Robust Approximation to Multiperiod Inventory Management
Operations Research
The Chvátal-Gomory Closure of a Strictly Convex Body
Mathematics of Operations Research
Lift-and-project cuts for mixed integer convex programs
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
The chvátal-gomory closure of an ellipsoid is a polyhedron
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
On interval-subgradient and no-good cuts
Operations Research Letters
Semidefinite relaxations for mixed 0-1 second-order cone program
ISCO'12 Proceedings of the Second international conference on Combinatorial Optimization
Intersection cuts for mixed integer conic quadratic sets
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
Facial structure and representation of integer hulls of convex sets
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
Communication: On families of quadratic surfaces having fixed intersections with two hyperplanes
Discrete Applied Mathematics
A computational study for common network design in multi-commodity supply chains
Computers and Operations Research
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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 second-order 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.