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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|The Brouwer Fixed Point Theorem
Many questions in optimization and analysis reduce to solving a nonlinear equation $h(x)=0$, for some function $h: \mathbf{E} \rightarrow \mathbf{E}$. Equivalently, if we define another map $f=I-h$ (where $I$ is the identity map), we seek a point $x$ in E satisfying $f(x)=x$; we call $x$ a fixed point of $f$.
The most potent fixed point existence theorems fall into three categories: “geometric” results, devolving from the Banach contraction principle (which we state below), “order-theoretic” results (to which we briefly return in Section 8.3), and “topological” results, for which the prototype is the theorem of Brouwer forming the main body of this section. We begin with Banach’s result.
Given a set $C \subset \mathbf{E}$ and a continuous self map $f: C \rightarrow C$, we ask whether $f$ has a fixed point. We call $f$ a contraction if there is a real constant $\gamma_{f}<1$ such that
$$
\|f(x)-f(y)\| \leq \gamma_{f}\|x-y\| \text { for all } x, y \in C .
$$
数学代考|凸分析作业代写Convex Analysis代考|Selection and the Kakutani-Fan Fixed Point Theorem
The Brouwer fixed point theorem in the previous section concerns functions from a nonempty compact convex set to itself. In optimization, as we have already seen in Section 5.4, it may be convenient to broaden our language to consider multifunctions $\Omega$ from the set to itself and seek a fixed point-a point $x$ satisfying $x \in \Omega(x)$. To begin this section we summarize some definitions for future reference.
We consider a subset $K \subset \mathbf{E}$, a Euclidean space $\mathbf{Y}$, and a multifunction $\Omega: K \rightarrow \mathbf{Y}$. We say $\Omega$ is $U S C$ at a point $x$ in $K$ if every open set $U$ containing $\Omega(x)$ also contains $\Omega(z)$ for all points $z$ in $K$ close to $x$.
Thus a multifunction $\Omega$ is USC if for any sequence of points $\left(x_{n}\right)$ approaching $x$, any sequence of elements $y_{n} \in \Omega\left(x_{n}\right)$ is eventually close to $\Omega(x)$. If $\Omega$ is USC at every point in $K$ we simply call it USC. On the other hand, as in Section $5.4$, we say $\Omega$ is $L S C$ if, for every $x$ in $K$, every neighbourhood $V$ of any point in $\Omega(x)$ intersects $\Omega(z)$ for all points $z$ in $K$ close to $x$.
We refer to the sets $\Omega(x)(x \in K)$ as the images of $\Omega$. The multifunction $\Omega$ is a cusco if it is USC with nonempty compact convex images. Clearly such multifunctions are locally bounded: any point in $K$ has a neighbourhood whose image is bounded. Cuscos appear in several important optimization contexts. For example, the Clarke subdifferential of a locally Lipschitz function is a cusco (Exercise 5 ).
To see another important class of examples we need a further definition. We say a multifunction $\Phi: \mathbf{E} \rightarrow \mathbf{E}$ is monotone if it satisfies the condition $\langle u-v, x-y\rangle \geq 0$ whenever $u \in \Phi(x)$ and $v \in \Phi(y) .$
In particular, any (not necessarily self-adjoint) positive semidefinite linear operator is monotone, as is the subdifferential of any convex function. One multifunction contains another if the graph of the first contains the graph of the second. We say a monotone multifunction is maximal if the only monotone multifunction containing it is itself. The subdifferentials of closed proper convex functions are examples (see Exercise 16). Zorn’s lemma (which lies outside our immediate scope) shows any monotone multifunction is contained in a maximal monotone multifunction.
数学代考|凸分析作业代写Convex Analysis代考|Variational Inequalities
At the very beginning of this book we considered the problem of minimizing a differentiable function $f: \mathbf{E} \rightarrow \mathbf{R}$ over a convex set $C \subset \mathbf{E}$. A necessary optimality condition for a point $x_{0}$ in $C$ to be a local minimizer is
$$
\left\langle\nabla f\left(x_{0}\right), x-x_{0}\right\rangle \geq 0 \text { for all points } x \text { in } C,
$$
or equivalently
$$
0 \in \nabla f\left(x_{0}\right)+N_{C}\left(x_{0}\right) .
$$
If the function $f$ is convex instead of differentiable, the necessary and sufficient condition for optimality (assuming a constraint qualification) is
$$
0 \in \partial f\left(x_{0}\right)+N_{C}\left(x_{0}\right),
$$
and there are analogous nonsmooth necessary conditions.
We call problems like (8.3.1) “variational inequalities”. Let us fix a multifunction $\Omega: C \rightarrow \mathbf{E}$. In this section we use the fixed point theory we have developed to study the multivalued variational inequality
$V I(\Omega, C): \quad$ Find points $x_{0}$ in $C$ and $y_{0}$ in $\Omega\left(x_{0}\right)$ satisfying $\left\langle y_{0}, x-x_{0}\right\rangle \geq 0$ for all points $x$ in $C$.
A more concise way to write the problem is this:
Find a point $x_{0}$ in $C$ satisfying $0 \in \Omega\left(x_{0}\right)+N_{C}\left(x_{0}\right)$.
Suppose the set $C$ is closed, convex, and nonempty. Recall that the projection $P_{C}: \mathbf{E} \rightarrow C$ is the (continuous) map that sends points in $\mathbf{E}$ to their unique nearest points in $C$ (see Section 2.1, Exercise 8). Using this notation we can also write the variational inequality as a fixed point problem:
Find a fixed point of $P_{C} \circ(I-\Omega): C \rightarrow C$.
This reformulation is useful if the multifunction $\Omega$ is single-valued, but less so in general because the composition will often not have convex images.
A more versatile approach is to define the (multivalued) normal mapping $\Omega_{C}=\left(\Omega \circ P_{C}\right)+I-P_{C}$, and repose the problem as follows:
Find a point $\bar{x}$ in $\mathbf{E}$ satisfying $0 \in \Omega_{C}(\bar{x})$.
Then setting $x_{0}=P_{C}(\bar{x})$ gives a solution to the original problem. Equivalently, we could phrase this as follows:
Find a fixed point of $(I-\Omega) \circ P_{C}: \mathbf{E} \rightarrow \mathbf{E}$.
数学代考|凸分析作业代写CONVEX ANALYSIS代考|THE BROUWER FIXED POINT THEOREM
优化和分析中的许多问题都归结为求解非线性方程H(X)=0, 对于某些函数H:和→和. 等效地,如果我们定义另一个地图F=一世−H 在H和r和$一世$一世s吨H和一世d和n吨一世吨是米一种p, 我们寻求一个点X在 E 满足F(X)=X; 我们称之为X的一个固定点F.
最有效的不动点存在定理分为三类: “几何”结果,源自巴拿赫收缩原理在H一世CH在和s吨一种吨和b和一世这在, “顺序理论”结果吨这在H一世CH在和br一世和F一世是r和吨你rn一世n小号和C吨一世这n8.3,以及“拓扑”结果,其原型是构成本节主体的 Brouwer 定理。我们从巴拿赫的结果开始。
给定一个集合C⊂和和一个连续的自我映射F:C→C, 我们问是否F有一个固定点。我们称之为F如果存在实常数则收缩CF<1这样
‖F(X)−F(是)‖≤CF‖X−是‖ 对全部 X,是∈C.
数学代考|凸分析作业代写CONVEX ANALYSIS代考|SELECTION AND THE KAKUTANI-FAN FIXED POINT THEOREM
上一节中的 Brouwer 不动点定理涉及从非空紧凸集到自身的函数。在优化方面,正如我们在 5.4 节中已经看到的,扩展我们的语言以考虑多功能可能会很方便Ω从集合到自身并寻找一个不动点——一个点X令人满意的X∈Ω(X). 在本节开始之前,我们总结了一些定义以供将来参考。
我们考虑一个子集到⊂和, 欧几里得空间是, 和一个多功能Ω:到→是. 我们说Ω是ü小号C在某一点X在到如果每个开集ü包含Ω(X)还包含Ω(和)对于所有点和在到相近X.
因此多功能Ω如果对于任何点序列,则为 USC(Xn)接近X, 任何元素序列是n∈Ω(Xn)最终接近Ω(X). 如果Ω是南加州大学的每一点到我们简称为 USC。另一方面,如第5.4, 我们说Ω是大号小号C如果,对于每个X在到, 每个街区五在任何一点Ω(X)相交Ω(和)对于所有点和在到相近X.
我们指的是集合Ω(X)(X∈到)作为图像Ω. 多功能Ω如果它是具有非空紧凑凸图像的 USC,则为 cusco。显然,这样的多功能是局部有界的:任何点到有一个图像有界的邻域。Cuscos 出现在几个重要的优化环境中。例如,局部 Lipschitz 函数的 Clarke 次微分是 cusco和X和rC一世s和5.
要查看另一类重要的示例,我们需要进一步定义。我们说多功能披:和→和如果满足条件则为单调的⟨你−v,X−是⟩≥0每当你∈披(X)和v∈披(是).
特别是,任何n这吨n和C和ss一种r一世一世是s和一世F−一种dj这一世n吨半正定线性算子是单调的,任何凸函数的次微分也是如此。如果第一个的图形包含第二个的图形,则一个多功能包含另一个。如果唯一包含它的单调多功能是它自己,我们就说一个单调多功能是最大的。闭真凸函数的次微分就是例子s和和和X和rC一世s和16. 佐恩引理在H一世CH一世一世和s这你吨s一世d和这你r一世米米和d一世一种吨和sC这p和表明任何单调多功能都包含在最大单调多功能中。
数学代考|凸分析作业代写CONVEX ANALYSIS代考|VARIATIONAL INEQUALITIES
在本书的开头,我们考虑了最小化可微函数的问题F:和→R在凸集上C⊂和. 一个点的必要最优性条件X0在C成为局部最小化器是
⟨∇F(X0),X−X0⟩≥0 对于所有点 X 在 C,
或等效地
0∈∇F(X0)+ñC(X0).
如果函数F是凸的而不是可微的,最优的充要条件一种ss你米一世nG一种C这ns吨r一种一世n吨q你一种一世一世F一世C一种吨一世这n是
0∈∂F(X0)+ñC(X0),
并且存在类似的非光滑必要条件。
我们称问题为8.3.1“变分不等式”。让我们修复一个多功能Ω:C→和. 在本节中,我们使用我们开发的不动点理论来研究多值变分不等式
五一世(Ω,C):找点X0在C和是0在Ω(X0)令人满意的⟨是0,X−X0⟩≥0对于所有点X在C.
写问题的更简洁的方法是:
找到一个点X0在C令人满意的0∈Ω(X0)+ñC(X0).
假设集合C是闭的、凸的、非空的。回想一下投影磷C:和→C是个C这n吨一世n你这你s发送点的地图和到它们唯一的最近点C s和和小号和C吨一世这n2.1,和X和rC一世s和8. 使用这个符号,我们也可以将变分不等式写成一个不动点问题
:磷C∘(一世−Ω):C→C.
如果多功能Ω是单值的,但通常较少,因为合成通常不会有凸图像。
更通用的方法是定义米你一世吨一世v一种一世你和d法线映射ΩC=(Ω∘磷C)+一世−磷C, 并将问题解决如下:
找到一个点X¯在和令人满意的0∈ΩC(X¯).
然后设置X0=磷C(X¯)给出了原始问题的解决方案。
等效地,我们可以这样表述:(一世−Ω)∘磷C:和→和.
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