(20 points)

1. Define the following:
   1. convex set in $\Re^n$
   2. convex function defined on $\Re^n$
   3. closed convex cone in $\Re^n$
   4. dual (nonnegative polar) cone of a set $C$ in $\Re^n$
2. Show that the following two sets are convex cones. Then derive an explicit form for the dual (polar) cone $C^+=C^*$.
   1. $C_1 = \left\{ (x,y) \mid 0 \le y\le x \right\}$.
   2. $C_2 = \left\{ (x,y) \mid y \ge \vert x\vert \right\}$.
3. Show that each of the above cones does not contain the point $(\bar{x},\bar{y})=(0,-1)$. For each of the above cones find a separating hyperplane with the point $(\bar{x},\bar{y})=(0,-1)$.
4. Consider the quadratic objective function
   
   $$f(x)=3x_1^2-4x_1x_2+2x_2^2+7x_1-x_2.$$
   
   
   1. Find appropriate $A,b$ and write $f(x)=x^T A x + b^T x$.
   2. Show that $f$ is a strictly convex function.
   3. Use the geometric characterization of optimality (Pshenichnyi condition). Show that the origin in $\Re^2$ minimizes $f$ (defined above) subject to $x \in C_1$ (defined in (2a) above).