Low-rank semidefinite programming for the MAX2SAT problem

@inproceedings{Wang2019LowrankSP,
  title={Low-rank semidefinite programming for the MAX2SAT problem},
  author={Po-Wei Wang and J. Zico Kolter},
  booktitle={AAAI},
  year={2019}
}
This paper proposes a new algorithm for solving MAX2SAT problems based on combining search methods with semidefinite programming approaches. Semidefinite programming techniques are well-known as a theoretical tool for approximating maximum satisfiability problems, but their application has traditionally been very limited by their speed and randomized nature. Our approach overcomes this difficult by using a recent approach to low-rank semidefinite programming, specialized to work in an… 

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References

SHOWING 1-10 OF 36 REFERENCES
Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations
TLDR
This approach uses Lagrangian duality to obtain a “nearly optimal” solution of the basic semidefinite Max-Cut relaxation, strengthened by triangle inequalities, and could prove optimality for several problems of the literature where, to the best of the knowledge, no other method is able to do so.
A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization
TLDR
A nonlinear programming algorithm for solving semidefinite programs (SDPs) in standard form that replaces the symmetric, positive semideFinite variable X with a rectangular variable R according to the factorization X=RRT.
The non-convex Burer-Monteiro approach works on smooth semidefinite programs
TLDR
It is shown that the low-rank Burer--Monteiro formulation of SDPs in that class almost never has any spurious local optima, including applications such as max-cut, community detection in the stochastic block model, robust PCA, phase retrieval and synchronization of rotations.
The Mixing method: low-rank coordinate descent for semidefinite programming with diagonal constraints
TLDR
This work shows that with a step size, the Mixing method converges to the global optimum of the semidefinite program almost surely in a locally linear rate under random initialization, and is the first low-rank semideFinite programming method to achieve a global optimum on the spherical manifold without assumption.
Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming
TLDR
This algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.
On semidefinite programming relaxations for the satisfiability problem
  • M. Anjos
  • Computer Science
    Math. Methods Oper. Res.
  • 2004
TLDR
A comparison of the relative practical performances of the SDP relaxations shows that, among these three relaxations, the new relaxation provides the best tradeoff between theoretical strength and practical performance within an enumerative algorithm.
Sums of squares based approximation algorithms for MAX-SAT
The Power of Semidefinite Programming Relaxations for MAX-SAT
TLDR
Semidefinite Programming (SDP) based relaxations are surprisingly powerful, providing much tighter bounds than LP relaxations, across different constrainedness regions, and this shows the effectiveness of SDP relaxations in providing heuristic guidance for iterative variable setting, significantly more accurate than the guidance based on LP relaxation.
SAT-based MaxSAT algorithms
SDP-based Max-2-Sat Decomposition
TLDR
The methods investigated were able to approach the lower bounds of SDP within 2% but do not provide performance guarantees and cannot compete with conventional local search methods.
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