Posted on Category:数学代写

# 数学代写|Nonlinear optimization代考

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APM462 Midterm Project
Due: Sat Mar 11 (before $9 \mathrm{pm}$ ) on Crowdmark

\begin{prob}

(1) Conside the convex set $C:=\left{(x, y) \in \mathbb{R}^2 \mid x^2+y^2 \leq 1\right.$ and $\left.x, y \geq 0\right}$, the unit disc intersect the first quadrant, and let $\delta_C(x, y)$ be its indicator function. Find $\partial \delta_C\left(x_0, y_0\right)$ where $\left(x_0, y_0\right)$ is a boundary point of $C$. Hint: Draw a picture. There should be 6 cases to consider. Note that $C=C_1 \cap C_2 \cap C_3$ where $C_1=\left{(x, y) \in \mathbb{R}^2 \mid x^2+y^2 \leq 1\right}$, $C_2=\left{(x, y) \in \mathbb{R}^2 \mid-x \leq 0\right}$, and $C_3=\left{(x, y) \in \mathbb{R}^2 \mid-y \leq 0\right}$.

\end{prob}

\begin{prob}

(2) Let $F(x, y)=f(x)+g(y)$ where $f, g: \mathbb{R}^1 \rightarrow \mathbb{R}$.
(a) Prove that $\partial F\left(x_0, y_0\right)=\partial f\left(x_0\right) \times \partial g\left(y_0\right)$. Recall that the product of two sets $A \times B={(a, b) \mid a \in A, b \in B}$.
(b) Use the formula in part (a) to compute $\partial F\left(x_0, y_0\right)$ for the function $F(x, y)=|x|+y^2$.

\end{prob}

\begin{prob}

(3) Consider the following optimization problem:
\begin{aligned} \text { minimize: } & f(x, y)=\frac{1}{2}(x-2)^2+\frac{1}{2}(y-3)^2 \ \text { subject to: } & g_1(x, y)=x \leq 1 \ & g_2(x, y)=y \leq 1 \end{aligned}
(a) Draw a diagram in $\mathbb{R}^2$ of the feasible set and the level sets of $f$. Then solve the problem graphically.
(b) Use the $1^{\text {st }}$ order necessary Kuhn-Tucker conditions to find the candidate(s) for minimizer(s).
(c) Check whether the candidates you found in part (b) satisfy the $2^{\text {nd }}$ order necessary condition for minimizer.
(d) Check whether the candidates you found in part (b) satisfy the $2^{\text {nd }}$ order sufficient condition for minimizer.

\end{prob}

\begin{prob}

(4) Consider the following optimization problem:
\begin{aligned} \text { minimize: } f(x, y) & =\frac{1}{2}(x-2)^2+\frac{1}{2}(y-3)^2 \ \text { subject to: } g(x, y) & =\max {|x|,|y|} \leq 1 \end{aligned}
(a) Draw a diagram in $\mathbb{R}^2$ of the feasible set $g \leq 1$ and the level sets of $f$. Then solve the problem graphically.
(b) Find $\partial f(x, y)$ and $\partial g(x, y)$. Note: for $\partial g(x, y)$ there are nine cases to consider.
(c) Solve the minimization problem using subdifferentials. Hint: does you solution to part (c) agree with your graphical solution in part (a)?
\end{prob}

\begin{prob}

(5) Solve the following convex minimization problem using subdifferentials where $f(x, y)=\max \left{|x|, y+\frac{1}{2}\right}$ and $b>0$ :
\begin{aligned} \text { minimize: } & f(x, y) \ \text { subject to: } & g_1(x, y)=x^2+y^2-1 \leq 0, \ & g_2(x, y)=|x|+y-b \leq 0 . \end{aligned}
Hint: the answer should depend on the parameter $b$.

\end{prob}

\begin{prob}

(6) Consider the following convex minimization problem where $g(x, y)=$ $|x|+y-1$ :
\begin{aligned} \text { minimize: } & f(x, y)=(x-3)^2+(y-1)^2 \ \text { subject to: } & (x, y) \in C:={g(x, y) \leq 0} \end{aligned}
(a) Solve this problem using subdifferentials.
(b) Convert this problem into an equivalent problem of inequality constraints and use the Kuhn-Tucker conditions to find candidates for minimizer(s). Hint: get rid of of the abslute value in the inequality $g(x, y) \leq 0$ by replacing the inequality with two inequalities.

\end{prob}

\begin{prob}

(7) Let $a \in \mathbb{R}$ be a number and consider the optimization problem:
\begin{aligned} \min f(x, y) & =x+a y \ \text { subject to: } g(x, y) & =y-|x| \geq 0 \end{aligned}
Note that ${g=0}$ is a cone with vertex at the origin. Also note that $g$ is not diffferentiable at the origin.
(a) Solve this problem using subdifferentials. Hint: after you draw a picture the solution should be pretty clear. Once you know what the solution should be, you will know what to aim for. Note that there are two cases to consider depending on the value of $a$.
(b) Find all the regular points of the set ${g \geq 0}$.
(c) Use the $1^{\text {st }}$ order Kuhn-Tucker neccessary conditions for a local min to find the candidate(s) for minimizer(s) and then solve the optimization problem. Hint: as in question (6) you should first replace the constraint $g \geq 0$ with two other constraints so as to get rid of the absolute value.

\end{prob}

## 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。