Optimization in Communications Engineering
WS 2015/16
1
2. Convex Functions
Problem 2.1 Definition of Convexity/Concavity
Determine whether the following functions are convex and/or concave:
a)
fA (x) = cT x, where x Rn and c Rn is constant.
b) fB (x) = x

Optimization in Communications Engineering
WS 2015/16
5. Optimality Conditions
Problem 5.1 Least Squares Problem
Consider the optimization problem
min kAx bk22 ,
x
where x Rn , A Rmn , and b Rm .
a)
Give a geometric interpretation for this optimization pr

Optimization in Communications Engineering
WS 2015/16
1
3. Extreme Points, Extreme Directions, and Polyhedral Sets
Problem 3.1 Standard Forms of Polyhedral Sets
Consider the following two standard forms of polyhedral sets, where A Rmn and b Rm :
(Primal)

Optimization in Communications Engineering
WS 2015/16
1
1. Convex Sets
Problem 1.1 Linear Spaces, Affine Sets, Cones, and Convex Sets
For each of the following sets, determine if it is a linear space, an affine set, a cone, and/or a convex
set.
a)
SA = cf

Optimization in Communications Engineering
WS 2015/16
1
4. Linear Programming and the Simplex Method
Problem 4.1 Simplex Method
Consider the linear program
min cT x s.t. Ax = b, x 0,
(4.1)
where A = [ 10 21 11 01 ] , b = [ 32 ], and c = [ 1 1 1 1 ]T . An