  Subcell limiting strategies for discontinuous Galerkin spectral element
methods do not provably satisfy a semi-discrete cell entropy inequality. In
this work, we introduce an extension to the subcell and monolithic convex
limiting strategies that satisfies the semi-discrete cell entropy inequality by
formulating the limiting factors as solutions to an optimization problem. The
optimization problem is efficiently solved using a deterministic greedy
algorithm. We also discuss the extension of the proposed subcell limiting
strategy to preserve general convex constraints. Numerical experiments confirm
that the proposed limiting strategy preserves high-order accuracy for smooth
solutions and satisfies the cell entropy inequality.
