You might wish to ask Michael Grant, the author of CVX, to discipline your function in the next release of CVX.

In the mean time, try IPOPT form COIN-OR, which is a fairly robust general purpose solver for nonlinear programming problems. To make it work, you need to remove the psd constraint by replacing each occurrence of $C$ by $R^TR$, where $R$ is an upper triangular matrix with nonnegative diagonal entries.

You might wish to ask Michael Grant, the author of CVX, to discipline your function in the next release of CVX.

In the mean time, try IPOPT form COIN-OR, which is a fairly robust general purpose solver for nonlinear programming problems. To make it work, you need to remove the psd constraint by replacing each occurrence of $C$ by $R^TR$, where $R$ is an upper triangular matrix with nonnegative diagonal entries.

Since IPOPT expects smooth constraints, you also need to replace the nonsmooth L1 constraint $\|S−C\|_1\le \alpha$ (assuming the 1-norm to be the column sum norm) by the smooth constraints $$-X\le R^TR-C \le X,~~~ e^TX\le \alpha e,$$ where $e$ is the all-one vector, and $X$ is a matrix of new variables.