What kinds of objectives do we commonly optimize in graphics? Also, are more sophisticated local searches than gradient ascent used? (I can see how, if the optimization were to be happening in lower level rendering loops or the like, this could be less than ideal.)
@Kurugama Far too many to name, but if you do a Google search for "variational SIGGRAPH" and click on any paper with the word "Variational" in the title, you'll find all sorts of examples. As for optimization algorithms, there are of course a huge number of techniques, and folks in graphics are inventing new ones every day that are tailored to the special features of the problems they're solving. That's why it's really important to understand the fundamentals behind these techniques, and not just treat optimization as a "black box."
If you're interested in learning more about optimization, a great place to start (IMO) is with convex optimization, since the theory is clean and well-established and yet it provides a good perspective on the basic principles and challenges of more general (nonconvex) optimization. There's a terrific free book on the subject: http://web.stanford.edu/~boyd/cvxbook/ (And I know at least one course at CMU uses this book; can't remember which one.)