Cassowary is an incremental version of simplex—it lets you efficiently add, remove, and modify constraints.
Gradient descent works for non-linear systems, but isn't guaranteed to find the global optimum.
And within convex problems there is a hierarchy of "difficulty": LP ⊂ QP ⊂ SOCP ⊂ SDP
The Simplex algorithm solves only LP. And it was shown that the Simplex has an exponential worst-case runtime (in the number of constraints). Whereas interior-point methods (basically variations of Gradient Descent) have a much better worst-case runtime.
See here for further reference: http://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf