Abstract We study solution approaches to a class of mixed-integer nonlinear programming problems that arise from recent developments in risk-averse stochastic optimization and contain second- and p -order cone programming as special cases. We explore possible applications of some of the solution techniques that have been successfully used in mixed-integer conic programming and show how they can be generalized to the problems under consideration. Particularly, we consider a branch-and-bound method based on outer polyhedral approximations, lifted nonlinear cuts, and linear disjunctive cuts. Results of numerical experiments with discrete portfolio optimization models are presented.