convex set

简明释义

凸集

英英释义

例句

1.In optimization problems, we often seek solutions within a convex set 凸集 to ensure that any local minimum is also a global minimum.

在优化问题中，我们通常希望在一个 convex set 凸集 中寻找解，以确保任何局部最小值也是全局最小值。

2.When analyzing economic models, we often assume preferences are represented by a convex set 凸集 to reflect diminishing marginal returns.

在分析经济模型时，我们通常假设偏好由一个 convex set 凸集 表示，以反映边际收益递减的情况。

3.The feasible region of a linear programming problem is always a convex set 凸集, allowing for efficient algorithms to find optimal solutions.

线性规划问题的可行区域总是一个 convex set 凸集，这使得有效算法能够找到最优解。

4.In geometry, a circle is an example of a convex set 凸集 because any line segment connecting two points within the circle lies entirely inside it.

在几何学中，圆是一个 convex set 凸集 的例子，因为连接圆内两个点的任何线段都完全位于圆内。

5.In machine learning, the decision boundary created by a support vector machine can be thought of as separating data points in a convex set 凸集.

在机器学习中，支持向量机创建的决策边界可以被视为在一个 convex set 凸集 中分离数据点。

作文

In mathematics, the concept of a convex set plays a crucial role in various fields, including geometry, optimization, and economics. A convex set is defined as a subset of a vector space that satisfies a particular property: for any two points within the set, the line segment connecting these points lies entirely within the set. This definition may seem abstract at first, but it has profound implications in practical applications. To understand this concept better, let us consider a simple example. Imagine a flat, two-dimensional surface, like a piece of paper. If we draw a circle on this paper, the area inside the circle represents a convex set. No matter which two points we choose inside the circle, the straight line connecting them will always remain inside the circle. However, if we take a crescent shape instead, this would not be a convex set because there exist points within the crescent where the connecting line would fall outside the shape. The significance of convex sets extends beyond mere geometry. In optimization problems, particularly in linear programming, many feasible regions are defined as convex sets. This means that if you are trying to find the best solution to a problem—say, maximizing profit or minimizing cost—you can be assured that the optimal solutions lie at the vertices of the convex set. This property simplifies the process of finding optimal solutions, making it easier to analyze and solve complex problems. Moreover, convex sets are essential in economic theory, particularly in consumer choice and production theory. For instance, when analyzing preferences, economists often assume that a consumer's set of preferred bundles of goods forms a convex set. This assumption implies that consumers prefer averages over extremes, leading to more stable demand functions. In production, firms may face constraints that can also be modeled as convex sets, allowing for efficient resource allocation. In conclusion, the concept of a convex set is fundamental in various mathematical and applied contexts. Its defining characteristic—that any line segment between two points within the set remains entirely within the set—provides a powerful framework for understanding and solving problems in geometry, optimization, and economics. By recognizing the properties and implications of convex sets, we can better analyze complex systems and make informed decisions in both theoretical and practical scenarios.

在数学中，凸集的概念在几何、优化和经济学等多个领域中扮演着至关重要的角色。凸集被定义为向量空间的一个子集，满足一个特定的属性：对于集合中的任意两个点，连接这两个点的线段完全位于该集合内。这个定义乍一看可能显得抽象，但在实际应用中却具有深远的意义。为了更好地理解这一概念，让我们考虑一个简单的例子。想象一下一个平坦的二维表面，就像一张纸。如果我们在这张纸上画一个圆，那么圆内的区域就代表了一个凸集。无论我们选择圆内的哪两个点，它们之间的直线连接将始终保持在圆内。然而，如果我们取一个新月形状，这就不是一个凸集，因为在新月的某些点之间，连接线会落在形状外部。凸集的重要性超越了单纯的几何学。在优化问题中，特别是在线性规划中，许多可行区域被定义为凸集。这意味着如果你试图找到一个问题的最佳解决方案——例如，最大化利润或最小化成本——你可以放心，最佳解决方案位于凸集的顶点。这一特性简化了寻找最优解的过程，使得分析和解决复杂问题变得更加容易。此外，凸集在经济理论中也是至关重要的，尤其是在消费者选择和生产理论中。例如，在分析偏好时，经济学家通常假设消费者所偏好的商品组合集形成一个凸集。这一假设意味着消费者更喜欢平均而非极端，从而导致需求函数更为稳定。在生产中，企业可能面临的约束也可以建模为凸集，从而实现资源的有效配置。总之，凸集的概念在多个数学和应用背景中是基础性的。其定义特征——即集合内任意两点之间的线段完全位于该集合内——为理解和解决几何、优化和经济学中的问题提供了强大的框架。通过认识凸集的属性和影响，我们可以更好地分析复杂系统，并在理论和实践场景中做出明智的决策。