Second-order cone programming problems (SOCPs) have been well studied in literature, and computationally efficient implementations of solution algorithms exist. In this paper, we study an extension: mixed-integer second-order cone programming problems (MISOCPs). Our focus is on designing an algorithm for solving the underlying SOCPs as nonlinear programming problems (NLPs) within two existing frameworks, branch-and-bound and outer approximation, for mixed-integer nonlinear programming (MINLP). We pay particular attention to resolving the nondifferentiability of the underlying SOCPs via a smoothing reformulation, as well as warmstarting and infeasibility detection using a regularization. We investigate the application of our proposed techniques to portfolio optimization problems that can be formulated as MISOCPs, and preliminary numerical results using the Matlab-based optimization package MILANO (Benson, MILANO—a Matlab-based code for mixed-integer linear and nonlinear optimization. http://www.pages.drexel.edu/~hvb22/milano) are provided.