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\begin{center} {\large \textbf{Math 582G -- Homework  1}} \smallskip \\ 
Due on Friday, January 21, 2021\ \\ \ \\
 \end{center}

Please indicate any sources you used for a given problem on the solution to that problem. 
For example, if you worked with another student to get the solution to a problem, please indicate who.
You are welcome to work together, but you should try the problems on your own first and write up your own solutions. 

\begin{flushleft}

\begin{prob} Let $K$ denote the convex cone of quadratic polynomials in $\R[x]$ that are nonnegative on $[-1,1]$, i.e.
\[ K \ = \ \mathcal{P}_2([-1,1]) \ \cong \ \{(a,b,c) \in \R^3 : ax^2 + bx + c \geq 0 \text{ for all } x\in [-1,1]\}. \]
\begin{itemize}
\item[(a)] Describe the dual cone $K^*$.   \smallskip 
\item[(b)] Give a semialgebraic description of $K$.  \smallskip 
\item[(c)] Draw the intersections of both $K$ and $K^*$ with the planes ``last coordinate''$=1$.  \smallskip 
\item[(d)] Does either $K$ or $K^*$ have a non-exposed face? 
\end{itemize}
\end{prob}
\bigskip 


\begin{prob} Let $\cH^n$ denote the real vector space of Hermitian matrices, 
\[
\cH^n = \{ A\in \C^{n\times n} : A = \overline{A}^T\},
\]
where $\ol{A}^T$ is the conjugate transpose of $A$. 
A Hermitian matrix $A$ is \emph{positive semidefinite} if $\ol{v}^TAv\geq 0$ for all  $v\in \C^n$. Let $\cH^n_+$ denote the set of positive semidefinite matrices in $ \cH^n $. \smallskip 
\begin{itemize}
\item[(a)] Show that $\cH^n_+$ is a closed, convex cone. \smallskip 
\item[(b)] Show that $\cH^n_+$ is self-dual under the inner product $\langle A, B\rangle = {\rm trace}(AB)$.  \smallskip 
\item[(c)] What are the extreme points of the convex set $\{A \in \cH^n_+ :  {\rm trace}(A) = 1\}$?
\end{itemize}
\end{prob}
\bigskip
\begin{prob}  For $A\in \R^{n\times n}_{\rm sym}$,  consider the maximization problem 
\[\max_{y\in \R} \ y \ \text{ s.t. } \  A- yI \in {\rm PSD}_n,   \]
where $I$ is the $n\times n$ identity matrix. \smallskip 
\begin{itemize}
\item[(a)] Write a minimization problem of which this is the dual.  \smallskip 
\item[(b)] Show that the primal and dual problems attain the same value. What is this optimal value in terms of the eigenvalues of the matrix $A$?
\end{itemize}
\end{prob}
\bigskip 

Here's another good to think about. Do not turn in the solution!
\begin{extraProb} Suppose that $C$, $K$ are closed convex cones and $L$ is a linear subspace in a finite dimensional real vectorspace $V$. Show the following: 
\begin{itemize}
\item[(a)] If $C\subseteq K$ then $C^*\supseteq K^*$.  \smallskip 
\item[(b)] $(C + K)^* = C^*\cap K^*$  \smallskip 
\item[(c)] $L^* = L^{\perp} = \{\ell\in V^*: \ell(v)=0 \text{ for all }v\in L\}$ \smallskip 
\item[(d)]  $K\subseteq L$,  if and only if  $K^* + L^{\perp}  = K^*$ 
\end{itemize}
Here $C+K = \{ u + v : u\in C, v\in K \}$ denotes the Minkowski sum of $C$ and $K$. 
\end{extraProb}



\end{flushleft}
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