Statistics-lab™可以为您提供harvard.edu AM221 Convex optimization凸优化课程的代写代考和辅导服务！
AM221 Convex optimization课程简介
This is a graduate-level course on optimization. The course covers mathematical programming and combinatorial optimization from the perspective of convex optimization, which is a central tool for solving large-scale problems. In recent years convex optimization has had a profound impact on statistical machine learning, data analysis, mathematical finance, signal processing, control, theoretical computer science, and many other areas. The first part will be dedicated to the theory of convex optimization and its direct applications. The second part will focus on advanced techniques in combinatorial optimization using machinery developed in the first part.
PREREQUISITES
Instructor: Yaron Singer (OH: Wednesdays 4-5pm, MD239)
Teaching fellows:
- Thibaut Horel (OH: Tuesdays 4:30-5:30pm, MD’s 2nd floor lounge)
- Rajko Radovanovic (OH: Mondays 5:30-6:30pm, MD’s 2nd floor lounge)
Time: Monday \& Wednesday, 2:30pm-4:00pm
Room: MD119
Sections: Wednesdays 5:30-7:00pm in MD221
AM221 Convex optimization HELP（EXAM HELP， ONLINE TUTOR）
Midpoint convexity. A set $C$ is midpoint convex if whenever two points $a, b$ are in $C$, the average or midpoint $(a+b) / 2$ is in $C$. Obviously a convex set is midpoint convex. It can be proved that under mild conditions midpoint convexity implies convexity. As a simple case, prove that if $C$ is closed and midpoint convex, then $C$ is convex. Solution. We have to show that $\theta x+(1-\theta) y \in C$ for all $\theta \in[0,1]$ and $x, y \in C$. Let $\theta^{(k)}$ be the binary number of length $k$, i.e., a number of the form $$ \theta^{(k)}=c_1 2^{-1}+c_2 2^{-2}+\cdots+c_k 2^{-k} $$ with $c_i \in\{0,1\}$, closest to $\theta$. By midpoint convexity (applied $k$ times, recursively), $\theta^{(k)} x+\left(1-\theta^{(k)}\right) y \in C$. Because $C$ is closed, $$ \lim _{k \rightarrow \infty}\left(\theta^{(k)} x+\left(1-\theta^{(k)}\right) y\right)=\theta x+(1-\theta) y \in C $$
To show that $C$ is convex, we need to show that for any $x, y \in C$ and $\theta \in [0,1]$, the point $\theta x + (1-\theta)y$ is also in $C$. Since $C$ is closed and $\theta$ is a fixed number, we can consider the sequence of points $(\theta^{(k)}x + (1-\theta^{(k)})y)$, where $\theta^{(k)}$ is the binary number of length $k$ closest to $\theta$.
By midpoint convexity, we know that $(\theta^{(k)}x + (1-\theta^{(k)})y) \in C$ for all $k$. Since $C$ is closed, the limit of this sequence exists and is in $C$, that is,
$$\lim_{k \rightarrow \infty} (\theta^{(k)}x + (1-\theta^{(k)})y) = \theta x + (1-\theta)y \in C.$$
Therefore, $C$ is convex.
Midpoint convexity. A set $C$ is midpoint convex if whenever two points $a, b$ are in $C$, the average or midpoint $(a+b) / 2$ is in $C$. Obviously a convex set is midpoint convex. It can be proved that under mild conditions midpoint convexity implies convexity. As a simple case, prove that if $C$ is closed and midpoint convex, then $C$ is convex.
Solution. We have to show that $\theta x+(1-\theta) y \in C$ for all $\theta \in[0,1]$ and $x, y \in C$. Let $\theta^{(k)}$ be the binary number of length $k$, i.e., a number of the form
$$
\theta^{(k)}=c_1 2^{-1}+c_2 2^{-2}+\cdots+c_k 2^{-k}
$$
with $c_i \in{0,1}$, closest to $\theta$. By midpoint convexity (applied $k$ times, recursively), $\theta^{(k)} x+\left(1-\theta^{(k)}\right) y \in C$. Because $C$ is closed,
$$
\lim _{k \rightarrow \infty}\left(\theta^{(k)} x+\left(1-\theta^{(k)}\right) y\right)=\theta x+(1-\theta) y \in C
$$
To show that $C$ is convex, we need to show that for any $x, y \in C$ and $\theta \in [0,1]$, the point $\theta x + (1-\theta)y$ is also in $C$. Since $C$ is closed and $\theta$ is a fixed number, we can consider the sequence of points $(\theta^{(k)}x + (1-\theta^{(k)})y)$, where $\theta^{(k)}$ is the binary number of length $k$ closest to $\theta$.
By midpoint convexity, we know that $(\theta^{(k)}x + (1-\theta^{(k)})y) \in C$ for all $k$. Since $C$ is closed, the limit of this sequence exists and is in $C$, that is,
$$\lim_{k \rightarrow \infty} (\theta^{(k)}x + (1-\theta^{(k)})y) = \theta x + (1-\theta)y \in C.$$
Therefore, $C$ is convex.
Textbooks
• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.
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