This talk will cover one of the more popular methods for unconstrained minimization and explore techniques to improve upon it.  In particular, I will cover the BFGS method, a type of quasi-Newton method, where one seeks the minimizer by using a sequence of approximate quadratic models. The most important technique to improve upon this method is developed from the following idea, which is the heart of the talk: can we minimize a function over a k-dimensional subspace in such a way as to make minimizing over k+1 dimensions a trivial task?  For quadratic functions, the answer turns out to be 'yes', which in turn motivates similar techniques for the nonlinear case.  Some benefits are: we operate with significantly smaller matrices, have a smaller memory footprint, and can reinitialize the curvature at each iteration at little to no cost, which dramatically improves convergence when the function is ill-conditioned near the solution.  Some knowledge of linear algebra will be helpful but not necessary - I will aim to make the talk as accessible as possible.