Theory of linear and integer programming
Theory of linear and integer programming
The generalized basis reduction algorithm
Mathematics of Operations Research
On the finite convergence of interior-point algorithms for linear programming
Mathematical Programming: Series A and B
On the complexity of approximating the maximal inscribed ellipsoid for a polytope
Mathematical Programming: Series A and B
Finding an interior point in the optimal face of linear programs
Mathematical Programming: Series A and B
A new algorithm for minimizing convex functions over convex sets
Mathematical Programming: Series A and B
Rounding of polytopes in the real number model of computation
Mathematics of Operations Research
Ellipsoidal Approximations of Convex Sets Based on the Volumetric Barrier
Mathematics of Operations Research
Solving a System of Linear Diophantine Equations with Lower and Upper Bounds on the Variables
Mathematics of Operations Research
Computational Optimization and Applications
Introduction to Linear Optimization
Introduction to Linear Optimization
Towards a Practical Volumetric Cutting Plane Method for Convex Programming
SIAM Journal on Optimization
Segment LLL-Reduction of Lattice Bases
CaLC '01 Revised Papers from the International Conference on Cryptography and Lattices
Market Split and Basis Reduction: Towards a Solution of the Cornuéjols-Dawande Instances
INFORMS Journal on Computing
Non-standard approaches to integer programming
Discrete Applied Mathematics
A new implementation of the generalized basis reduction algorithm for convex integer programming
A new implementation of the generalized basis reduction algorithm for convex integer programming
Hard Equality Constrained Integer Knapsacks
Mathematics of Operations Research
Cuts for Conic Mixed-Integer Programming
IPCO '07 Proceedings of the 12th international conference on Integer Programming and Combinatorial Optimization
Branching on general disjunctions
Mathematical Programming: Series A and B
Improved strategies for branching on general disjunctions
Mathematical Programming: Series A and B
Column basis reduction and decomposable knapsack problems
Discrete Optimization
On the structure of reduced kernel lattice bases
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
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We present branching-on-hyperplane methods for solving mixed integer linear and mixed integer convex programs. In particular, we formulate the problem of finding a good branching hyperplane using a novel concept of adjoint lattice. We also reformulate the problem of rounding a continuous solution to a mixed integer solution. A worst case complexity of a Lenstra-type algorithm is established using an approximate log-barrier center to obtain an ellipsoidal rounding of the feasible set. The results for the mixed integer convex programming also establish a complexity result for the mixed integer second order cone programming and mixed integer semidefinite programming feasibility problems as a special case. Our results motivate an alternative reformulation technique and a branching heuristic using a generalized (e.g., ellipsoidal) norm reduced basis available at the root node.