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### 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  ...  We show how the extended formulation can be algorithmically projected to the original space by solving linear programs.  ...  We are grateful to Sam Burer for providing us the box-QP instances and their MIQCP-SDP formulations.  ...

### 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.  ...

### 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.  ...

### 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 ).  ...  (M QP ) belongs to the class of Mixed Integer Non Linear Programs (MINLP). These problems are N P-hard .  ...  Introduction Consider the following linearly-constrained mixed-integer quadratic program: (M QP ) 8 > > > > > > > > < > > > > > > > > : M in h(x) s.t.  ...

### 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).  ...

### 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).  ...

### 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.  ...

### A low-cost alternating projection approach for a continuous formulation of convex and cardinality constrained optimization [article]

Nataša Krejić, Evelin H. M. Krulikovski, Marcos Raydan
2022 arXiv   pre-print
We consider convex constrained optimization problems that also include a cardinality constraint.  ...  We assume that the region defined by the convex constraints can be written as the intersection of a finite collection of convex sets, such that it is easy and inexpensive to project onto each one of them  ...  The second author was financially supported by Fundação para a Ciência e a Tecnologia (FCT) (Portuguese Foundation for Science and Technology) under the scope of the projects UIDB/MAT/00297/2020, UIDP/  ...

### 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.  ...  Acknowledgements The work was performed with financial support of research project NPU I No.  ...

### 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.  ...

### 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  . • 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  ...

### 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.  ...

### Cuts for mixed 0-1 conic programming

M.T. Çezik, G. Iyengar
2005 Mathematical programming
We show that many of the techniques developed for generating linear cuts for mixed 0-1 linear programs, such as the Gomory cuts, the lift-and-project cuts, and cuts from other hierarchies of tighter relaxations  ...  We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts.  ...  For a more detailed discussion of convex quadratic cuts for MCPs see  . Stubbs and Mehrotra  investigate convex quadratic inequalities in the context of mixed 0-1 convex programs.  ...

### Protein Threading Based on Nonlinear Integer Programming [article]

Wajeb Gharibi, Marwah Mohammed Bakri
2013 arXiv   pre-print
Actually, the existing linear integer programming is directly linearized from the original quadratic integer programming. We then develop corresponding efficient algorithms.  ...  The most popular algorithm is based on linear integer programming. In this paper, we consider methods based on nonlinear integer programming.  ...  Linearization strategies are to reformulate the zero-one quadratic programs as equivalent mixed-integer programming problems (1.1) and (1.3) with additional binary variables and/or continuous variables  ...

### Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

Fani Boukouvala, Ruth Misener, Christodoulos A. Floudas
2016 European Journal of Operational Research
This manuscript reviews recent advances in deterministic global optimization for Mixed-Integer Nonlinear Programming (MINLP), as well as Constrained Derivative-Free Optimization (CDFO).  ...  We show their individual prerequisites, formulations and applicability, but also point out possible points of interaction in problems which contain hybrid characteristics.  ...  MIQCQP, MIQCP, QAP, Box-Constrained MIQP A mixed-integer quadratically-constrained quadratic program (MIQCQP) is defined as a MINLP where all of the nonlinearities are quadratic; symbols have the same  ...
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