直到找到最优解。使用切割平面的一个主要优点是能够提高优化过程的效率。传统方法可能需要检查所有潜在解，这在大规模问题中可能计算开销巨大。相比之下，切割平面使我们能够专注于解空间中最有希望的区域，大大减少了所需计算的数量。此外，切割平面可以从各种来源派生，包括有效不等式和原始问题的放松。例如，在典型的整数规划场景中，人们可能会从允许分数解的问题放松开始。一旦获得这些解，就可以生成切割平面来排除那些不对应于整数值的分数解，从而指导搜索朝向可行的整数解。切割平面的另一个重要方面是它在现实世界问题中的应用。许多行业，从物流到金融，都依赖于优化技术来做出明智的决策。例如，在供应链管理中，公司必须确定以最小化成本的最有效方式分配产品。通过采用切割平面，他们可以快速得出考虑各种约束（如交货时间和车辆容量）的最优分配策略。总之，切割平面的概念是现代优化技术的核心。它允许在复杂数学问题中有效缩小解空间，使得更容易在各种应用中找到最优解。随着计算能力的不断提升，切割平面的使用可能会变得更加普遍，推动多个领域的创新，并增强我们有效解决复杂问题的能力。理解和掌握切割平面的概念对于任何希望深入研究优化和计算数学的人来说都是必不可少的，因为它提供了强大的工具来应对各种挑战。