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To prove the lemma, it suffices to show that function F is convex. The Hessian is: 

<img src='content_image/299203.jpg'>

Because z$_{i}$ is negative, ∇ $^{2}$F is positive semidefinite. Therefore, F is convex. 

The next step is to show that the objective function µ ln ( ∑ N i $_{=1}$θ$_{i}$ e$^{ω}$ i µ ) is convex. Define a new function. N 

<img src='content_image/299202.jpg'>

Before proceeding with the proof, we present a very useful result from Boyd and Vandenberghe (2004). $^{11}$. 

Lemma 4 Let the function f ( y ) be defined as (73) where a$_{k}$ , y ∈ R $^{n}$, b$_{k}$ ∈ R . 

f ( y ) is a convex function. 

Therefore, the Hessian of f ( y ) must be positive semidefinite. 

Lemma 5 Function G , which is defined as (72), is convex. 

<img src='content_image/299201.jpg'>

Proof. Note that function G can be expressed as (74). With µ fixed, the structure of G as a function of ω$_{1}$, ω$_{2}$, ..., ω$_{N}$ is exactly analogous to (73). 

<img src='content_image/299200.jpg'>

With µ fixed, G is a convex function of ω$_{1}$, ω$_{2}$, ..., ω$_{N}$ . Hence all the principal minors of the matrix in (75) are nonnegative . 

<img src='content_image/299199.jpg'>

