Nonlinear Optimization Methods with Financial Applications

Financial applications have a long history of including optimization, starting with Markowitz’s origin of the quadratic optimization model for determining an efficient portfolio to minimize variance for a given return. Portfolio optimization continues to be an active area with most applications focused on linear and quadratic optimization. General nonlinear optimization arises in this area, however, as models begin to acknowledge and capture the nonlinearity, asymmetry, and non-normality associated with returns in practice. In addition, complex financial products often involve a variety of nonlinear relationships that lead to nonlinear optimization in parameter estimation, tracking, and hedging. Credit instruments and their risk management also introduce nonlinearities that are difficult to include in linear or quadratic models. This note summarizes some of these financial applications and the role that nonlinear optimization methods can play in their solution.