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\textbf{CSE 203B W25 Homework 3} \\
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\noindent{\textbf{Due Time : 11:50pm, Thursday Feb. 6, 2025}} 
\textbf{Submit to Gradescope \\ Gradescope: \url{https://gradescope.com/} }\\
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{\bf In this homework, we work on exercises from text book including level sets of convex, concave, quasi-convex, quasi-concave functions (3.1, 3.2), second-order conditions for convexity on affine sets (3.9), Kullback-Leibler divergence (3.13), saddle points of convex-concave functions (3.14) determination of convex, concave, quasi-convex, quasi-concave functions (3.16), conjugate functions (3.36), and gradient and Hessian of conjugate functions (3.40). Extra assignments are given on Kullback-Leibler divregence, and softmax functions.\\ 

Please make two separate submissions on Gradescope: \\ \\
\underline{Exercises:}
\begin{itemize}
    \item Graded by completion, you may work in a group of up to four students.
    \item Submit a single PDF file and add all group members to the submission.
    \item Describe each member's contributions at the beginning of your report.
\end{itemize}
\underline{Assignments:}
\begin{itemize}
    \item Graded by content and must be completed individually.
    \item Submit a single PDF file.
\end{itemize}
}

\noindent\textbf{I. Exercises from textbook chapter 3 (8 pts)} \\[0.05in]

3.1, 3.2, 3.9, 3.13, 3.14, 3.16, 3.36, 3.40. \\

\noindent\textbf{II. Assignments (42 pts)} \\%\\[0.2in]

\noindent
II. 1. Convex Functions. (19 pts)\\
\noindent
I.1.1. Determine if the following functions are convex or concave, and provide proofs. (12 pts) 
\begin{enumerate}
    \item[(a)] $f(x) = \sqrt{x}, \quad dom \, f = \mathbb{R}_{++}$. (3 pts)
    \item[(b)] $f(x) = \sum_{i=1}^n e^{x_i}, dom \, f =\mathbb{R}^n$. (3 pts)
    \item[(c)] $f(x_1, x_2) = x_1^\alpha x_2^{1-\alpha}, \quad 0 \leq \alpha \leq 1, dom \, f = \mathbb{R}^2_{++}$. (3 pts)
    \item[(d)] $f(x) = \max\{a_1^T x + b_1, \dots, a_k^T x + b_k\}, dom \, f = \mathbb{R}^n$. (3 pts)

\end{enumerate}

\noindent
II.1.2. Consider the function:
\[
f(x) = \left( \sum_{i=1}^n x_i^p \right)^{1/p}, dom \, f = \mathbb{R}^n_{++}, \quad p < 1, p \neq 0.
\]

\noindent
Answer the following questions related to this function:

\begin{enumerate}
    \item[(a)] Derive the Hessian matrix of $f(x)$. (3 pts)
    \item[(b)] Is the function $f(x)$ convex or concave? Show your proof. (4 pts)
\end{enumerate}



\noindent
II. 2. Conjugate Functions. (23 pts)

Find the conjugate function of the following functions.

\noindent  
II.2.1. $f_1(x) = 2x + 1$, where $x \in \mathbb{R}$. (2 pts)

\noindent  
II.2.2. Let $f_2(x) = \frac{1}{3} x^\top Q x$, where $Q \in S_{++}^n$ (a symmetric positive definite matrix) and $x \in \mathbb{R}^n$.  (3 pts)

\noindent  
II.2.3. Let $f_3(x) = -\log(a x^2 + b x + c)$, where $x \in \mathbb{R}$, $a > 0$, $b \in \mathbb{R}$, $c > 0$.  (6 pts)

\noindent  
II.2.4. Let $f_4(x) = \sum_{i=1}^n e^{a_i x_i + b_i}$, where $x \in \mathbb{R}^n$, $a_i > 0$, and $b_i \in \mathbb{R}$ for all $i$. (6 pts)

\noindent  
II.2.5. Let $f_5(x) = \log \sum_{i=1}^n \exp\left(\frac{x_i}{\gamma}\right)$, where $x \in \mathbb{R}^n$. (6 pts)




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