dumpfile may not necessarily meet your needs, but is about all you can do with CVX
help cvx_solver_settings
...
cvx_solver_settings( 'dumpfile', <filename> ) is a setting supported by
all solvers. If set, it will save a .MAT file containing the exact
input arguments delivered to the solver. This file will be created
immediately before the solver is called, so you will be able to examine
their contents even if the solver fails with an error. This feature is
to be used primarily by solver developers.