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我们提供的凸分析Convex Analysis及其相关学科的代写，服务范围广, 其中包括但不限于:、
- 优化理论 optimization theory
- 变分法 Calculus of variations
- 最优控制理论 Optimal control
- 动态规划 Dynamic programming
- 鲁棒优化 Robust optimization
- 随机优化 Stochastic programming
- 组合优化 Combinatorial optimization
数学代考|凸分析作业代写Convex Analysis代考|Continuity of Convex Functions
We have already seen that linear functions are always continuous. More generally, a remarkable feature of convex functions on $\mathbf{E}$ is that they must be continuous on the interior of their domains. Part of the surprise is that an algebraic/geometric assumption (convexity) leads to a topological conclusion (continuity). It is this powerful fact that guarantees the usefulness of regularity conditions like $A$ dom $f \cap \operatorname{cont} g \neq \emptyset(3.3 .9)$, which we studied in the previous section.
Clearly an arbitrary function $f$ is bounded above on some neighbourhood of any point in cont $f$. For convex functions the converse is also true, and in a rather strong sense, needing the following definition. For a real $L \geq 0$, we say that a function $f: \mathbf{E} \rightarrow(\infty,+\infty]$ is Lipschitz (with constant $L)$ on a subset $C$ of $\operatorname{dom} f$ if $|f(x)-f(y)| \leq L\|x-y\|$ for any points $x$ and $y$ in $C$. If $f$ is Lipschitz on a neighbourhood of a point $z$ then we say that $f$ is locally Lipschitz around $z$. If $\mathbf{Y}$ is another Euclidean space we make analogous definitions for functions $F: \mathbf{E} \rightarrow \mathbf{Y}$, with $\|F(x)-F(y)\|$ replacing $|f(x)-f(y)|$.
数学代考|凸分析作业代写Convex Analysis代考|Fenchel Biconjugation
We have seen that many important convex functions $h: \mathbf{E} \rightarrow(\infty,+\infty]$ agree identically with their biconjugates $h^{* *}$. Table $3.1$ in Section $3.3$ lists many one-dimensional examples, and the Bipolar cone theorem (3.3.14) shows $\delta_{K}=\delta_{K}^{* *}$ for any closed convex cone $K$. In this section we isolate exactly the circumstances when $h=h^{* *}$.
We can easily check that $h^{* *}$ is a minorant of $h$ (that is, $h^{* *} \leq h$ pointwise). Our specific aim in this section is to find conditions on a point $x$ in $\mathbf{E}$ guaranteeing $h^{* *}(x)=h(x)$. This becomes the key relationship for the study of duality in optimization. As we see in this section, the conditions we need are both geometric and topological. This is neither particularly surprising or stringent. Since any conjugate function must have a closed convex epigraph, we cannot expect a function to agree with its biconjugate unless the function itself has a closed convex epigraph. On the other hand, this restriction is not particularly strong since, as we saw in the previous section, convex functions automatically have strong continuity properties.
We say the function $h: \mathbf{E} \rightarrow[-\infty,+\infty]$ is closed if its epigraph is a closed set. We say $h$ is lower semicontinuous at a point $x$ in $\mathbf{E}$ if
$$
\liminf h\left(x^{r}\right)\left(=\lim _{s \rightarrow \infty} \inf _{r \geq s} h\left(x^{r}\right)\right) \geq h(x)
$$
for any sequence $x^{r} \rightarrow x$. A function $h: \mathbf{E} \rightarrow[-\infty,+\infty]$ is lower semicontinuous if it is lower semicontinuous at every point in $\mathbf{E}$; this is in fact equivalent to $h$ being closed, which in turn holds if and only if $h$ has closed level sets. Any two functions $h$ and $g$ satisfying $h \leq g$ (in which case we call $h$ a minorant of $g$ ) must satisfy $h^{*} \geq g^{*}$, and hence $h^{* *} \leq g^{* *}$.
数学代考|凸分析作业代写Convex Analysis代考|Lagrangian Duality
The duality between a convex function $h$ and its Fenchel conjugate $h^{*}$ which we outlined earlier is an elegant piece of theory. The real significance, however, lies in its power to describe duality theory for convex programs, one of the most far-reaching ideas in the study of optimization.
We return to the convex program that we studied in Section 3.2:
$$
\inf \{f(x) \mid g(x) \leq 0, x \in \mathbf{E}\}
$$
Here the function $f$ and the components $g_{1}, g_{2}, \ldots, g_{m}: \mathbf{E} \rightarrow(\infty,+\infty]$ are convex, and satisfy $\emptyset \neq \operatorname{dom} f \subset \cap_{1}^{m} \operatorname{dom} g_{i}$. As before, the Lagrangian function $L: \mathbf{E} \times \mathbf{R}_{+}^{m} \rightarrow(\infty,+\infty]$ is defined by $L(x ; \lambda)=f(x)+\lambda^{T} g(x)$.
Notice that the Lagrangian encapsulates all the information of the primal problem (4.3.1): clearly
$$
\sup _{\lambda \in \mathbf{R}_{+}^{m}} L(x ; \lambda)= \begin{cases}f(x) & \text { if } x \text { is feasible } \\ +\infty & \text { otherwise }\end{cases}
$$
so if we denote the optimal value of (4.3.1) by $p \in[-\infty,+\infty]$, we could rewrite the problem in the following form:
$$
p=\inf _{x \in \mathbf{E}} \sup _{\lambda \in \mathbf{R}_{+}^{m}} L(x ; \lambda) .
$$
This makes it rather natural to consider an associated problem
$$
d=\sup _{\lambda \in \mathbf{R}_{+}^{m}} \inf _{x \in \mathbf{E}} L(x ; \lambda)
$$
where $d \in[-\infty,+\infty]$ is called the dual value. Thus the dual problem consists of maximizing over vectors $\lambda$ in $\mathbf{R}_{+}^{m}$ the dual function $\Phi(\lambda)=$ $\inf _{x} L(x ; \lambda)$. This dual problem is perfectly well-defined without any assumptions on the functions $f$ and $g$. It is an easy exercise to show the “weak duality inequality” $p \geq d$. Notice $\Phi$ is concave.
It can happen that the primal value $p$ is strictly larger than the dual value $d$ (Exercise 5 ). In this case we say there is a duality gap. We next investigate conditions ensuring there is no duality gap. As in Section 3.2, the chief tool in our analysis is the primal value function $v: \mathbf{R}^{m} \rightarrow[-\infty,+\infty]$, defined by
$$
v(b)=\inf \{f(x) \mid g(x) \leq b\}
$$
数学代考|凸分析作业代写CONVEX ANALYSIS代考|CONTINUITY OF CONVEX FUNCTIONS
我们已经看到线性函数总是连续的。更一般地，凸函数的一个显着特征是和是它们在其域的内部必须是连续的。部分惊喜是代数/几何假设C这nv和X一世吨是导致拓扑结论C这n吨一世n你一世吨是. 正是这个强有力的事实保证了规律性条件的有用性，例如一种判断F∩帐户G≠∅(3.3.9)，我们在上一节中研究过。
显然是任意函数F在 cont 中任意点的某个邻域之上F. 对于凸函数，反之亦然，并且在相当强烈的意义上，需要以下定义。对于一个真正的 $L \geq 0$, we say that a function $f: \mathbf{E} \rightarrow(\infty,+\infty]$ is Lipschitz (with constant $L)$ on a subset $C$ of $\operatorname{dom} f$ if $|f(x)-f(y)| \leq L|x-y|$ for any points $x$ and $y$ in $C$. If $f$ is Lipschitz on a neighbourhood of a point $z$ then we say that $f$ is locally Lipschitz around $z$. If $\mathbf{Y}$ is another Euclidean space we make analogous definitions for functions $F: \mathbf{E} \rightarrow \mathbf{Y}$, with $|F(x)-F(y)|$ replacing $|f(x)-f(y)|$.
数学代考|凸分析作业代写CONVEX ANALYSIS代考|FENCHEL BICONJUGATION
我们已经看到了许多重要的凸函数H:和→(∞,+∞]与他们的双共轭完全一致H∗∗. 桌子3.1在部分3.3列出了许多一维示例，以及双极锥定理3.3.14节目d到=d到∗∗对于任何封闭的凸锥到. 在本节中，我们准确地隔离了以下情况：H=H∗∗.
我们可以很容易地检查H∗∗是未成年人H 吨H一种吨一世s,$H∗∗≤H$p这一世n吨在一世s和. 我们在本节中的具体目标是找到一个点的条件X在和保证H∗∗(X)=H(X). 这成为优化对偶性研究的关键关系。正如我们在本节中看到的，我们需要的条件既是几何条件又是拓扑条件。这既不是特别令人惊讶也不是特别严格。由于任何共轭函数都必须有一个闭凸上标，我们不能期望一个函数与它的双共轭一致，除非函数本身有一个闭凸上标。另一方面，这个限制并不是特别强，因为正如我们在上一节中看到的，凸函数自动具有很强的连续性。
我们说函数H:和→[−∞,+∞]如果它的题词是一个闭集，则它是闭集。我们说H在一点下半连续X在和如果
林infH(Xr)(=林s→∞信息r≥sH(Xr))≥H(X)
对于任何序列Xr→X. 一个函数H:和→[−∞,+∞]如果它在每个点上都是下半连续的，则它是下半连续的和; 这实际上相当于H被关闭，而当且仅当H有封闭的水平集。任意两个函数H和G令人满意的H≤G 一世n在H一世CHC一种s和在和C一种一世一世$H$一种米一世n这r一种n吨这F$G$必须满足H∗≥G∗， 因此H∗∗≤G∗∗.
数学代考|凸分析作业代写CONVEX ANALYSIS代考|LAGRANGIAN DUALITY
凸函数之间的对偶H及其芬切尔共轭H∗我们之前概述的是一个优雅的理论。然而，真正的意义在于它描述凸程序的对偶理论的能力，这是优化研究中影响最深远的思想之一。
我们回到我们在 3.2 节中研究的凸程序：
信息{F(X)∣G(X)≤0,X∈和}
这里的功能F和组件G1,G2,…,G米:和→(∞,+∞]是凸的，并且满足∅≠判断F⊂∩1米判断G一世. 和以前一样，拉格朗日函数大号:和×R+米→(∞,+∞]定义为大号(X;λ)=F(X)+λ吨G(X).
请注意，拉格朗日函数封装了原始问题的所有信息4.3.1： 清楚地
支持λ∈R+米大号(X;λ)={F(X) 如果 X 是可行的 +∞ 除此以外
所以如果我们表示最优值4.3.1经过p∈[−∞,+∞]，我们可以将问题改写为以下形式：
p=信息X∈和支持λ∈R+米大号(X;λ).
这使得考虑相关问题变得相当自然
d=支持λ∈R+米信息X∈和大号(X;λ)
在哪里d∈[−∞,+∞]称为对偶值。因此，对偶问题包括最大化向量λ在R+米双重功能披(λ)= 信息X大号(X;λ). 这个对偶问题是完美定义的，无需对函数进行任何假设F和G. 展示“弱对偶不等式”是一个简单的练习p≥d. 注意披是凹的。
原始值可能会发生p严格大于对偶值d 和X和rC一世s和5. 在这种情况下，我们说存在对偶性差距。我们接下来研究确保没有对偶间隙的条件。与第 3.2 节一样，我们分析的主要工具是原始价值函数v:R米→[−∞,+∞]， 被定义为
v(b)=信息{F(X)∣G(X)≤b}
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