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A combinatorial, primal-dual approach to semidefinite programs

Sanjeev Arora, Satyen Kale
2007 Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC '07  
We develop a general primal-dual approach to solve SDPs using a generalization of the well-known multiplicative weights update rule to symmetric matrices.  ...  The design of our primal-dual algorithms is guided by a robust analysis of rounding algorithms used to obtain integer solutions from fractional ones.  ...  Vijay Vazirani suggested several years ago that primal-dual methods be investigated in the SDP context.  ... 
doi:10.1145/1250790.1250823 dblp:conf/stoc/AroraK07 fatcat:vejflz5yercrtcoaflkn2wizom

A Combinatorial, Primal-Dual Approach to Semidefinite Programs

Sanjeev Arora, Satyen Kale
2016 Journal of the ACM  
We develop a general primal-dual approach to solve SDPs using a generalization of the well-known multiplicative weights update rule to symmetric matrices.  ...  The design of our primal-dual algorithms is guided by a robust analysis of rounding algorithms used to obtain integer solutions from fractional ones.  ...  Vijay Vazirani suggested several years ago that primal-dual methods be investigated in the SDP context.  ... 
doi:10.1145/2837020 fatcat:66pm5wv4hzgmjavsjowgbgpb7m

Cutting Plane Methods and Subgradient Methods [chapter]

John E. Mitchell
2009 Decision Technologies and Applications  
Semidefinite programs are expensive to solve directly, so we also consider cutting surface approaches to solving them.  ...  Semidefinite programming relaxations of combinatorial optimization problems are often tighter than linear programming relaxations.  ...  These decomposition approaches to solving semidefinite programs and other conic programs are the topic of §4.  ... 
doi:10.1287/educ.1090.0064 fatcat:yc24hzo62rfyxaemsrwt6jed3a

Recent Developments in Interior-Point Methods [chapter]

Stephen J. Wright
2000 IFIP Advances in Information and Communication Technology  
In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite  ...  The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming.  ...  Powell and the other organizers of the IFIP TC7 '99 conference for arranging a most enjoyable and interesting meeting, and for a dose reading of the paper which resulted in many improvements.  ... 
doi:10.1007/978-0-387-35514-6_14 fatcat:chzlfupe7bbhbdyynhatlckjcu

The application of semidefinite programming for detection in CDMA

Peng Hui Tan, L.K. Rasmussen
2001 IEEE Journal on Selected Areas in Communications  
The simulated bit error rate performance demonstrates that this approach provides a good approximation to the ML performance.  ...  In this paper, a detection strategy based on a semidefinite relaxation of the CDMA maximum-likelihood (ML) problem is investigated. Cutting planes are introduced to strengthen the approximation.  ...  A semidefinite program (7) can be solved by employing the primal-dual path-following algorithm of [19] as a basic optimization tool.  ... 
doi:10.1109/49.942507 fatcat:jvyu2fvcbzdxhmbfsyurz3wtue

Semidefinite Programming Relaxations and Algebraic Optimization in Control

Pablo A. Parrilo, Sanjay Lall
2003 European Journal of Control  
We present an overview of the essential elements of semidefinite programming as a computational tool for the analysis of systems and control problems.  ...  These developments are illustrated with examples of applications to control systems.  ...  The standard dual associated with the primal program (3) is also a semidefinite program, given by maximize b T y subject to m i=1 y i A i C, (7) where the dual variable y ∈ R m .  ... 
doi:10.3166/ejc.9.307-321 fatcat:rh3a4ufic5bujku67elivso4gu

Semidefinite programming bounds for error-correcting codes [article]

Frank Vallentin
2019 arXiv   pre-print
This chapter is written for the forthcoming book "A Concise Encyclopedia of Coding Theory" (CRC press), edited by W. Cary Huffman, Jon-Lark Kim, and Patrick Sol\'e.  ...  From this it follows easily that taking the dual of the dual conic program gives a conic program which is equivalent to the primal.  ...  Then x is optimal for the primal and y is optimal for the dual if and only if x, A T y − c E = 0 holds. (3) strong duality: Suppose that primal and dual conic programs both have a strictly feasible solution  ... 
arXiv:1902.01253v1 fatcat:fsh4yvdbvre5dcitmx6h2xgffy

Dualize it: software for automatic primal and dual conversions of conic programs

Johan Löfberg
2009 Optimization Methods and Software  
Many optimization problems gain from being interpreted and solved in either primal or dual form.  ...  For a user with a particular application, one of these forms is usually much more natural to use, but this is not always the most efficient one.  ...  Acknowledgment The author would like to acknowledge inspiring discussions and motivation from Nathan Srebro, Michael Tsuk and Erik Wernholt.  ... 
doi:10.1080/10556780802553325 fatcat:irewkza5lvbohkw7r6m5mkj324

On verified numerical computations in convex programming

Christian Jansson
2009 Japan journal of industrial and applied mathematics  
A discussion of important problem transformations to special types of convex problems and convex relaxations is included.  ...  Especially, we consider the computation of verified error bounds for non-smooth convex conic optimization in the framework of functional analysis, for linear programming, and for semidefinite programming  ...  Linear and semidefinite programs play a very useful role in global and combinatorial optimization.  ... 
doi:10.1007/bf03186539 fatcat:epentouc4bgm7azndvdwlfzpdy

First-order semidefinite programming for the two-electron treatment of many-electron atoms and molecules

David A. Mazziotti
2007 Mathematical Modelling and Numerical Analysis  
The constraints in the variational calculation of the 2-RDM require a special optimization known as a semidefinite programming.  ...  Recent work on 2-RDM methods will be reviewed and illustrated with particular emphasis on the importance of advances in large-scale semidefinite programming.  ...  The author expresses his appreciation to Dudley Herschbach, Herschel Rabitz, John Coleman, and Alexander Mazziotti for their support and encouragement.  ... 
doi:10.1051/m2an:2007021 fatcat:skaehz3djvbopmwz44fmwdhb6m

A recurrent neural network for real-time semidefinite programming

Danchi Jiang, Jun Wang
1999 IEEE Transactions on Neural Networks  
First, an auxiliary cost function is introduced to minimize the duality gap between the admissible points of the primal problem and the corresponding dual problem.  ...  Index Terms-Linear matrix inequalities, recurrent neural networks, semidefinite programming.  ...  CONCLUSION We have proposed a recurrent neural network approach to real-time semidefinite programming.  ... 
doi:10.1109/72.737496 pmid:18252506 fatcat:7czfbfxzdrbgpguaqrha4ajixi

Interior Point Methods for Combinatorial Optimization [chapter]

John E. Mitchell, Panos M. Pardalos, Mauricio G. C. Resende
1998 Handbook of Combinatorial Optimization  
In this paper, we review recent interior point approaches for solving combinatorial optimization problems.  ...  We discuss in detail tecniques for linear and network programming, branch and bound and branch and cut methods, nonconvex potential function minimization, lower bounding techniques, and semidefinite programming  ...  ACKNOWLEDGEMENT The first author acknowledges support in part by ONR grant N00014-94-1-0391, and by a grant from the Dutch NWO and by Delft University of Technology for 1997-98, while visiting TWI/SSOR  ... 
doi:10.1007/978-1-4613-0303-9_4 fatcat:jqlwni3hrbggze4w74hww2gsre

Computational experience with a bundle approach for semidefinite cutting plane relaxations of Max-Cut and Equipartition

Ilse Fischer, Gerald Gruber, Franz Rendl, Renata Sotirov
2005 Mathematical programming  
This leads to function evaluations requiring to solve a relatively simple semidefinite program.  ...  We propose a dynamic version of the bundle method to get approximate solutions to semidefinite programs with a nearly arbitrary number of linear inequalities.  ...  Acknowledgement: We thank two anonymous referees for their constructive comments and suggestions to improve an earlier version of the paper.  ... 
doi:10.1007/s10107-005-0661-9 fatcat:2yxzp7updfgurccimeyganpmva

Interior Point Methods for Combinatorial Optimization [chapter]

John E. Mitchell
1996 Applied Optimization  
In this paper, we review recent interior point approaches for solving combinatorial optimization problems.  ...  We discuss in detail tecniques for linear and network programming, branch and bound and branch and cut methods, nonconvex potential function minimization, lower bounding techniques, and semidefinite programming  ...  ACKNOWLEDGEMENT The first author acknowledges support in part by ONR grant N00014-94-1-0391, and by a grant from the Dutch NWO and by Delft University of Technology for 1997-98, while visiting TWI/SSOR  ... 
doi:10.1007/978-1-4613-3449-1_11 fatcat:5utlmdqf4fbxpbrlvxdmblrgde

Exploiting Low-Rank Structure in Semidefinite Programming by Approximate Operator Splitting [article]

Mario Souto, Joaquim D. Garcia, Alvaro Veiga
2018 arXiv   pre-print
This work aims to reduce this scalability gap by proposing a novel proximal algorithm for solving general semidefinite programming problems.  ...  The main contribution of this work is to achieve a substantial speedup by effectively adjusting the proposed algorithm in order to exploit the low-rank property inherent to several semidefinite programming  ...  We extend many thanks to all members of LAMPS (Laboratory of Applied Mathematical Programming and Statistics), in special Thuener Silva and Raphael Saavedra, for the daily support and fruitful discussions  ... 
arXiv:1810.05231v3 fatcat:3uk6cjpedfcofjdcc5dahzkxjm
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