We study the distance between the optimal solutions (resp. optimal values) of mixed-integer conic programs and the optimal solutions (resp. optimal values) of their continuous relaxations. We show that these values can be upper bounded in terms of the recession cone of the feasible region of the continuous relaxation when the recession cone is regular. We then specialize our analysis to mixed-integer second order cone programs (SOCPs). In the case of SOCPs whose feasible regions are defined by a single Lorentz cone constraint, we show that the distance can be upper bounded in terms of the data of the problem (the objective function vector, the matrix defining the SOCP, and the right-hand side). We also study when this bound can be made independent of the right-hand side, akin to the mixed-integer programming case. Finally, in the case the feasible region of the SOCP is defined by two or more Lorentz cone constraints, then we show that, in general, the distance cannot be bounded independently of the right-hand side defining the SOCP, not even when its feasible region is compact or one of the Lorentz cones constraints defines a polyhedral set.

We consider the traveling salesman problem (TSP) with neighborhoods, a generalization of the TSP where we have the flexibility of visiting a neighborhood of each node instead of visiting the node itself. To solve this problem, two formulations based on second-order conic programming are proposed along decomposition schemes. In addition, linear approximations arbitrarily close to the initial formulations are proposed. Finally, computational experiments are performed to test the effectiveness of the proposed formulations.

The advancement of domain reduction techniques, such as bound tightening, convexification, and redundant variables elimination, has significantly enhanced the performance of Branch-and-Bound-based solvers in mathematical programming. However, decomposition-based MINLP solvers have not yet fully harnessed the potential of domain reduction techniques. This work delves into the impact of integrating convexification and domain reduction techniques within the Outer-Approximation method. We propose a refined convexification-based Outer-Approximation method alongside a Branch-and-Bound method for both convex and nonconvex Mixed-Integer Nonlinear Programming problems. In both methods, these domain reduction techniques can be employed both in the presolve phase and during solving integer-fixed NLP subproblems. However, this work focuses on the impact of domain reduction methods at the method's initialization stage, including feasibility-based and optimality-based bound tightening, the auxiliary variable method, and other convex reformulations. The proposed methods have been developed and incorporated into the open-source Mixed-Integer Nonlinear Decomposition Toolbox for Pyomo-MindtPy. Comprehensive benchmark tests were conducted, validating the effectiveness and reliability of our proposed algorithms. These tests highlight the improvements achieved by incorporating convexification and domain reduction techniques into the Outer-Approximation and Branch-and-Bound methods for MINLP problems.

An M-decomposable set is a closed convex set which is the sum of a compact convex set and a closed convex cone. We present properties of these sets in the context of integer programming. In particular, we present properties of their integer hull, some cutting plane closures and subadditive duality.