# Gradient Descent with nonlinear constraint on Symmetric positive definite matrix space

I would like to find the stationary point $S_*$ (global minimum) that minimizes the function $f(S)=\mathrm{trace}(S)+m^2\mathrm{trace}(S^{-2})$ which has been proven to be convex in Convexity of $\mathrm{trace}(S) + m^2\mathrm{trace}(S^{-2})$ for $S\in \mathcal{M}_{m,m}$ symmetric positive definite. I have also a nonlinear constraint in this optimization problem $\|S-C\|_1\leq \epsilon$ where $\epsilon$ and $C\in \mathcal{M}_{m,m}$ are given.

Actually I wasn't able to find a good optimization solver for this function. And so I think that this could be solvable using the gradient descent algorithm with projection which would lead to the global minimum of $f$ by derinving the expression of the update rule: $$S_{k+1}=S_k-\alpha_k \nabla_Sf(S)$$

This would lead to have an update rule $S_{k+1}\leftarrow h(S_k)$ where $h(S_k)$ is a function that depends on $S$ at the step $k$. And so I would like to find the expression of $h(S_k)$ for the gradient descent algorithm taking into account the projection of the gradient in the Symmetric Positive Definite space and the nonlinear constraint space.

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How are $S^{\operatorname{step}(k)}$ and $S_k$ related? –  joriki Mar 11 '13 at 3:56
@joriki $S^{step(k)}$ is $S_k$ I've just corrected in my question –  user2987 Mar 11 '13 at 4:23
I think that $h$ will also depend on $\epsilon$, $C$, $\alpha_k$ and $m$. –  user2987 Mar 11 '13 at 4:30