# Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations

@article{Saxena2010ConvexRO, title={Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations}, author={Anureet Saxena and Pierre Bonami and Jon Lee}, journal={Mathematical Programming}, year={2010}, volume={124}, pages={383-411} }

This paper addresses the problem of generating strong convex relaxations of Mixed Integer Quadratically Constrained Programming (MIQCP) problems. MIQCP problems are very difficult because they combine two kinds of non- convexities: integer variables and non-convex quadratic constraints. To produce strong relaxations of MIQCP problems, we use techniques from disjunctive programming and the lift-and-project methodology. In particular, we propose new methods for generating valid inequalities from…

## 144 Citations

### Global solution of non-convex quadratically constrained quadratic programs

- Mathematics, Computer ScienceOptim. Methods Softw.
- 2019

This paper proposes an extension of MIQCR which applies to any QCQP and proposes to solve by a branch-and-bound algorithm based on the relaxation of the additional quadratic constraints and of the integrality constraints.

### Exact quadratic convex reformulations of mixed-integer quadratically constrained problems

- MathematicsMath. Program.
- 2016

The resolution is based on the reformulation of the original problem (QP) into an equivalent quadratic problem whose continuous relaxation is convex, so that it can be effectively solved by a branch-and-bound algorithm based onquadratic convex relaxation.

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

- Computer Science, MathematicsMathematical Programming
- 2021

This work presents a new class of convex quadratic relaxations which are derived viaquadratic cuts, which are an outer-approximation of a semi-infinite convex program which under certain conditions is equivalent to a well-known semidefinite program relaxation.

### A Branch and Bound algorithm for general mixed-integer quadratic programs based on quadratic convex relaxation

- MathematicsJ. Comb. Optim.
- 2014

The results show that the solution time of most of the considered instances with up to 60 variables is improved by the Branch and Bound algorithm in comparison with the approach of Billionnet et al. (2012) and with the general mixed-integer nonlinear solver BARON.

### Disjunctive Cuts for Nonconvex MINLP

- Computer Science
- 2012

An application to MINLP of a well-known separation method for disjunctive cuts that has shown to be very effective in Mixed Integer Linear Programming (MILP) is described and experimental results show encouraging results.

### Compact mixed-integer programming relaxations in quadratic optimization

- Computer Science, Mathematics
- 2020

This work leverages an elegant piecewise linear approximation for univariate quadratic functions due to Yarotsky, formulating this (simple) approximation using mixed-integer programming to produce valid dual bounds for nonconvex Quadratic optimization problems.

### Dynamically generated cutting planes for mixed-integer quadratically constrained quadratic programs and their incorporation into GloMIQO 2

- MathematicsOptim. Methods Softw.
- 2015

The branch-and-cut framework integrated into GloMIQO 2, which addresses mixed-integer quadratically constrained quadratic programs (MIQCQP) to ε-global optimality, is documents.

### Semidefinite relaxations for non-convex quadratic mixed-integer programming

- Computer ScienceMathematical Programming
- 2012

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.

### Enhancing Semidefinite Relaxation for Quadratically Constrained Quadratic Programming via Penalty Methods

- Computer ScienceJ. Optim. Theory Appl.
- 2019

It is shown that the conditionally quasi-convex relaxation can provide tighter bounds than the standard semidefinite relaxation, and is introduced as a special penalty method for quadratically constrained linear programming based on its semideFinite relaxation.

### Enhancing Semidefinite Relaxation for Quadratically Constrained Quadratic Programming via Penalty Methods

- Computer ScienceJournal of Optimization Theory and Applications
- 2018

It is shown that the conditionally quasi-convex relaxation can provide tighter bounds than the standard semidefinite relaxation, and is introduced as a special penalty method for quadratically constrained linear programming based on its semideFinite relaxation.

## References

SHOWING 1-10 OF 45 REFERENCES

### Disjunctive Cuts for Non-convex Mixed Integer Quadratically Constrained Programs

- Computer ScienceIPCO
- 2008

This paper proposes new methods for generating valid inequalities by using the equation Y = xxT and uses the concave constraint 0 ≥ Y - xxT to derive disjunctions of two types that are obtained by combining several eigenvectors in order to minimize the width of the disjunction.

### Disjunctive Programming: Properties of the Convex Hull of Feasible Points

- MathematicsDiscret. Appl. Math.
- 1998

### A global optimization algorithm for nonconvex generalized disjunctive programming and applications to process systems

- Computer Science, Mathematics
- 2001

### A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations

- Computer ScienceMath. Program.
- 2008

Semidefinite programming relaxations are proposed and studied, which are bounded and hence suitable for use with finite KKT-branching and demonstrate the practical effectiveness of the method.

### Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

- Computer ScienceMath. Program.
- 2004

The development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs is addressed and novel relaxation schemes, range reduction tests, and branching strategies are developed which are incorporated into the prototypical branch-and-bound algorithm.

### A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems

- Computer Science
- 1998

This paper presents RLT-Based Global Optimization Algorithms for Nonconvex Polynomial Programming Problems and Reformulation-Convexification Technique for Polynomials Programs: Design and Implementation, and some special applications to Discrete and Continuous Non Convex Programs.

### Enhancing RLT relaxations via a new class of semidefinite cuts

- Computer ScienceJ. Glob. Optim.
- 2002

In this paper, we propose a mechanism to tighten Reformulation-Linearization Technique (RLT) based relaxations for solving nonconvex programming problems by importing concepts from semidefinite…

### Second order cone programming relaxation of nonconvex quadratic optimization problems

- Computer Science
- 2001

A SOCP relaxation method is proposed, which strengthens the lift-and-project LP (linear programming) relaxation method by adding convex quadratic valid inequalities for the positive semidefinite cone involved in the SDP relaxation.

### An algorithmic framework for convex mixed integer nonlinear programs

- Computer Science, MathematicsDiscret. Optim.
- 2008

### Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming

- Computer ScienceJ. Glob. Optim.
- 2009

This work considers relaxations for nonconvex quadratically constrained quadratic programming (QCQP) based on semidefinite programming (SDP) and the reformulation-linearization technique (RLT) and shows that the use of SDP and RLT constraints together can produce bounds that are substantially better than either technique used alone.