SDP-quality bounds via convex quadratic relaxations for global optimization of mixed-integer quadratic programs

  title={SDP-quality bounds via convex quadratic relaxations for global optimization of mixed-integer quadratic programs},
  author={Carlos J. Nohra and Arvind U. Raghunathan and Nikolaos V. Sahinidis},
  journal={Mathematical Programming},
We consider the global optimization of nonconvex mixed-integer quadratic programs with linear equality constraints. In particular, we present a new class of convex quadratic relaxations which are derived via quadratic cuts. To construct these quadratic cuts, we solve a separation problem involving a linear matrix inequality with a special structure that allows the use of specialized solution algorithms. Our quadratic cuts are nonconvex, but define a convex feasible set when intersected with the… 
Homogeneous Formulation of Convex Quadratic Programs for Infeasibility Detection
  • A. Raghunathan
  • Computer Science
    2021 60th IEEE Conference on Decision and Control (CDC)
  • 2021
A novel Homogeneous QP (HQP) formulation which is obtained by embedding the original QP in a larger space and presenting an Infeasible Interior Point Method (IIPM) for the HQP and showing polynomial iteration complexity when applied to HQP.


Spectral Relaxations and Branching Strategies for Global Optimization of Mixed-Integer Quadratic Programs
A family of convex quadratic relaxations which are derived by convexifying nonconvex quadRatic functions through perturbations of thequadratic matrix are presented.
An efficient compact quadratic convex reformulation for general integer quadratic programs
This paper proposes a convex reformulation less general than MIQCR because it is limited to the general integer case, but that has a significantly smaller size, and calls this approach Compact Quadratic Convex Reformulation (CQCR).
Relaxing Nonconvex Quadratic Functions by Multiple Adaptive Diagonal Perturbations
  • Hongbo Dong
  • Mathematics, Computer Science
    SIAM J. Optim.
  • 2016
A cutting-surface method based on multiple diagonal perturbations to derive convex quadratic relaxations for nonconvex Quadratic problems with separable constraints for MIQCPs.
Semidefinite relaxations for non-convex quadratic mixed-integer programming
Semidefinite relaxations for unconstrained non-convex quadratic mixed-integer optimization problems are presented and are computationally easy to solve for medium-sized instances, even if some of the variables are integer and unbounded.
Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations
This paper proposes new methods for generating valid inequalities from the equation Y =  xxT with the non-convex constraint and uses the convex SDP constraint to derive convex quadratic cuts, and combines both approaches in a cutting plane algorithm.
Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations
This paper study methods to build low-dimensional relaxations of MIQCP that capture the strength of the extended formulations and uses projection techniques pioneered in the context of the lift-and-project methodology to illustrate the efficiency of the proposed techniques.
Exploiting integrality in the global optimization of mixed-integer nonlinear programming problems with BARON
The paper describes BARON's dynamic strategy for deciding under what conditions to activate integer programming relaxations in the course of branch-and-bound, and describes cutting plane and probing techniques that originate from the literature of integer linear programming and have been adapted in BARON to solve nonlinear problems.
Semidefinite relaxations for quadratically constrained quadratic programming: A review and comparisons
Using theoretical analysis, it is shown that the recently developed doubly nonnegative relaxation is equivalent to the Shor relaxation, when the latter is enhanced with a partial first-order relaxation-linearization technique.
Partial Lagrangian relaxation for general quadratic programming
It is shown that one can express as a semidefinite program the dual of the partial Lagrangian relaxation of (Pb) where the linear constraints are not relaxed.
Global optimization of nonconvex factorable programming problems
A branch-and-bound approach based on linear programming relaxations generated through various approximation schemes that utilize, for example, the Mean-Value Theorem and Chebyshev interpolation polynomials coordinated with a Reformulation-Linearization Technique (RLT).