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## 数学代写|凸优化代写Convex Optimization代考|Executive summary of convex-concave saddle point problems

The results to follow are absolutely standard, and their proofs can be found in all textbooks on the subject, see, e.g., [221] or [15, Section D.4].

Let $U$ and $V$ be nonempty sets, and let $\Phi: U \times V \rightarrow \mathbf{R}$ be a function. These data define an antagonistic game of two players, I and II, where player I selects a point $u \in U$, and player II selects a point $v \in V$; as an outcome of these selections, player I pays to player II the sum $\Phi(u, v)$. Clearly, player I is interested in minimizing this payment, and player II in maximizing it. The data $U, V, \Phi$ are known to the players in advance, and the question is, what should be their selections?

When player I makes his selection $u$ first, and player II makes his selection $v$ with $u$ already known, player I should be ready to pay for a selection $u \in U$ a toll as large as
$$\bar{\Phi}(u)=\sup {v \in V} \Phi(u, v) .$$ In this situation, a risk-averse player I would select $u$ by minimizing the above worst-case payment, by solving the primal problem $$\operatorname{Opt}(P)=\inf {u \in U} \bar{\Phi}(u)=\inf {u \in U} \sup {v \in V} \Phi(u, v)$$
associated with the data $U, V, \Phi$.

## 数学代写|凸优化代写Convex Optimization代考|Main result

Theorem 2.23. Let
$$\mathcal{O}=\left(\Omega, \Pi ;\left{p_{\mu}: \mu \in \mathcal{M}\right} ; \mathcal{F}\right)$$
be a simple observation scheme, and let $M_{1}, M_{2}$ be nonempty compact convex subsets of $\mathcal{M}$. Then
(i) The function
$$\Phi(\phi,[\mu ; \nu])=\frac{1}{2}\left[\ln \left(\int_{\Omega} \mathrm{e}^{-\phi(\omega)} p_{\mu}(\omega) \Pi(d \omega)\right)+\ln \left(\int_{\Omega} \mathrm{e}^{\phi(\omega)} p_{\nu}(\omega) \Pi(d \omega)\right)\right]:$$
is continuous on its domain, convex in $\phi(\cdot) \in \mathcal{F}$, concave in $[\mu ; \nu] \in M_{1} \times M_{2}$, and possesses a saddle point $\left(\min\right.$ in $\phi \in \mathcal{F}, \max$ in $\left.[\mu ; \nu] \in M_{1} \times M_{2}\right)\left(\phi_{}(\cdot),\left[\mu_{} ; \nu_{}\right]\right)$ on $\mathcal{F} \times\left(M_{1} \times M_{2}\right)$. W.l.o.g. $\phi_{}$ can be assumed to satisfy the relation ${ }^{8}$
$$\int_{\Omega} \exp \left{-\phi_{}(\omega)\right} p_{\mu_{}}(\omega) \Pi(d \omega)=\int_{\Omega} \exp \left{\phi_{}(\omega)\right} p_{\nu_{}}(\omega) \Pi(d \omega) .$$

## 数学代写|凸优化代写Convex Optimization代考|Executive summary of convex-concave saddle point problems

$$\bar{\Phi}(u)=\sup v \in V \Phi(u, v) .$$

$$\operatorname{Opt}(P)=\inf u \in U \bar{\Phi}(u)=\inf u \in U \sup v \in V \Phi(u, v)$$

## 数学代写|凸优化代写Convex Optimization代考|Main result

\left 的分隔符缺失或无法识别

(i) 函数
$$\Phi(\phi,[\mu ; \nu])=\frac{1}{2}\left[\ln \left(\int_{\Omega} \mathrm{e}^{-\phi(\omega)} p_{\mu}(\omega) \Pi(d \omega)\right)+\ln \left(\int_{\Omega} \mathrm{e}^{\phi(\omega)} p_{\nu}(\omega) \Pi(d \omega)\right)\right]:$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。