Abstract. A conic integer program is an integer programming problem with conic constraints. Many problems in finance, engineering, statistical learning, and probabilistic optimization are modeled using conic constraints.Here we study mixed-integer sets defined by second-order conic constraints. We introduce general-purpose cuts for conic mixed-integer programming based on polyhedral conic substructures of second-order conic sets. These cuts can be readily incorporated in branch-and-bound algorithms that solve either secondorder conic programming or linear programming relaxations of conic integer programs at the nodes of the branch-and-bound tree.Central to our approach is a reformulation of the second-order conic constraints with polyhedral second-order conic constraints in a higher dimensional space. In this representation the cuts we develop are linear, even though they are nonlinear in the original space of variables. This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming. The reformulation also allows the use of polyhedral methods for conic integer programming.We report computational results on solving unstructured second-order conic mixed-integer problems as well as mean-variance capital budgeting problems and least-squares estimation problems with binary inputs. Our computational experiments show that conic mixed-integer rounding cuts are very effective in reducing the integrality gap of continuous relaxations of conic mixed-integer programs and, hence, improving their solvability.

Abstract. In financial markets high levels of risk are associated with large returns as well as large losses, whereas with lower levels of risk, the potential for either return or loss is small. Therefore, risk management is fundamentally concerned with finding an optimal tradeoff between risk and return matching an investor's risk tolerance. Managing risk is studied mostly in a financial context; nevertheless, it is relevant in any area with a significant source of uncertainty.The mean-risk tradeoff is well-studied for problems with a convex feasible set. However, this is not the case in the discrete setting, even though, in practice, portfolios are often restricted to discrete choices. In this paper we study mean-risk minimization for problems with discrete decision variables. In particular, we consider discrete optimization problems with a submodular mean-risk minimization objective. We show the connection between extended polymatroids and the convex lower envelope of this mean-risk objective. For 0-1 problems a complete linear characterization of the convex lower envelope is given. For mixed 0-1 problems we derive an exponential class of conic quadratic inequalities.We report computational experiments on a risk-aware capital budgeting problem with uncertain returns on investments with discrete choices. The results show significant improvements in the solvability of the problem with the introduced convexification method.

No abstract

Lifting is a procedure for deriving valid inequalities for mixed-integer sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to solve such problems with branch-and-cut algorithms. Here we generalize the theory of lifting to conic integer programming, i.e., integer programs with conic constraints. We show how to derive conic valid inequalities for a conic integer program from conic inequalities valid for its lowerdimensional restrictions. In order to simplify the computations, we also discuss sequence-independent lifting for conic integer programs. When the cones are restricted to nonnegative orthants, conic lifting reduces to the lifting for linear integer programming as one may expect.

No abstract