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Semidefinite Programming for Min-Max Problems and Games [article]

Rida Laraki
2009 arXiv   pre-print
This provides a unified approach and a class of algorithms to approximate all Nash equilibria and min-max strategies of many static and dynamic games.  ...  We introduce two min-max problems: the first problem is to minimize the supremum of finitely many rational functions over a compact basic semi-algebraic set whereas the second problem is a 2-player zero-sum  ...  Let P be the min-max problem defined in (3.2) . Let λ * d be the optimal value of the semidefinite program (5.9), and suppose that with r := max j=1,...  ... 
arXiv:0810.3150v2 fatcat:qwenkmtwbfanfosr75fqgmfbi4

Semidefinite programming for min–max problems and games

R. Laraki, J. B. Lasserre
2010 Mathematical programming  
Key words and phrases. N -player games; Nash equlibria; min-max optimization problems; semidefinite programming. We would like to thank Bernhard von Stengel and the referees for their comments.  ...  We consider two min-max problems: (1) minimizing the supremum of finitely many rational functions over a compact basic semi-algebraic set and (2) solving a 2-player zero-sum polynomial game in randomized  ...  Let P be the min-max problem defined in (3.2) . Let λ * d be the optimal value of the semidefinite program (5.9) , and suppose that with r := max j=1,...  ... 
doi:10.1007/s10107-010-0353-y fatcat:skzmuaws35ahhctyxzsv7jxr5q

Condition numbers of stochastic mean payoff games and what they say about nonarchimedean semidefinite programming [article]

Xavier Allamigeon, Stéphane Gaubert, Ricardo D. Katz, Mateusz Skomra
2018 arXiv   pre-print
Recently, a correspondence has been established between nonarchimedean semidefinite programs and stochastic mean payoff games with perfect information.  ...  Nonarchimedean semidefinite programs encode parametric families of classical semidefinite programs, for sufficiently large values of the parameter.  ...  Acknowledgement The second author thanks Vladimir Gurvich for enlightening discussions on the pumping algorithm of [GKK88] and its extension to stochastic games in [BEGM15] .  ... 
arXiv:1802.07712v1 fatcat:qypn46dt6vhdfdtxzqrtictitu

Parallel Approximation of Non-interactive Zero-sum Quantum Games

Rahul Jain, John Watrous
2009 2009 24th Annual IEEE Conference on Computational Complexity  
We prove that an equilibrium point of any such game can be approximated by means of an efficient parallel algorithm, which implies that one-turn quantum refereed games, wherein the referee is specified  ...  This paper studies a simple class of zero-sum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to  ...  We therefore thank Rohit Khandekar for his implicit and indirect contribution to this work, and for allowing us to include it in this paper.  ... 
doi:10.1109/ccc.2009.26 dblp:conf/coco/JainW09 fatcat:qsani3o7tfc5bolt5tjlwrbn3e

Parallel approximation of non-interactive zero-sum quantum games [article]

Rahul Jain, John Watrous
2008 arXiv   pre-print
We prove that an equilibrium point of any such game can be approximated by means of an efficient parallel algorithm, which implies that one-turn quantum refereed games, wherein the referee is specified  ...  This paper studies a simple class of zero-sum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to  ...  We therefore thank Rohit Khandekar for his implicit and indirect contribution to this work, and for allowing us to include it in this paper.  ... 
arXiv:0808.2775v1 fatcat:mekzd65ykrfmhk7apdc2anpepa

Solving generic nonarchimedean semidefinite programs using stochastic game algorithms

Xavier Allamigeon, Stéphane Gaubert, Mateusz Skomra
2018 Journal of symbolic computation  
This allows us to solve nonarchimedean semidefinite feasibility problems using algorithms for stochastic games. These algorithms are of a combinatorial nature and work for large instances.  ...  A general issue in computational optimization is to develop combinatorial algorithms for semidefinite programming. We address this issue when the base field is nonarchimedean.  ...  Acknowledgments We thank the referees for their detailed comments. References  ... 
doi:10.1016/j.jsc.2017.07.002 fatcat:qvmjhlcyl5fb5pdqsgr3ya4dyi

A Spectral Generalization of Von Neumann Minimax Theorem [article]

Bahman Kalantari
2019 arXiv   pre-print
For diagonal A_i's this reduces to the classic minimax.  ...  ., A_m, the following spectral minimax property holds: min_X ∈Δ_nmax_y ∈ S_m∑_i=1^m y_iA_i ∙ X=max_y ∈ S_mmin_X ∈Δ_n∑_i=1^m y_iA_i ∙ X, where S_m is the simplex and Δ_n the spectraplex.  ...  Furthermore, it is well known that in semidefinite programming, as a conic linear programming, if there exists a feasible X ≻ 0 and feasible (y, S) with S ≻ 0, the optimal objective value of both problems  ... 
arXiv:1905.09762v1 fatcat:y4d76ng7oram5avjuydkozvu3m

Polynomial games and sum of squares optimization

Pablo A. Parrilo
2006 Proceedings of the 45th IEEE Conference on Decision and Control  
We show that the value of the game, and the corresponding optimal strategies, can be obtained by solving a single semidefinite programming problem.  ...  We study two-person zero-sum games, where the payoff function is a polynomial expression in the actions of the players. This class of games was introduced by Dresher, Karlin, and Shapley in 1950.  ...  Alexandre Megretski, for his help with the translation of [20] , and to Noah Stein, for bringing Karlin's quote in Section II-B to my attention.  ... 
doi:10.1109/cdc.2006.377261 dblp:conf/cdc/Parrilo06 fatcat:nyitmlx4l5ewbnw2rchk2o2g3m

On Khot's unique games conjecture

Luca Trevisan
2012 Bulletin of the American Mathematical Society  
The conjecture has inspired a remarkable body of work, which has clarified the computational complexity of several optimization problems and the effectiveness of "semidefinite programming" convex relaxations  ...  In 2002, Subhash Khot formulated the Unique Games Conjecture, a conjecture about the computational complexity of certain optimization problems.  ...  Semidefinite Programming and Unique Games.  ... 
doi:10.1090/s0273-0979-2011-01361-1 fatcat:olwc5dausved3fhbgb5fn6rx5y

A Direct Product Theorem for Discrepancy

Troy Lee, Adi Shraibman, Robert Špalek
2008 2008 23rd Annual IEEE Conference on Computational Complexity  
The main tool for our results is semidefinite programming, in particular a recent characterization of discrepancy in terms of a semidefinite programming quantity by Linial and Shraibman (2006) .  ...  As a consequence we obtain a strong direct product theorem for distributional complexity, and direct sum theorems for worst-case complexity, for bounds shown by the discrepancy method.  ...  A (now) common approach for dealing with NP-hard problems is to consider a semidefinite relaxation of the problem.  ... 
doi:10.1109/ccc.2008.25 dblp:conf/coco/LeeSS08 fatcat:o2ugisl5ejddxhejo2ztg6vcry

Multireference alignment using semidefinite programming

Afonso S. Bandeira, Moses Charikar, Amit Singer, Andy Zhu
2014 Proceedings of the 5th conference on Innovations in theoretical computer science - ITCS '14  
Inspired by existing Unique-Games approximation algorithms, we provide a semidefinite program (SDP) based relaxation which approximates the maximum likelihood estimator (MLE) for the multireference alignment  ...  Although we show that the MLE problem is Unique-Games hard to approximate within any constant, we observe that our poly-time approximation algorithm for the MLE appears to perform quite well in typical  ...  Acknowledgements The authors thank Yutong Chen for valuable assistance with the implementation of our algorithm. A. S. Bandeira was supported by AFOSR Grant No. FA9550-12-1-0317. M.  ... 
doi:10.1145/2554797.2554839 dblp:conf/innovations/BandeiraCSZ14 fatcat:5nl2fjexbvgk3h3wemz4lrixea

A Semidefinite Hierarchy for Disjointly Constrained Multilinear Programming [article]

Kai Kellner
2016 arXiv   pre-print
For nondegenerate bimatrix games, a Nash equilibrium can be computed by the sum of squares approach in finitely many steps.  ...  Based on a reformulation of the problem in terms of sum-of-squares polynomials, we study a hierarchy of semidefinite relaxations to the problem.  ...  The multilinear problem (1.1) can equivalently be stated as a nonnegativity question f * = max {f (x 1 , . . . , x l ) | x i ∈ P i , i ∈ [l]} = min {µ | µ − f (x 1 , . . . , x l ) ≥ 0 for (x 1 , . . .  ... 
arXiv:1603.03634v1 fatcat:happcknkmzacfbb4q2c774zxky

In SDP relaxations, inaccurate solvers do robust optimization [article]

Jean-Bernard Lasserre, Victor Magron
2019 arXiv   pre-print
In other words the resulting procedure can be viewed as a 'max-min' robust optimization problem with two players (the solver which maximizes on B_∞(f,ε) and the user who minimizes over the original decision  ...  We interpret some wrong results (due to numerical inaccuracies) already observed when solving SDP-relaxations for polynomial optimization on a double precision floating point SDP solver.  ...  But in fact, as we are in the convex case, Theorem 2.1 implies that this maxmin game is also equivalent to the minmax game.  ... 
arXiv:1811.02879v3 fatcat:4oz6nbfhnbfqtbrrf4vsa3m6d4

Product theorems via semidefinite programming [article]

Troy Lee, Rajat Mittal
2008 arXiv   pre-print
classes of games.  ...  The tendency of semidefinite programs to compose perfectly under product has been exploited many times in complexity theory: for example, by Lovasz to determine the Shannon capacity of the pentagon; to  ...  Acknowledgements We would like to thank Mario Szegedy for many insightful conversations. We would also like to thank the anonymous referees of ICALP 2008 for their helpful comments.  ... 
arXiv:0803.4206v2 fatcat:oyfcxdqfvnca5fny2nsic6t5ay

Product Theorems Via Semidefinite Programming [chapter]

Troy Lee, Rajat Mittal
2008 Lecture Notes in Computer Science  
classes of games.  ...  The tendency of semidefinite programs to compose perfectly under product has been exploited many times in complexity theory: for example, by Lovász to determine the Shannon capacity of the pentagon; to  ...  Acknowledgements We would like to thank Mario Szegedy for many insightful conversations. We would also like to thank the anonymous referees of ICALP 2008 for their helpful comments.  ... 
doi:10.1007/978-3-540-70575-8_55 fatcat:a4o4vabbw5aqrageattra5lm2m
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