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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|Introduction
We have chosen to finish this book by indicating many of the ways in which finite dimensionality has played a critical role in the previous chapters. While our list is far from complete it should help illuminate the places in which care is appropriate when “generalizing”. Many of our main results (on subgradients, variational principles, open mappings, Fenchel duality, metric regularity) immediately generalize to at least reflexive Banach spaces. When they do not, it is principally because the compactness properties and support properties of convex sets have become significantly more subtle. There are also significantly many properties that characterize Hilbert space. The most striking is perhaps the deep result that a Banach space $X$ is (isomorphic to) Hilbert space if and only if every closed vector subspace is complemented in $X$. Especially with respect to best approximation properties, it is Hilbert space that best captures the properties of Euclidean space.
Since this chapter will be primarily helpful to those with some knowledge of Banach space functional analysis, we make use of a fair amount of standard terminology without giving details. In the exercises more specific cases are considered.
数学代考|凸分析作业代写Convex Analysis代考|Finite Dimensionality
We begin with a compendium of standard and relatively easy results whose proofs may be pieced together from many sources. Sometimes the separable version of these results is simpler.
Theorem 10.2.1 (Closure, continuity, and compactness) The following statements are equivalent:
(i) $X$ is finite-dimensional.
(ii) Every vector subspace of $X$ is closed.
(iii) Every linear map taking values in $X$ has closed range.
(iv) Every linear functional on $X$ is continuous.
(v) Every convex function $f: X \rightarrow \mathbf{R}$ is continuous.
(vi) The closed unit ball in $X$ is (pre-)compact.
(vii) For each nonempty closed set $C$ in $X$ and for each $x$ in $X$, the distance
$$
d_{C}(x)=\inf \{\|x-y\| \mid y \in C\}
$$
is attained.
(viii) The weak and norm topologies coincide on $X$.
(ix) The weak-star and norm topologies coincide on $X^{*}$.
Turning from continuity to tangency properties of convex sets we have the following result.
数学代考|凸分析作业代写Convex Analysis代考|Counterexamples and Exercises
In the previous section we saw that nonempty closed convex subsets $S$ of the Euclidean space $\mathbf{E}$ are characterized by the attractive global property that every point in $\mathbf{E}$ has a unique nearest point in $S$. The corresponding local property is also a useful tool: we begin with a condition guaranteeing this property.
We call the closed set $S$ prox-regular at a point $\bar{x}$ in $S$ if there exists a constant $\rho>0$ such that all distinct points $x, x^{\prime} \in S$ near $\bar{x}$ and small vectors $v \in N_{S}(x)$ satisfy the inequality
$$
\left\langle v, x^{\prime}-x\right\rangle<\rho\left\|x^{\prime}-x\right\|^{2}
$$
Geometrically, this condition states that the ball centered at the point $x+\frac{1}{2 \rho} v$ containing the point $x$ on its boundary has interior disjoint from $S$.1. (Absorbing sets) A convex set $C$ satisfying $X=\cup\{t C \mid t \geq 0\}$ is said to be absorbing (and zero is said to be in the core of $C$ ).
(a) A normed space is said to be barreled if every closed convex absorbing subset $C$ has zero in its interior. Use the Baire category theorem to show that Banach spaces are barreled. (There are normed spaces which are barreled but in which the Baire category theorem fails, and there are Baire normed spaces which are not complete: appropriate dense hyperplanes and countable codimension subspaces will do the job.)
(b) Let $f$ be proper lower semicontinuous and convex. Suppose that zero lies in the core of the domain of $f$. By considering the set
$$
C=\{x \in X \mid f(x) \leq 1\},
$$
deduce that $f$ is continuous at zero.
(c) Show that an infinite-dimensional Banach space cannot be written as a countable union of finite-dimensional subspaces, and so cannot have a countable but infinite vector space basis.
(d) Let $X=\ell_{2}(\mathbf{N})$ and let $C=\left\{x \in X|| x_{n} \mid \leq 2^{-n}\right\}$. Show $X \neq \bigcup\{t C \mid t \geq 0\}$ but $X=\operatorname{cl} \bigcup\{t C \mid t \geq 0\} .$
(e) Let $X=\ell_{p}(\mathbf{N})$ for $1 \leq p<\infty$. Let
$$
C=\left\{x \in X|| x_{n} \mid \leq 4^{-n}\right\},
$$
and let
$$
D=\left\{x \in X \mid x_{n}=2^{-n} t, t \geq 0\right\}
$$
Show $C \cap D=\{0\}$, and so
$$
T_{C \cap D}(0)=\{0\}
$$
but
$$
T_{C}(0) \cap T_{D}(0)=D .
$$
数学代考|凸分析作业代写CONVEX ANALYSIS代考|INTRODUCTION
我们选择通过指出有限维在前几章中发挥关键作用的许多方式来结束本书。虽然我们的清单还远未完成，但它应该有助于阐明在“概括”时需要注意的地方。我们的许多主要成果这ns你bGr一种d一世和n吨s,v一种r一世一种吨一世这n一种一世pr一世nC一世p一世和s,这p和n米一种pp一世nGs,F和nCH和一世d你一种一世一世吨是,米和吨r一世Cr和G你一世一种r一世吨是立即推广到至少自反巴拿赫空间。当他们不这样做时，主要是因为凸集的紧致特性和支持特性变得更加微妙。希尔伯特空间也有很多特性。最引人注目的可能是 Banach 空间的深层结果X是一世s这米这rpH一世C吨这希尔伯特空间当且仅当每个闭合向量子空间都在X. 特别是关于最佳逼近性质，希尔伯特空间最能捕捉欧几里得空间的性质。
由于本章主要对那些对 Banach 空间泛函分析有一定了解的人有所帮助，因此我们使用了相当多的标准术语而不提供细节。在练习中会考虑更具体的情况。
数学代考|凸分析作业代写CONVEX ANALYSIS代考|FINITE DIMENSIONALITY
我们从一个标准的和相对简单的结果概要开始，其证明可以从许多来源拼凑起来。有时这些结果的可分离版本更简单。
定理 10.2.1C一世这s你r和,C这n吨一世n你一世吨是,一种ndC这米p一种C吨n和ss以下语句是等效的：
一世 X是有限维的。
一世一世的每个向量子空间X已经关闭。
一世一世一世每个线性映射都取值X有封闭范围。
一世v每个线性泛函X是连续的。
v每个凸函数F:X→R是连续的。
v一世封闭单元球X是pr和−袖珍的。
v一世一世对于每个非空闭集C在X并且对于每个X在X， 距离
dC(X)=信息{‖X−是‖∣是∈C}
达到。
v一世一世一世弱拓扑和规范拓扑重合X.
一世X弱星拓扑和范数拓扑重合X∗.
从凸集的连续性转向相切性，我们得到以下结果。
数学代考|凸分析作业代写CONVEX ANALYSIS代考|COUNTEREXAMPLES AND EXERCISES
在上一节中，我们看到非空闭凸子集小号欧几里得空间和其特点是具有吸引力的全局属性，每个点和有一个唯一的最近点小号. 相应的局部属性也是一个有用的工具：我们从保证该属性的条件开始。
我们称闭集小号prox-regular at a pointX¯在小号如果存在一个常数ρ>0这样所有不同的点X,X′∈小号靠近X¯和小向量v∈ñ小号(X)满足不等式
⟨v,X′−X⟩<ρ‖X′−X‖2
从几何上看，这个条件表明球以该点为中心X+12ρv包含点X在其边界上与小号.1.一种bs这rb一世nGs和吨s凸集C令人满意的X=∪{吨C∣吨≥0}据说很吸睛一种nd和和r这一世ss一种一世d吨这b和一世n吨H和C这r和这F$C$.
一种如果每个封闭的凸吸收子集，则称规范空间是桶化的C其内部为零。使用 Baire 范畴定理证明 Banach 空间是桶形的。吨H和r和一种r和n这r米和dsp一种C和s在H一世CH一种r和b一种rr和一世和db你吨一世n在H一世CH吨H和乙一种一世r和C一种吨和G这r是吨H和这r和米F一种一世一世s,一种nd吨H和r和一种r和乙一种一世r和n这r米和dsp一种C和s在H一世CH一种r和n这吨C这米p一世和吨和:一种ppr这pr一世一种吨和d和ns和H是p和rp一世一种n和s一种ndC这你n吨一种b一世和C这d一世米和ns一世这ns你bsp一种C和s在一世一世一世d这吨H和j这b.
b让F是适当的下半连续和凸的。假设零位于域的核心F. 通过考虑集合
$$
C={x \in X \mid f(x) \leq 1},
$$
deduce that $f$ is continuous at zero.
(c) Show that an infinite-dimensional Banach space cannot be written as a countable union of finite-dimensional subspaces, and so cannot have a countable but infinite vector space basis.
(d) Let $X=\ell_{2}(\mathbf{N})$ and let $C=\left{x \in X|| x_{n} \mid \leq 2^{-n}\right}$. Show $X \neq \bigcup{t C \mid t \geq 0}$ but $X=\operatorname{cl} \bigcup{t C \mid t \geq 0} .$
(e) Let $X=\ell_{p}(\mathbf{N})$ for $1 \leq p<\infty$. Let
$$
C=\left{x \in X|| x_{n} \mid \leq 4^{-n}\right},
$$
and let
$$
D=\left{x \in X \mid x_{n}=2^{-n} t, t \geq 0\right}
$$
Show $C \cap D={0}$, and so
$$
T_{C \cap D}(0)={0}
$$
but
$$
T_{C}(0) \cap T_{D}(0)=D .
$$
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