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# 数学代考| Convex Cones 运筹学代写

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## 运筹学代写

Conic Linear Programming, hereafter CLP, is a natural extension of Linear programming (LP). In LP, the variables form a vector which is required to be component-wise nonnegative, while in CLP they are points in a pointed convex cone (see Appendix B.1) of an Euclidean space, such as vectors as well as matrices of finite dimensions. For example, Semidefinite programming (SDP) is a kind of CLP, where the variable points are symmetric matrices constrained to be positive semidefinite. Both types of problems may have linear equality constraints as well. Although CLPs have long been known to be convex optimization problems, no efficient solution algorithm was known until about two decades ago, when it was discovered that interior-point algorithms for LP discussed in Chap. 5 , can be adapted to solve certain CLPs with both theoretical and practical efficiency. During the same to solve certain CLPs with both theoretical and practical efficiency. During the same period, it was discovered that CLP, especially SDP, is representative of a wide assortment of applications, including combinatorial optimization, statistical comoptimal control, etc. CLP is now widely recognized as a powerful mathematical computation model of general importance.

First, we illustrate several convex cones popularly used in conic linear optimization.
Example 1 The followings are all (closed) convex cones.

• The $n$-dimensional nonnegative orthant, $E_{+}^{n}=\left{\mathbf{x} \in E^{n}: \mathbf{x} \geq 0\right}$, is a convex cone.
( Springer Nature Switzerland AG 2021 D. G. Luenberger, Y. Ye, Linear and Nonlinear Programming, International Series in Operations Research \& Management Science 228, https://doi.org/10.1007/978-3-030-85450-8_6 – The set $\left{(u ; \mathbf{x}) \in E^{n+1}: u \geq|\mathbf{x}|_{p}\right}$ is a convex cone in $E^{n+1}$, called the $p$-order
• The set of all $n$-dimensional symmetric positive semidefinite matrices, denoted by $\mathcal{S}_{+}^{n}$, is a convex cone, called the positive semidefinite matrix cone. When $\mathrm{X}$ is positive semidefinite (positive definite), we often write the property as $\mathbf{X} \succeq(\succ) 0$. cone where $1 \leq p<\infty$. When $p=2$, the cone is called second-order cone or “Ice-cream” cone.

Sometimes, we use the notion of conic inequalities $\mathbf{P} \succeq_{K} \mathbf{Q}$ or $\mathbf{Q} \preceq_{K} \mathbf{P}$, in which cases we simply mean $\mathbf{P}-\mathbf{Q} \in K$.
Suppose $\mathbf{A}$ and $\mathbf{B}$ are $k \times n$ matrices. We define the inner product
$$\mathbf{A} \bullet \mathbf{B}=\operatorname{trace}\left(\mathbf{A}^{T} \mathbf{B}\right)=\sum_{i, j} a_{i j} b_{i j}$$
When $k=1$, they become $n$-dimensional vectors and the inner product is the standard dot product of two vectors. In SDP, this definition is almost always used for the case where the matrices are both square and symmetric. The matrix norm associated with the inner product is called Frobenius norm:
$$|\mathbf{X}|_{f}=\sqrt{\mathbf{X} \bullet \mathbf{X}}$$
For a cone $K$, the dual of $K$ is the cone
$$K^{*}:={\mathbf{Y}: \mathbf{X} \bullet \mathbf{Y} \geq 0 \text { for all } \mathbf{X} \in K}$$
It is not difficult to see that the dual cones of the first two cones in Example 1 are all them self, respectively; while the dual cone of the $p$-order cone is the $q$-order cone where
$$\frac{1}{p}+\frac{1}{q}=1$$
One can see that when $p=2, q=2$ as well; that is, they are both 2 -order cones. For a closed convex cone $K$, the dual of the dual cone is itself.

• $n$ 维非负正数，$E_{+}^{n}=\left{\mathbf{x} \in E^{n}: \mathbf{x} \geq 0\right}$ , 是一个凸锥。
（Springer Nature Switzerland AG 2021 DG Luenberger, Y. Ye，线性和非线性规划，国际运筹学与管理科学系列 228，https://doi.org/10.1007/978-3-030-85450-8_6 – The设置 $\left{(u ; \mathbf{x}) \in E^{n+1}: u \geq|\mathbf{x}|_{p}\right}$ 是 $中的凸锥E^{n+1}$，称为$p$-order
• 所有 $n$ 维对称正半定矩阵的集合，记为 $\mathcal{S}_{+}^{n}$，是一个凸锥，称为正半定矩阵锥。当 $\mathrm{X}$ 为半正定（正定）时，我们常将性质写为 $\mathbf{X} \succeq(\succ) 0$。锥，其中 $1 \leq p<\infty$。当 $p=2$ 时，圆锥称为二阶圆锥或“冰淇淋”圆锥。

$$\mathbf{A} \bullet \mathbf{B}=\operatorname{trace}\left(\mathbf{A}^{T} \mathbf{B}\right)=\sum_{i, j} a_{ij} b_{ij}$$

$$|\mathbf{X}|_{f}=\sqrt{\mathbf{X} \bullet \mathbf{X}}$$

$$K^{*}:={\mathbf{Y}: \mathbf{X} \bullet \mathbf{Y} \geq 0 \text { for all } \mathbf{X} \in K}$$

$$\frac{1}{p}+\frac{1}{q}=1$$

## 什么是运筹学代写

• 确定需要解决的问题。
• 围绕问题构建一个类似于现实世界和变量的模型。
• 使用模型得出问题的解决方案。
• 在模型上测试每个解决方案并分析其成功。
• 实施解决实际问题的方法。

## 运筹学代写的三个特点

• 优化——运筹学的目的是在给定的条件下达到某一机器或者模型的最佳性能。优化还涉及比较不同选项和缩小潜在最佳选项的范围。
• 模拟—— 这涉及构建模型，以便在应用解决方案刀具体的复杂大规模问题之前之前尝试和测试简单模型的解决方案。
• 概率和统计——这包括使用数学算法和数据挖掘来发现有用的信息和潜在的风险，做出有效的预测并测试可能的解决方法。