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A Strong Dual for Conic Mixed-Integer Programs

Diego A. Morán R., Santanu S. Dey, Juan Pablo Vielma
2012 SIAM Journal on Optimization  
In particular, we construct a subadditive dual for mixed-integer conic programming problems. Under a simple condition on the primal problem, we are able to prove strong duality. *  ...  Mixed-integer conic programming is a generalization of mixed-integer linear programming.  ...  Strong duality for conic programming In mixed-integer linear programming, the proof of strong duality for the corresponding subadditive dual relies on the existence of a strong duality result for linear  ... 
doi:10.1137/110840868 fatcat:37kt7pvcfbffhd2v5wezmwtrha

On Subadditive Duality for Conic Mixed-Integer Programs [article]

Burak Kocuk, Diego Moran
2019 arXiv   pre-print
In this paper, we show that the subadditive dual of a feasible conic mixed-integer program (MIP) is a strong dual whenever it is feasible.  ...  In addition, we prove that all known conditions and other 'natural' conditions for strong duality, such as strict mixed-integer feasibility, boundedness of the feasible set or essentially strict feasibility  ...  Introduction Duality for mixed-integer programs (MIPs): Duality is an important concept in mathematical programming for both analyzing the properties of optimization problems and constructing solution  ... 
arXiv:1808.10419v2 fatcat:mzxbbs6mbvc2dahfx5xjppvdwe

Extended Formulations in Mixed-integer Convex Programming [article]

Miles Lubin and Emre Yamangil and Russell Bent and Juan Pablo Vielma
2015 arXiv   pre-print
We present a unifying framework for generating extended formulations for the polyhedral outer approximations used in algorithms for mixed-integer convex programming (MICP).  ...  For mixed-integer conic-representable problems, we provide the first outer approximation algorithm with finite-time convergence guarantees, opening a path for the use of conic solvers for continuous relaxations  ...  We have implemented a solver, Pajarito, which currently accepts input as mixed-integer conic programming problems with a mix of second-order and exponential cones.  ... 
arXiv:1511.06710v1 fatcat:csbh77daivasncpbsnodb7ay44

Polyhedral approximation in mixed-integer convex optimization

Miles Lubin, Emre Yamangil, Russell Bent, Juan Pablo Vielma
2017 Mathematical programming  
In order to automate this extended formulation we rely on the algebraic modeling technique of disciplined convex programming (DCP), and for generality and ease of implementation we use conic representations  ...  Generalizing both mixed-integer linear optimization and convex optimization, mixed-integer convex optimization possesses broad modeling power but has seen relatively few advances in general-purpose solvers  ...  The aim of this paper is to develop methodologies for solving the more general class of mixed-integer convex optimization, or mixed-integer convex programming (MICP), problems by reducing them to a sequence  ... 
doi:10.1007/s10107-017-1191-y fatcat:zpzgtx2xfjgpjfg2qyoitquiwy

CBLIB 2014: a benchmark library for conic mixed-integer and continuous optimization

Henrik A. Friberg
2015 Mathematical Programming Computation  
It is the first extensive benchmark library for the advancing field of conic mixed-integer and continuous optimization, which is already supported by all major commercial solvers and spans a wide range  ...  The library addresses the particular need for public test sets mixing cone types and allowing integer variables, but has all types of conic optimization in target.  ...  Thanks to Ambros Gleixner and Thorsten Koch for maintaining the benchmarking library at Zuse Institute Berlin, and to Mathias Stolpe for his guidance. The author, Henrik A.  ... 
doi:10.1007/s12532-015-0092-4 fatcat:fmqqk4oc4zfbjjhbwacvsvbrnu

Lifting for conic mixed-integer programming

Alper Atamtürk, Vishnu Narayanan
2009 Mathematical programming  
Lifting is a procedure for deriving valid inequalities for mixed-integer sets from valid inequalities for suitable restrictions of those sets.  ...  We show how to derive conic valid inequalities for a conic integer program from conic inequalities valid for its lowerdimensional restrictions.  ...  Acknowledgments We are thankful to an anonymous referee for several valuable comments, especially the ones that led to Propositions 6 and 7.  ... 
doi:10.1007/s10107-009-0282-9 fatcat:2nc7jumc3rcjhhge6btjw45cry

Conic mixed-integer rounding cuts

Alper Atamtürk, Vishnu Narayanan
2008 Mathematical programming  
This feature leads to a computationally efficient implementation of nonlinear cuts for conic mixed-integer programming.  ...  A conic integer program is an integer programming problem with conic constraints.  ...  In this section we describe conic mixed-integer rounding cuts for conic mixed-integer programming.  ... 
doi:10.1007/s10107-008-0239-4 fatcat:cgfxok6ydfga5mll2amqb5vgoa

Cuts for mixed 0-1 conic programming

M.T. Çezik, G. Iyengar
2005 Mathematical programming  
, extend in a straightforward manner to mixed 0-1 conic programs.  ...  In this we paper we study techniques for generating valid convex constraints for mixed 0-1 conic programs.  ...  Thus, for all L p -norms, (28) is a conic program.  ... 
doi:10.1007/s10107-005-0578-3 fatcat:myzj3jb2dfegjacyimdhnudtji

Outer Approximation With Conic Certificates For Mixed-Integer Convex Problems [article]

Chris Coey, Miles Lubin, Juan Pablo Vielma
2018 arXiv   pre-print
The polyhedral relaxations are refined with K^* cuts derived from conic certificates for continuous primal-dual conic subproblems.  ...  A mixed-integer convex (MI-convex) optimization problem is one that becomes convex when all integrality constraints are relaxed.  ...  ., 2018] B&B-NL implementation for mixed-integer semidefinite (MISDP) problems uses a primal-dual conic interior-point solver for the SDP subproblems.  ... 
arXiv:1808.05290v1 fatcat:7h3tmjbrpbelnfpciaul6gy5ku

Symmetry-exploiting cuts for a class of mixed-0/1 second-order cone programs

Sarah Drewes, Sebastian Pokutta
2014 Discrete Optimization  
In [2], Gomory mixed-integer rounding cuts for second-order cone programs have been devised and in [3] lifting for conic mixed-integer programming was investigated.  ...  For example in [7] lift-and-project based cuts for mixed 0/1 conic programming problems have been studied.  ...  ACKNOWLEDGEMENTS The authors are indebted to the two anonymous referees for the helpful comments.  ... 
doi:10.1016/j.disopt.2014.04.002 fatcat:tmaguogruzcibn5kbwbj5xyah4

Subgradient Based Outer Approximation for Mixed Integer Second Order Cone Programming [chapter]

Sarah Drewes, Stefan Ulbrich
2011 IMA Volumes in Mathematics and its Applications  
This paper deals with outer approximation based approaches to solve mixed integer second order cone programs.  ...  This enables us to extend convergence results valid for continuously differentiable mixed integer nonlinear problems to subdifferentiable constraint functions.  ...  We would like to thank the referees for their constructive comments that were very helpful to improve this paper.  ... 
doi:10.1007/978-1-4614-1927-3_2 fatcat:mryheqfcqvb43ie5h3f6aykk64

Strong duality in conic linear programming: facial reduction and extended duals [article]

Gabor Pataki
2013 arXiv   pre-print
The facial reduction algorithm of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program (P) | Ax ≤_K b in the absence of any constraint qualification.  ...  Ramana's dual is applicable when (P) is a semidefinite program (SDP) and is an explicit SDP itself.  ...  Acknowledgement Thanks are due to an anonymous referee for his/her helpful comments on an earlier version of the paper.  ... 
arXiv:1301.7717v3 fatcat:x6gca7skjrbfde5ogghz7atm7y

A Decomposition Method for Distributionally-Robust Two-stage Stochastic Mixed-integer Cone Programs [article]

Fengqiao Luo, Sanjay Mehrotra
2019 arXiv   pre-print
mixed-integer second order cone programs.  ...  We develop a decomposition algorithm for distributionally-robust two-stage stochastic mixed-integer convex cone programs, and its important special case of distributionally-robust two-stage stochastic  ...  Generalization of the Decomposition Method for DR-TSS Mixed-integer Conic Programs The decomposition method from the previous section can be generalized for solving DR-TSS mixed-integer convex conic programs  ... 
arXiv:1911.08713v1 fatcat:bwnczzskrvhnnbk2qiourqnbwi

Conic reformulations for Kullback-Leibler divergence constrained distributionally robust optimization and applications

Burak Kocuk
2021 An International Journal of Optimization and Control: Theories & Applications  
The resulting conic reformulation of the original optimization problem can be directly solved by a commercial conic programming solver.  ...  dual exponential cone constrained program under mild assumptions on the underlying optimization problem.  ...  Beste Basciftci for her comments on an earlier version of this paper.  ... 
doi:10.11121/ijocta.01.2021.001001 fatcat:apxhcak4lbftfkpthby76d4l7a

Strong Duality in Conic Linear Programming: Facial Reduction and Extended Duals [chapter]

Gábor Pataki
2013 Computational and Analytical Mathematics  
The facial reduction algorithm of Borwein and Wolkowicz and the extended dual of Ramana provide a strong dual for the conic linear program in the absence of any constraint qualification.  ...  Ramana's dual is applicable when (P ) is a semidefinite program (SDP) and is an explicit SDP itself.  ...  Acknowledgement Thanks are due to an anonymous referee for his/her helpful comments on an earlier version of the paper.  ... 
doi:10.1007/978-1-4614-7621-4_28 fatcat:c3wpzassqjeq3hcvchzfrmut5i
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