Quasiconvex functions

A function \(f\) is quasiconvex iff \(\mathbf{dom} f\) is convex and for any \(x,y \in \mathbf{dom} f\) and \(0\leq \theta \leq 1\)
\[  f(\theta x + (1-\theta) y) \leq \max\{ f(x), f(y) \}\]  A continuous function \(f: \mathbf{R} \rightarrow \mathbf{R}\) is quasiconvex iff at least one of the following conditions holds

• \(f\) is nondecreasing
• \(f\) is nonincreasing
• there is a point \(c\in \mathbf{dom} f\) such that for \(t \leq c\), \(f\) is nonincreasing, and for \(t \ge c\), \(f\) is nondecreasing.

First order conditions

A continuous differentiable function \(f: \mathbf{R}^n \rightarrow \mathbf{R}\) is quasiconvex iff \(\mathbf{dom} f\) is convex and for all \(x,y\in \mathbf{dom}f\) 

\[ f(y) \leq f(x) \Rightarrow \nabla f(x)^T (y-x) \leq 0 \]

Operations that preserve quasiconvexity

• nonnegative weighted maximum
  The function \[ f(x) = \sup_{y\in C} (w_y g_y(x) ) \]  where \(w_y \ge 0\) and \(g_y(x)\) is a parameterized family of quasiconvex functions.
• composition
• if \(g\) is quasiconvex and \(h\) is nondecreasing then, \(f = h \circ g\) is quasiconvex
• composition of a quasiconvex function with an affine or linear-fractional transformation yields a quasiconvex function.
• minimization
  if \(f(x,y)\) is quasiconvex jointly in \(x\) and \(y\) and \(C\) is a convex set, then the function
  \[ g(x) = \inf_{y\in C} f(x,y)\] is quasiconvex.

Gaussian interference channel

The sum capacity is known for two special cases:

Strong interference: Capacity is achieved by decoding and canceling the interference before decoding the desired signal.  The condition is defined as \(I_2 \ge S_1\) and \(I_1 \ge S_2\)
\begin{align*}
R_1 & \leq C(S_1), \\
R_2 & \leq C(S_2), \\
R_1 + R_2 & \leq \min \lbrace C(S_1+I_1), C(S_2+I_2)  \rbrace
\end{align*}

Weak interference:  Capacity is achieved by treating interference as Gaussian noise

\[  C_{sum} = C \left( \frac{S_1}{1+I_1} \right) + C \left( \frac{S_2}{1+I_2} \right)  \]

Gaussian broadcast channel

Two users case

Scalar Gaussian broadcast channel belongs to the class of degraded broadcast channel.  (The users can be absolutely ranked by their channel strength)  

When the transmitter has more than one antenna, the Gaussian broadcast channel is non-degraded.

Sender of power \(P\) transmit to two receivers, one with Gaussian noise power \(N_1\) and the other with \(N_2\).  assume \(N_1 < N_2\).   The capacity region is
\begin{align*}
R_1 &< C \left(\frac{\alpha P}{N_1}\right) \\
R_2 &< C \left(\frac{(1-\alpha) P}{\alpha P + N_2}\right) \\
\end{align*} where \(0 \leq \alpha \leq 1 \)

The weaker receiver \(Y_2\) decodes its own message treating the other user's message as interference.  The stronger receiver \(Y_1\) decodes \(Y_2\)'s message and cancels it before decoding his own.

Note:  The role of transmitter side information reduces with the growth in the number of TX antennas.  With either CSIT or CDIT and under the ZMSW model the asymptotic growth is linear as \( \mathcal{C} \min(M,K) \).

weighted arithmetic-geometric mean inequality

Generalized form of the AM-GM inequality
\[ \sum_i \alpha_i v_i \ge \prod_i v_i^{\alpha_i} \] where \(\mathbf{v} \succ 0\) and \(\mathbf{\alpha} \succeq 0,\; \mathbf{1}^T\mathbf{\alpha} = 1\).