What will be the Gradient and Hessian of the log sum of an inverse of the vector,?

For example, if $x$ is a vector, then what will be the gradient and hessian w.r.t $x$ for: $$ \log\sum_i(\dfrac{c_i}{x_i}) $$ where $c$ is a vector of constants.


Let's use a colon (:) to denote the trace/Frobenius product, i.e. $A:B={\rm tr}(A^TB)$
and $\odot$ to denote elementwise/Hadamard multiplication
and $\oslash$ to denote elementwise/Hadamard division.

Define a new scalar variable and its differential $$\eqalign{ \sigma &= 1:c\oslash x \cr &= (1\oslash x):c \cr\cr d\sigma &= -dx\oslash (x\odot x):c \cr &=-c\oslash (x\odot x):dx }$$ Then the function, differential, and gradient are $$\eqalign{ \lambda &= \log(\sigma) \implies \sigma = \exp(\lambda) \cr \cr d\lambda &= \frac{d\sigma}{\sigma} = -\frac{1}{\sigma}\,c\oslash (x\odot x):dx \cr \cr \frac{\partial \lambda}{\partial x} &= g = -\frac{1}{\sigma}\,c\oslash (x\odot x) = -e^{-\lambda}\,c\oslash (x\odot x)\cr }$$ If we let $$G={\rm Diag}(g),\,\,X={\rm Diag}(x)$$ then we can write the Hessian as $$H=\frac{\partial g}{\partial x}=-(2GX^{-1}+gg^T)$$


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