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

Anureet Saxena, Pierre Bonami, Jon Lee
2010 Mathematical programming  
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.  ...  student at Tepper School of Business, Carnegie Mellon University.  ... 
doi:10.1007/s10107-010-0371-9 fatcat:w4c5rkcadrh4po6bsj7lgeiqrm

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

Anureet Saxena, Pierre Bonami, Jon Lee
2010 Mathematical programming  
A common way to produce a convex relaxation of a Mixed Integer Quadratically Constrained Program (MIQCP) is to lift the problem into a higher dimensional space by introducing variables Y ij to represent  ...  One advantage of such extended relaxations is that they can be efficiently strengthened by using the (convex) SDP constraint Y − xx T 0 and disjunctive programming.  ...  We are grateful to Sam Burer for providing us the box-QP instances and their MIQCP-SDP formulations.  ... 
doi:10.1007/s10107-010-0340-3 fatcat:vajnwliitjcb7mrc4p7dya6jq4

Extended formulations in mixed integer conic quadratic programming

Juan Pablo Vielma, Iain Dunning, Joey Huchette, Miles Lubin
2016 Mathematical Programming Computation  
In this paper we consider the use of extended formulations in LP-based algorithms for mixed integer conic quadratic programming (MICQP).  ...  mixed integer programming problems that include general MICQP problems.  ...  Acknowledgemensts We thank the review team for their thoughtful and constructive comments, which improved the exposition of the paper.  ... 
doi:10.1007/s12532-016-0113-y fatcat:kznbqi3uing7feu6uac36iu5km

Compact extended formulations for low-rank functions with indicator variables [article]

Shaoning Han, Andrés Gómez
2021 arXiv   pre-print
We study the mixed-integer epigraph of a low-rank convex function with non-convex indicator constraints, which are often used to impose logical constraints on the support of the solutions.  ...  Extended formulations describing the convex hull of such sets can easily be constructed via disjunctive programming, although a direct application of this method often yields prohibitively large formulations  ...  Such extended formulations are the strongest possible for the mixed-integer set under study.  ... 
arXiv:2110.14884v1 fatcat:w7gkgc44bnbmbg64qomtfkxevq

Non-convex mixed-integer nonlinear programming: A survey

Samuel Burer, Adam N. Letchford
2012 Surveys in Operations Research and Management Science  
A wide range of problems arising in practical applications can be formulated as Mixed-Integer Nonlinear Programs (MINLPs).  ...  We survey the literature on non-convex MINLP, discussing applications, algorithms and software. Special attention is paid to the case in which the objective and constraint functions are quadratic.  ...  [126, 127] have derived strong cutting planes for non-convex MIQCQPs (mixed-integer quadratically constrained quadratic programs).  ... 
doi:10.1016/j.sorms.2012.08.001 fatcat:ykgvnv2a6jd57lrhjh4qtxd4pi

Convex Mixed-Integer Nonlinear Programs Derived from Generalized Disjunctive Programming using Cones [article]

David E. Bernal, Ignacio E. Grossmann
2021 arXiv   pre-print
We propose the formulation of convex Generalized Disjunctive Programming (GDP) problems using conic inequalities leading to conic GDP problems.  ...  We then show the reformulation of conic GDPs into Mixed-Integer Conic Programming (MICP) problems through both the Big-M and Hull Reformulations.  ...  , Simulation Based Engineering Program through the Institute for the Design of Advanced Energy Systems (IDAES).  ... 
arXiv:2109.09657v1 fatcat:yqvnk5nokrajzkbkwl7lkv7ree

A Tight SDP Relaxation for MIQCQP Problems in Power Systems Based on Disjunctive Programming [article]

Qifeng Li
2015 arXiv   pre-print
An optimization problem considering AC power flow constraints and integer decision variables can usually be posed as a mixed-integer quadratically constrained quadratic program (MIQCQP) problem.  ...  that an enhanced mixed-integer SDP (MISDP) relaxation, which is a mixed-integer convex problem, is obtained.  ...  Mixed-integer convex Mixed-integer convex Convex Convex Convex     MIQCQP GDP    Mixed-integer convex = Mixed-integer convex E.  ... 
arXiv:1509.05141v1 fatcat:cqeefwerhfd2jbkqo3pifqtcbq

Extending the QCR method to general mixed-integer programs

Alain Billionnet, Sourour Elloumi, Amélie Lambert
2010 Mathematical programming  
This reformulation, that we call MIQCR (Mixed Integer Quadratic Convex Reformulation), is based on the solution of an SDP relaxation of (M QP ).  ...  Our contribution in this work is to extend the ideas of QCR (Quadratic Convex Reformulation) [2, 3] from the binary case to the general mixed-integer case.  ...  Introduction Consider the following linearly-constrained mixed-integer quadratic program: (M QP ) 8 > > > > > > > > < > > > > > > > > : M in h(x) s.t.  ... 
doi:10.1007/s10107-010-0381-7 fatcat:7uqbfsabfnc2jb63kveu2fy7ga

The N-K Problem in Power Grids: New Models, Formulations and Numerical Experiments (extended version) [article]

Daniel Bienstock, Abhinav Verma
2009 arXiv   pre-print
The case where k is small is of practical interest. We present theoretical and computational results involving a mixed-integer model and a continuous nonlinear model related to this question.  ...  a set of k or fewer arcs whose removal will cause the system to fail.  ...  A better mixed-integer programming formulation As before, let A be an attack and C a (given) configuration of generators.  ... 
arXiv:0912.5233v2 fatcat:c4inm2je6zd4pnky2e5pcngvii

Traces of the XII Aussois Workshop on Combinatorial Optimization

Michael Jünger, Thomas M. Liebling, Denis Naddef, William R. Pulleyblank, Gerhard Reinelt, Giovanni Rinaldi, Laurence A. Wolsey
2010 Mathematical programming  
"Convex Relaxations of Non-Convex Mixed Integer Quadratically Constrained Programs: Extended formulations" by Anureet Saxena, Pierre Bonami, and Jon Lee proposes a method to generate strong convex relaxations  ...  of Mixed Integer Quadratically Constrained programming problems.  ... 
doi:10.1007/s10107-010-0369-3 fatcat:b7slvsqv4ncy7j45l3u4dcilz4

Introductory mathematical programming for EC

Ofer M. Shir
2022 Proceedings of the Genetic and Evolutionary Computation Conference Companion  
• Global optimization has been for several decades addressed by algorithms and Mathematical Programming (MP) -branded as Operations Research (OR), yet rooted at Theoretical CS [1] . • Also -it has been  ...  (LP) 2 integer programming (IP) 3 mixed-integer programming (MIP) 4 quadratic programming (QP) and mixed-integer QP (MIQP) Ofer Shir Introductory MathProg for EC GECCO'22 7 / 56 Ofer Shir Ofer Shir Introductory  ...  Program (QCP) has quadratic terms in its constraints (possibly no quadratic terms in the objective) • Mixed-integer QP and QCP involve also integer decision variables • Renowned MIQP: the quadratic assignment  ... 
doi:10.1145/3520304.3533630 fatcat:oq47iyyzlvf5vevwyquxnwo6py

Implementation of Mixed-integer Programming on Embedded System

Jakub Novak, Petr Chalupa
2015 Procedia Engineering  
For systems with binary input variables the quadratic problem is replaced by more challenging Mixed-Integer Quadratic Programming (MIQP) problem.  ...  One of the most widespread modern control strategies is the discrete-time Model Predictive Control (MPC) method which requires the solution of the quadratic programming problem.  ...  MSMT-7778/2014 by the Ministry of Education of the Czech Republic and also by the European Regional Development Fund under the Project CEBIA-Tech No. CZ.1.05/2.1.00/03.0089.  ... 
doi:10.1016/j.proeng.2015.01.539 fatcat:qjlfcbcjwfasjfwhohf4nho344

Conic mixed-integer rounding cuts

Alper Atamtürk, Vishnu Narayanan
2008 Mathematical programming  
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  ...  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  ...  such as convex quadratically constrained quadratic programs and semidefinite programs.  ... 
doi:10.1007/s10107-008-0239-4 fatcat:cgfxok6ydfga5mll2amqb5vgoa

QPLIB: a library of quadratic programming instances

Fabio Furini, Emiliano Traversi, Pietro Belotti, Antonio Frangioni, Ambros Gleixner, Nick Gould, Leo Liberti, Andrea Lodi, Ruth Misener, Hans Mittelmann, Nikolaos V. Sahinidis, Stefan Vigerske (+1 others)
2018 Mathematical Programming Computation  
.: Semidefinite relaxations for non-convex quadratic mixed-integer programming. Mathematical Programming 141(1), 435-452 (2013) 28. Burer, S.: Copositive programming. In: F.M. Anjos, B.J.  ...  This is illustrated by the fact that the class of problems that we simply refer to here as "QP" might more accurately be called Mixed-Integer Quadratically-Constrained Quadratic Programming (MIQCQP) in  ...  Any blank lines, or lines starting with any of the characters !, % or # are ignored. Each term in the first column of Table 8 denotes a required value.  ... 
doi:10.1007/s12532-018-0147-4 fatcat:7q7dunjewffb3ca2am6kmuf4im

An Iterative Method for Nonconvex Quadratically Constrained Quadratic Programs [article]

Chuangchuang Sun, Ran Dai
2016 arXiv   pre-print
This paper examines the nonconvex quadratically constrained quadratic programming (QCQP) problems using an iterative method.  ...  Each iteration of IRM is formulated as a convex problem with semidefinite constraints.  ...  Mixed-Boolean Quadratic Programming In this subsection, the proposed IRM method is applied to solve mixed-boolean quadratic programming problems formulated as, J = min x T Q 0 x s.t. x T Q i x + c i ≤  ... 
arXiv:1609.02609v1 fatcat:lv2ayxbhabajtgoovitzggmg6a
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