Here is a recent and interesting paper connecting GJK and convex optimization [1]. They show that GJK is equivalent to the iterations of the Frank-Wolfe algorithm applied to the QP, and that recent improvements to Frank-Wolfe can be applied to improve GJK.
Frank-Wolfe is a somewhat less well-known (compared to simplex, interior point, etc) convex optimization algorithm with many interesting properties [2, Section 3.3].