Many problems in science and engineering can be formulated as nonlinear optimization problems in which the goal is to find an acceptable input vector x that minimizes the value of a scalar objective function f(x). Although many numerical algorithms for solving such problems have been extensively analyzed, surprisingly few achieve satisfactory performance when the objective is defined by expensive computer simulations of complex physical processes. Such objective functions pose peculiar challenges that must be addressed in new ways; accordingly, we will discuss the development of algorithms for numerical optimization in this environment that are computationally tractable and analytically sound.