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): 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 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 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 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 C A convexity result 26 f . B / D r log ∣ $_{∣}$B B T ∣ GLYPH<0> p tr $^{(}$QB B > ) GLYPH<0> log$^{(}$ d ∑ j D 1 exp $^{{}$a$_{j}$ C c > j B B $^{>}$c$_{j}$ } ) (48) @ @ GLYPH<22> D s GLYPH<0> $^{1}$x $^{>}$e x GLYPH<22> C 1 2 x > $^{[(}$s GLYPH<0> $^{1}$diag $^{{}$e x GLYPH<22> } GLYPH<0> s GLYPH<0> $^{2}$e x GLYPH<22> $^{(}$e x GLYPH<22> ) > $^{)}$. x GLYPH<12> x / C 2 $^{(}$s GLYPH<0> $^{3}$e x GLYPH<22> $^{{(}$x $^{>}$e x GLYPH<22> ) GLYPH<12> $^{(}$x $^{>}$e x GLYPH<22> )} > GLYPH<0> s GLYPH<0> $^{2}$diag $^{{}$e x GLYPH<22> $^{}}$x diag $^{{}$x $^{>}$e x GLYPH<22> } $^{)]}$exp .GLYPH<27> / : (47) 2.GLYPH<22>/ D s GLYPH<0> $^{1}$. x GLYPH<12> x / $^{>}$exp . x GLYPH<22>/ GLYPH<0> s GLYPH<0> 2 $^{(}$x $^{>}$exp . x GLYPH<22>/ ) GLYPH<12> $^{(}$x $^{>}$exp . x GLYPH<22>/ ) ; (46) ❊ log $^{(}$1 $^{>}$exp . xV / ) GLYPH<25> log $^{(}$1 $^{>}$exp . x GLYPH<22>/ ) C 1 2 2.GLYPH<22>/ $^{>}$exp .GLYPH<27> / ; (45)