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# 数学代写|凸优化代写Convex Optimization代考|SS2022 Simple observation schemes—Main result

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

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