content_image/298094.jpg content_image/298093.jpg content_image/298092.jpg content_image/298091.jpg
where x is a J GLYPH<2> K matrix whose rows are the vectors x > j . Let V GLYPH<24> N$_{K}$ .GLYPH<22>; 6 / , and restrict 6 such that 6 D diag f exp .GLYPH<27> / g for GLYPH<27> 2 R $^{K}$. We can now rewrite (43): 

<img src='content_image/298094.jpg'>

where 2.GLYPH<22>/ is the diagonal of the Hessian of f , evaluated at the point GLYPH<22> . Define s D 1 $^{>}$exp . x GLYPH<22>/ . Using matrix differentials, it can be shown that 

<img src='content_image/298093.jpg'>

where GLYPH<12> denotes the Hadamard product. 

To use the approximation (45) in an optimization over GLYPH<22> , we need to compute the gradient. The formula for 2.GLYPH<22>/ makes this a more extensive but still mechanical exercise in differentials. One obtains 

<img src='content_image/298092.jpg'>

## C A convexity result 

Let a$_{1}$ ; : : : ; a$_{d}$ be scalars, c$_{1}$ ; : : : ; c$_{d}$ be n -vectors, p ; r > 0, and Q GLYPH<23> 0. We show here that the function 

<img src='content_image/298091.jpg'>

is concave on the set of full-rank n GLYPH<2> n matrices. 

We argue that each of the three constituent terms, from left to right, is concave. The 

